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id="toc-Jain_mathematics_(400_BCE_–_200_CE)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jain_mathematics_(400_BCE_–_200_CE)"> <div class="vector-toc-text"> 4 Jain mathematics (400 BCE – 200 CE) </div> </a> <ul id="toc-Jain_mathematics_(400_BCE_–_200_CE)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Oral_tradition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Oral_tradition"> <div class="vector-toc-text"> 5 Oral tradition </div> </a> <button aria-controls="toc-Oral_tradition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Oral tradition subsection </button> <ul id="toc-Oral_tradition-sublist" class="vector-toc-list"> <li id="toc-Styles_of_memorisation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Styles_of_memorisation"> <div class="vector-toc-text"> 5.1 Styles of memorisation </div> </a> <ul id="toc-Styles_of_memorisation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Sutra_genre" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Sutra_genre"> <div class="vector-toc-text"> 5.2 The Sutra genre </div> </a> <ul id="toc-The_Sutra_genre-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_written_tradition:_prose_commentary" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_written_tradition:_prose_commentary"> <div class="vector-toc-text"> 6 The written tradition: prose commentary </div> </a> <ul id="toc-The_written_tradition:_prose_commentary-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerals_and_the_decimal_number_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numerals_and_the_decimal_number_system"> <div class="vector-toc-text"> 7 Numerals and the decimal number system </div> </a> <ul id="toc-Numerals_and_the_decimal_number_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bakhshali_Manuscript" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bakhshali_Manuscript"> <div class="vector-toc-text"> 8 Bakhshali Manuscript </div> </a> <ul id="toc-Bakhshali_Manuscript-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_period_(400–1300)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classical_period_(400–1300)"> <div class="vector-toc-text"> 9 Classical period (400–1300) </div> </a> <button aria-controls="toc-Classical_period_(400–1300)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Classical period (400–1300) subsection </button> <ul id="toc-Classical_period_(400–1300)-sublist" class="vector-toc-list"> <li id="toc-Fourth_to_sixth_centuries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourth_to_sixth_centuries"> <div class="vector-toc-text"> 9.1 Fourth to sixth centuries </div> </a> <ul id="toc-Fourth_to_sixth_centuries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seventh_and_eighth_centuries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seventh_and_eighth_centuries"> <div class="vector-toc-text"> 9.2 Seventh and eighth centuries </div> </a> <ul id="toc-Seventh_and_eighth_centuries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ninth_to_twelfth_centuries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ninth_to_twelfth_centuries"> <div class="vector-toc-text"> 9.3 Ninth to twelfth centuries </div> </a> <ul id="toc-Ninth_to_twelfth_centuries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Medieval_and_early_modern_mathematics_(1300–1800)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Medieval_and_early_modern_mathematics_(1300–1800)"> <div class="vector-toc-text"> 10 Medieval and early modern mathematics (1300–1800) </div> </a> <button aria-controls="toc-Medieval_and_early_modern_mathematics_(1300–1800)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Medieval and early modern mathematics (1300–1800) subsection </button> <ul id="toc-Medieval_and_early_modern_mathematics_(1300–1800)-sublist" class="vector-toc-list"> <li id="toc-Navya-Nyaya" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Navya-Nyaya"> <div class="vector-toc-text"> 10.1 Navya-Nyaya </div> </a> <ul id="toc-Navya-Nyaya-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kerala_School" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kerala_School"> <div class="vector-toc-text"> 10.2 Kerala School </div> </a> <ul id="toc-Kerala_School-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Others" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Others"> <div class="vector-toc-text"> 10.3 Others </div> </a> <ul id="toc-Others-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Charges_of_Eurocentrism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Charges_of_Eurocentrism"> <div class="vector-toc-text"> 11 Charges of Eurocentrism </div> </a> <ul id="toc-Charges_of_Eurocentrism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 15 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Source_books_in_Sanskrit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Source_books_in_Sanskrit"> <div class="vector-toc-text"> 15.1 Source books in Sanskrit </div> </a> <ul id="toc-Source_books_in_Sanskrit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 16 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Indian mathematics</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 31 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-31" aria-hidden="true" > 31 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA_%D9%81%D9%8A_%D8%A7%D9%84%D9%87%D9%86%D8%AF_%D8%A7%D9%84%D9%82%D8%AF%D9%8A%D9%85%D8%A9" title="الرياضيات في الهند القديمة – Arabic" lang="ar" hreflang="ar" data-title="الرياضيات في الهند القديمة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%9C%E0%A5%8D%E0%A4%9E" title="भारतीय गणितज्ञ – Awadhi" lang="awa" hreflang="awa" data-title="भारतीय गणितज्ञ" data-language-autonym="अवधी" data-language-local-name="Awadhi" class="interlanguage-link-target">अवधी</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A6%BE%E0%A6%B0%E0%A6%A4%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="ভারতীয় গণিত – Bangla" lang="bn" hreflang="bn" data-title="ভারতীয় গণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Indijska_matematika" title="Indijska matematika – Bosnian" lang="bs" hreflang="bs" data-title="Indijska matematika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matem%C3%A0tiques_a_l%27%C3%8Dndia" title="Matemàtiques a l'Índia – Catalan" lang="ca" hreflang="ca" data-title="Matemàtiques a l'Índia" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Indische_Mathematik" title="Indische Mathematik – German" lang="de" hreflang="de" data-title="Indische Mathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_india" title="Matemática india – Spanish" lang="es" hreflang="es" data-title="Matemática india" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques_indiennes" title="Mathématiques indiennes – French" lang="fr" hreflang="fr" data-title="Mathématiques indiennes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B8%EB%8F%84%EC%9D%98_%EC%88%98%ED%95%99" title="인도의 수학 – Korean" lang="ko" hreflang="ko" data-title="인도의 수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%B5%D5%AB_%D5%BA%D5%A1%D5%BF%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%A8_%D5%80%D5%B6%D5%A4%D5%AF%D5%A1%D5%BD%D5%BF%D5%A1%D5%B6%D5%B8%D6%82%D5%B4" title="Մաթեմատիկայի պատմությունը Հնդկաստանում – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկայի պատմությունը Հնդկաստանում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="भारतीय गणित – Hindi" lang="hi" hreflang="hi" data-title="भारतीय गणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Indijska_matematika" title="Indijska matematika – Croatian" lang="hr" hreflang="hr" data-title="Indijska matematika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matematika_India" title="Matematika India – Indonesian" lang="id" hreflang="id" data-title="Matematika India" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%BE%E0%B2%9A%E0%B3%80%E0%B2%A8_%E0%B2%AD%E0%B2%BE%E0%B2%B0%E0%B2%A4%E0%B3%80%E0%B2%AF_%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಪ್ರಾಚೀನ ಭಾರತೀಯ ಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ಪ್ರಾಚೀನ ಭಾರತೀಯ ಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="भारतीय गणित – Marathi" lang="mr" hreflang="mr" data-title="भारतीय गणित" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matematik_India" title="Matematik India – Malay" lang="ms" hreflang="ms" data-title="Matematik India" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indiase_wiskunde" title="Indiase wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Indiase wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BE%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%9E%E0%A4%A4%E0%A5%87%E0%A4%97%E0%A5%81_%E0%A4%A7%E0%A4%B2%E0%A4%83" title="भारतीय गणितिज्ञतेगु धलः – Newari" lang="new" hreflang="new" data-title="भारतीय गणितिज्ञतेगु धलः" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A4%E3%83%B3%E3%83%89%E3%81%AE%E6%95%B0%E5%AD%A6" title="インドの数学 – Japanese" lang="ja" hreflang="ja" data-title="インドの数学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Indisk_matematikk" title="Indisk matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Indisk matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AD%E0%A8%BE%E0%A8%B0%E0%A8%A4%E0%A9%80_%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਭਾਰਤੀ ਗਣਿਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਭਾਰਤੀ ਗਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matem%C3%A1tica_indiana" title="Matemática indiana – Portuguese" lang="pt" hreflang="pt" data-title="Matemática indiana" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D1%81%D1%82%D0%BE%D1%80%D0%B8%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8_%D0%B2_%D0%98%D0%BD%D0%B4%D0%B8%D0%B8" title="История математики в Индии – Russian" lang="ru" hreflang="ru" data-title="История математики в Индии" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%9A%E0%A5%80%E0%A4%A8%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D" title="प्राचीनगणितम् – Sanskrit" lang="sa" hreflang="sa" data-title="प्राचीनगणितम्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matematika_indase" title="Matematika indase – Albanian" lang="sq" hreflang="sq" data-title="Matematika indase" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D0%B8%D1%98%D1%81%D0%BA%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Индијска математика – Serbian" lang="sr" hreflang="sr" data-title="Индијска математика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Indijska_matematika" title="Indijska matematika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Indijska matematika" data-language-autonym="Srpskohrvatski / српскохрватски" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Development of mathematics in South Asia</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Mathematics in India" redirects here. For the 2009 monograph by Kim Plofker, see <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)">Mathematics in India (book)</a>.</div> Indian mathematics emerged in the <a href="/wiki/Indian_subcontinent" title="Indian subcontinent">Indian subcontinent</a><a href="#cite_note-plofker-1">[1]</a> from 1200 BCE<a href="#cite_note-hayashi2005-p360-361-2">[2]</a> until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>, <a href="/wiki/Bhaskara_II" class="mw-redirect" title="Bhaskara II">Bhaskara II</a>, <a href="/wiki/Var%C4%81hamihira" title="Varāhamihira">Varāhamihira</a>, and <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava</a>. The <a href="/wiki/Decimal" title="Decimal">decimal number system</a> in use today<a href="#cite_note-irfah346-3">[3]</a> was first recorded in Indian mathematics.<a href="#cite_note-4">[4]</a> Indian mathematicians made early contributions to the study of the concept of <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a> as a number,<a href="#cite_note-bourbaki46-5">[5]</a> <a href="/wiki/Negative_numbers" class="mw-redirect" title="Negative numbers">negative numbers</a>,<a href="#cite_note-bourbaki49-6">[6]</a> <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, and <a href="/wiki/Algebra" title="Algebra">algebra</a>.<a href="#cite_note-concise-britannica-7">[7]</a> In addition, <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a><a href="#cite_note-8">[8]</a> was further advanced in India, and, in particular, the modern definitions of <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> were developed there.<a href="#cite_note-9">[9]</a> These mathematical concepts were transmitted to the Middle East, China, and Europe<a href="#cite_note-concise-britannica-7">[7]</a> and led to further developments that now form the foundations of many areas of mathematics. Ancient and medieval Indian mathematical works, all composed in <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a>, usually consisted of a section of <a href="/wiki/Sutra" title="Sutra">sutras</a> in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved.<a href="#cite_note-plofker-1">[1]</a><a href="#cite_note-filliozat-p140to143-10">[10]</a> All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark <a href="/wiki/Bakhshali_Manuscript" class="mw-redirect" title="Bakhshali Manuscript">Bakhshali Manuscript</a>, discovered in 1881 in the village of <a href="/wiki/Bakhshali" title="Bakhshali">Bakhshali</a>, near <a href="/wiki/Peshawar" title="Peshawar">Peshawar</a> (modern day <a href="/wiki/Pakistan" title="Pakistan">Pakistan</a>) and is likely from the 7th century CE.<a href="#cite_note-hayashi95-11">[11]</a><a href="#cite_note-plofker-brit6-12">[12]</a> A later landmark in Indian mathematics was the development of the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> expansions for <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> (sine, cosine, and <a href="/wiki/Arc_tangent" class="mw-redirect" title="Arc tangent">arc tangent</a>) by mathematicians of the <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school</a> in the 15th century CE. Their work, completed two centuries before the invention of <a href="/wiki/Calculus" title="Calculus">calculus</a> in Europe, provided what is now considered the first example of a <a href="/wiki/Power_series" title="Power series">power series</a> (apart from geometric series).<a href="#cite_note-13">[13]</a> However, they did not formulate a systematic theory of <a href="/wiki/Derivative" title="Derivative">differentiation</a> and <a href="/wiki/Integral" title="Integral">integration</a>, nor is there any evidence of their results being transmitted outside <a href="/wiki/Kerala" title="Kerala">Kerala</a>.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Prehistory">Prehistory</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=1" title="Edit section: Prehistory">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Poids_cubiques_harapp%C3%A9ens_-_BM.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Poids_cubiques_harapp%C3%A9ens_-_BM.jpg/220px-Poids_cubiques_harapp%C3%A9ens_-_BM.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Poids_cubiques_harapp%C3%A9ens_-_BM.jpg/330px-Poids_cubiques_harapp%C3%A9ens_-_BM.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Poids_cubiques_harapp%C3%A9ens_-_BM.jpg/440px-Poids_cubiques_harapp%C3%A9ens_-_BM.jpg 2x" data-file-width="5847" data-file-height="3899" /></a><figcaption>Cubical weights standardised in the Indus Valley civilisation</figcaption></figure> Excavations at <a href="/wiki/Harappa" title="Harappa">Harappa</a>, <a href="/wiki/Mohenjo-daro" title="Mohenjo-daro">Mohenjo-daro</a> and other sites of the <a href="/wiki/Indus_Valley_civilisation" class="mw-redirect" title="Indus Valley civilisation">Indus Valley civilisation</a> have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass-produced weights in regular <a href="/wiki/Geometrical" class="mw-redirect" title="Geometrical">geometrical</a> shapes, which included <a href="/wiki/Hexahedron" title="Hexahedron">hexahedra</a>, <a href="/wiki/Barrel" title="Barrel">barrels</a>, <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cones</a>, and <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinders</a>, thereby demonstrating knowledge of basic <a href="/wiki/Geometry" title="Geometry">geometry</a>.<a href="#cite_note-18">[18]</a> The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> Hollow cylindrical objects made of shell and found at <a href="/wiki/Lothal" title="Lothal">Lothal</a> (2200 BCE) and <a href="/wiki/Dholavira" title="Dholavira">Dholavira</a> are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.<a href="#cite_note-21">[21]</a> <div class="mw-heading mw-heading2"><h2 id="Vedic_period">Vedic period</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=2" title="Edit section: Vedic period">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt 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subject</th></tr><tr><td class="sidebar-content hlist" style="padding-top: 0.2em;"> <ul><li><a class="mw-selflink selflink">Mathematics</a></li> <li><a href="/wiki/Indian_astronomy" title="Indian astronomy">Astronomy</a></li> <li><a href="/wiki/Indian_national_calendar" title="Indian national calendar">Calendar</a></li> <li><a href="/wiki/History_of_measurement_systems_in_India" title="History of measurement systems in India">Measurement systems</a> <ul><li><a href="/wiki/Indian_units_of_measurement" title="Indian units of measurement">Units of measurement</a></li></ul></li> <li><a href="/wiki/Cartography_of_India" title="Cartography of India">Cartography</a></li> <li><a href="/wiki/Indian_geography" class="mw-redirect" title="Indian geography">Geography</a></li> <li><a href="/wiki/History_of_printing#In_India" title="History of printing">Printing</a></li> <li><a href="/wiki/History_of_metallurgy_in_the_Indian_subcontinent" title="History of metallurgy in the Indian subcontinent">Metallurgy</a></li> <li><a href="/wiki/Indian_coinage" class="mw-redirect" title="Indian coinage">Coinage</a></li> <li><a href="/wiki/History_of_metallurgy_in_the_Indian_subcontinent" title="History of metallurgy in the Indian subcontinent">Indian Alchemy</a></li> <li><a href="/wiki/Ayurveda" title="Ayurveda">Traditional medicine</a></li> <li><a href="/wiki/History_of_agriculture_in_India" class="mw-redirect" title="History of agriculture in India">Agriculture</a></li> <li><a href="/wiki/History_of_education_in_the_Indian_subcontinent" title="History of education in the Indian subcontinent">Education</a></li> <li><a href="/wiki/Indian_architecture" class="mw-redirect" title="Indian architecture">Architecture</a> <ul><li><a href="/wiki/List_of_bridges_in_India" title="List of bridges in India">Bridges</a></li></ul></li> <li><a href="/wiki/Transport_in_India" title="Transport in India">Transport</a> <ul><li><a href="/wiki/Indian_maritime_history" title="Indian maritime history">Maritime history</a></li> <li><a href="/wiki/Naval_history_of_India" class="mw-redirect" title="Naval history of India">Navigation</a></li></ul></li> <li><a href="/wiki/Military_history_of_India" title="Military history of India">Military</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Science_and_technology_in_India" title="Template:Science and technology in India"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Science_and_technology_in_India" title="Template talk:Science and technology in India"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Science_and_technology_in_India" title="Special:EditPage/Template:Science and technology in India"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Vedanga" title="Vedanga">Vedanga</a> and <a href="/wiki/Vedas" title="Vedas">Vedas</a></div> <div class="mw-heading mw-heading3"><h3 id="Samhitas_and_Brahmanas">Samhitas and Brahmanas</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=3" title="Edit section: Samhitas and Brahmanas">edit</a>]</div> The texts of the <a href="/wiki/Vedic_Period" class="mw-redirect" title="Vedic Period">Vedic Period</a> provide evidence for the use of <a href="/wiki/History_of_large_numbers" title="History of large numbers">large numbers</a>. By the time of the <a href="/wiki/Yajurveda" title="Yajurveda">Yajurvedasaṃhitā-</a> (1200–900 BCE), numbers as high as 1012 were being included in the texts.<a href="#cite_note-hayashi2005-p360-361-2">[2]</a> For example, the <a href="/wiki/Mantra" title="Mantra">mantra</a> (sacred recitation) at the end of the annahoma ("food-oblation rite") performed during the <a href="/wiki/Ashvamedha" title="Ashvamedha">aśvamedha</a>, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:<a href="#cite_note-hayashi2005-p360-361-2">[2]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Hail to śata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail to niyuta ("hundred thousand," 105), hail to prayuta ("million," 106), hail to arbuda ("ten million," 107), hail to nyarbuda ("hundred million," 108), hail to samudra ("billion," 109, literally "ocean"), hail to madhya ("ten billion," 1010, literally "middle"), hail to anta ("hundred billion," 1011, lit., "end"), hail to parārdha ("one trillion," 1012 lit., "beyond parts"), hail to the uṣas (dawn) , hail to the vyuṣṭi (twilight), hail to udeṣyat (the one which is going to rise), hail to udyat (the one which is rising), hail udita (to the one which has just risen), hail to svarga (the heaven), hail to martya (the world), hail to all.<a href="#cite_note-hayashi2005-p360-361-2">[2]</a></blockquote> The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">With three-fourths Puruṣa went up: one-fourth of him again was here.</blockquote> The <a href="/wiki/Satapatha_Brahmana" class="mw-redirect" title="Satapatha Brahmana">Satapatha Brahmana</a> (<abbr title="circa">c.</abbr> 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.<a href="#cite_note-22">[22]</a> <div class="mw-heading mw-heading3"><h3 id="Śulba_Sūtras">Śulba Sūtras</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=4" title="Edit section: Śulba Sūtras">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a></div> The <a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Śulba Sūtras</a> (literally, "Aphorisms of the Chords" in <a href="/wiki/Vedic_Sanskrit" title="Vedic Sanskrit">Vedic Sanskrit</a>) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.<a href="#cite_note-23">[23]</a> Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"<a href="#cite_note-hayashi2003-p118-24">[24]</a> that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.<a href="#cite_note-hayashi2003-p118-24">[24]</a> According to Hayashi, the Śulba Sūtras contain "the earliest extant verbal expression of the <a href="/wiki/Pythagorean_Theorem" class="mw-redirect" title="Pythagorean Theorem">Pythagorean Theorem</a> in the world, although it had already been known to the <a href="/wiki/First_Babylonian_dynasty" class="mw-redirect" title="First Babylonian dynasty">Old Babylonians</a>." <blockquote>The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."<a href="#cite_note-hayashi2005-p363-25">[25]</a></blockquote> Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.<a href="#cite_note-hayashi2005-p363-25">[25]</a> They contain lists of <a href="/wiki/Pythagorean_triples" class="mw-redirect" title="Pythagorean triples">Pythagorean triples</a>,<a href="#cite_note-26">[26]</a> which are particular cases of <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a>.<a href="#cite_note-cooke198-27">[27]</a> They also contain statements (that with hindsight we know to be approximate) about <a href="/wiki/Squaring_the_circle" title="Squaring the circle">squaring the circle</a> and "circling the square."<a href="#cite_note-cooke199-200-28">[28]</a> <a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a> (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37),<a href="#cite_note-joseph229-29">[29]</a> as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."<a href="#cite_note-joseph229-29">[29]</a><a href="#cite_note-30">[30]</a> It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."<a href="#cite_note-joseph229-29">[29]</a> Baudhayana gives an expression for the <a href="/wiki/Square_root_of_two" class="mw-redirect" title="Square root of two">square root of two</a>:<a href="#cite_note-cooke200-31">[31]</a> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}=1.4142156\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>34</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.4142156</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}=1.4142156\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0be7f431a09bc860f840316312504241192a605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.719ex; height:5.343ex;" alt="{\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}=1.4142156\ldots }" /></dd></dl></dd></dl> The expression is accurate up to five decimal places, the true value being 1.41421356...<a href="#cite_note-32">[32]</a> This expression is similar in structure to the expression found on a Mesopotamian tablet<a href="#cite_note-33">[33]</a> from the Old Babylonian period (1900–1600 <a href="/wiki/BCE" class="mw-redirect" title="BCE">BCE</a>):<a href="#cite_note-cooke200-31">[31]</a> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\approx 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>24</mn> <mn>60</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>51</mn> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1.41421297</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab02a36548a68463f14c1cbd8f706dcd1733adf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.79ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2}}\approx 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297\ldots }" /></dd></dl></dd></dl> which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places. According to mathematician S. G. Dani, the Babylonian cuneiform tablet <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> written c. 1850 BCE<a href="#cite_note-34">[34]</a> "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,<a href="#cite_note-35">[35]</a> indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."<a href="#cite_note-dani-36">[36]</a> Dani goes on to say: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.<a href="#cite_note-dani-36">[36]</a></blockquote> In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by <a href="/wiki/Manava" title="Manava">Manava</a> (fl. 750–650 BCE) and the Apastamba Sulba Sutra, composed by <a href="/wiki/Apastamba" class="mw-redirect" title="Apastamba">Apastamba</a> (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra. <dl><dt>Vyakarana</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vyakarana" class="mw-redirect" title="Vyakarana">Vyakarana</a></div> The Vedic period saw the work of Sanskrit grammarian <a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a> (c. 520–460 BCE). His grammar includes a precursor of the <a href="/wiki/Backus%E2%80%93Naur_form" title="Backus–Naur form">Backus–Naur form</a> (used in the description <a href="/wiki/Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a>).<a href="#cite_note-37">[37]</a> <div class="mw-heading mw-heading2"><h2 id="Pingala_(300_BCE_–_200_BCE)">Pingala (300 BCE – 200 BCE)</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=5" title="Edit section: Pingala (300 BCE – 200 BCE)">edit</a>]</div> Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is <a href="/wiki/Pingala" title="Pingala">Pingala</a> (piṅgalá) (<a href="/wiki/Floruit" title="Floruit">fl.</a> 300–200 BCE), a <a href="/wiki/Music_theory" title="Music theory">music theorist</a> who authored the <a href="/wiki/Chhandas" class="mw-redirect" title="Chhandas">Chhandas</a> <a href="/wiki/Shastra" title="Shastra">Shastra</a> (chandaḥ-śāstra, also Chhandas Sutra chhandaḥ-sūtra), a <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> treatise on <a href="/wiki/Sanskrit_prosody" title="Sanskrit prosody">prosody</a>. Pingala's work also contains the basic ideas of <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> (called maatraameru). Although the Chandah sutra hasn't survived in its entirety, a 10th-century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as <a href="/wiki/Mount_Meru_(mythology)" class="mw-redirect" title="Mount Meru (mythology)">Meru</a>-prastāra (literally "the staircase to Mount Meru"), has this to say: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ...<a href="#cite_note-fowler96-38">[38]</a></blockquote> The text also indicates that Pingala was aware of the <a href="/wiki/Combinatorics" title="Combinatorics">combinatorial</a> identity:<a href="#cite_note-singh36-39">[39]</a> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose 0}+{n \choose 1}+{n \choose 2}+\cdots +{n \choose n-1}+{n \choose n}=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 0}+{n \choose 1}+{n \choose 2}+\cdots +{n \choose n-1}+{n \choose n}=2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c0b9285e3435dbdc53de3091cb85b6b818c396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.487ex; height:6.176ex;" alt="{\displaystyle {n \choose 0}+{n \choose 1}+{n \choose 2}+\cdots +{n \choose n-1}+{n \choose n}=2^{n}}" /></dd></dl></dd></dl> <dl><dt>Kātyāyana</dt></dl> <a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a> (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much <a href="/wiki/Geometry" title="Geometry">geometry</a>, including the general <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> and a computation of the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> correct to five decimal places. <div class="mw-heading mw-heading2"><h2 id="Jain_mathematics_(400_BCE_–_200_CE)">Jain mathematics (400 BCE – 200 CE)</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=6" title="Edit section: Jain mathematics (400 BCE – 200 CE)">edit</a>]</div> Although <a href="/wiki/Jainism" title="Jainism">Jainism</a> as a religion and philosophy predates its most famous exponent, the great <a href="/wiki/Mahavira" title="Mahavira">Mahaviraswami</a> (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. <a href="/wiki/Jain" class="mw-redirect" title="Jain">Jain</a> mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period." A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and <a href="/wiki/Infinity" title="Infinity">infinities</a> led them to classify numbers into three classes: enumerable, innumerable and <a href="/wiki/Infinity" title="Infinity">infinite</a>. Not content with a simple notion of infinity, their texts define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple <a href="/wiki/Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a> (bījagaṇita samīkaraṇa). Jain mathematicians were apparently also the first to use the word shunya (literally void in <a href="/wiki/Sanskrit_language" class="mw-redirect" title="Sanskrit language">Sanskrit</a>) to refer to zero. This word is the ultimate <a href="/wiki/0_(number)#Etymology" class="mw-redirect" title="0 (number)">etymological origin of the English word "zero"</a>, as it was <a href="/wiki/Calque" title="Calque">calqued</a> into Arabic as ṣifr and then subsequently borrowed into <a href="/wiki/Medieval_Latin" title="Medieval Latin">Medieval Latin</a> as zephirum, finally arriving at English after passing through one or more <a href="/wiki/Romance_languages" title="Romance languages">Romance languages</a> (c.f. French zéro, Italian zero).<a href="#cite_note-40">[40]</a> In addition to Surya Prajnapti, important Jain works on mathematics included the <a href="/wiki/Sth%C4%81n%C4%81%E1%B9%85ga_S%C5%ABtra" class="mw-redirect" title="Sthānāṅga Sūtra">Sthānāṅga Sūtra</a> (c. 300 BCE – 200 CE); the Anuyogadwara Sutra (c. 200 BCE – 100 CE), which includes the earliest known description of <a href="/wiki/Factorial" title="Factorial">factorials</a> in Indian mathematics;<a href="#cite_note-41">[41]</a> and the <a href="/wiki/%E1%B9%A2a%E1%B9%ADkha%E1%B9%85%E1%B8%8D%C4%81gama" class="mw-redirect" title="Ṣaṭkhaṅḍāgama">Ṣaṭkhaṅḍāgama</a> (c. 2nd century CE). Important Jain mathematicians included <a href="/wiki/Bhadrabahu" class="mw-redirect" title="Bhadrabahu">Bhadrabahu</a> (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called <a href="/wiki/Tiloya_Panatti" title="Tiloya Panatti">Tiloyapannati</a>; and <a href="/wiki/Umasvati" class="mw-redirect" title="Umasvati">Umasvati</a> (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>, composed a mathematical work called the <a href="/wiki/Tattv%C4%81rtha_S%C5%ABtra" class="mw-redirect" title="Tattvārtha Sūtra">Tattvārtha Sūtra</a>. <div class="mw-heading mw-heading2"><h2 id="Oral_tradition">Oral tradition</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=7" title="Edit section: Oral tradition">edit</a>]</div> Mathematicians of ancient and early medieval India were almost all <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> <a href="/wiki/Pandit" title="Pandit">pandits</a> (paṇḍita "learned man"),<a href="#cite_note-filliozat-p137-42">[42]</a> who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (<a href="/wiki/Vyakarana" class="mw-redirect" title="Vyakarana">vyākaraṇa</a>), <a href="/wiki/Exegesis" title="Exegesis">exegesis</a> (<a href="/wiki/Mimamsa" class="mw-redirect" title="Mimamsa">mīmāṃsā</a>) and logic (<a href="/wiki/Nyaya" title="Nyaya">nyāya</a>)."<a href="#cite_note-filliozat-p137-42">[42]</a> Memorisation of "what is heard" (<a href="/wiki/%C5%9Aruti" title="Śruti">śruti</a> in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia."<a href="#cite_note-pingree1988a-43">[43]</a> <div class="mw-heading mw-heading3"><h3 id="Styles_of_memorisation">Styles of memorisation</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=8" title="Edit section: Styles of memorisation">edit</a>]</div> Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.<a href="#cite_note-44">[44]</a> For example, memorisation of the sacred <a href="/wiki/Veda" class="mw-redirect" title="Veda">Vedas</a> included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated in the original order.<a href="#cite_note-filliozat-p139-45">[45]</a> The recitation thus proceeded as: <div style="text-align: center;"> word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...</div> In another form of recitation, dhvaja-pāṭha<a href="#cite_note-filliozat-p139-45">[45]</a> (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as: <div style="text-align: center;"> word1word2, wordN − 1wordN; word2word3, wordN − 2wordN − 1; ..; wordN − 1wordN, word1word2;</div> The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to Filliozat,<a href="#cite_note-filliozat-p139-45">[45]</a> took the form: <div style="text-align: center;">word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ... </div> That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the <a href="/wiki/Rigveda" title="Rigveda">Ṛgveda</a> (c. 1500 BCE), as a single text, without any variant readings.<a href="#cite_note-filliozat-p139-45">[45]</a> Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the <a href="/wiki/Vedic_period" title="Vedic period">Vedic period</a> (c. 500 BCE). <div class="mw-heading mw-heading3"><h3 id="The_Sutra_genre">The Sutra genre</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=9" title="Edit section: The Sutra genre">edit</a>]</div> Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred <a href="/wiki/Veda" class="mw-redirect" title="Veda">Vedas</a>, which took the form of works called <a href="/wiki/Vedanga" title="Vedanga">Vedāṇgas</a>, or, "Ancillaries of the Veda" (7th–4th century BCE).<a href="#cite_note-filliozat2004-p140-141-46">[46]</a> The need to conserve the sound of sacred text by use of <a href="/wiki/Shiksha" title="Shiksha">śikṣā</a> (<a href="/wiki/Phonetics" title="Phonetics">phonetics</a>) and <a href="/wiki/Chhandas" class="mw-redirect" title="Chhandas">chhandas</a> (<a href="/wiki/Metre_(poetry)" title="Metre (poetry)">metrics</a>); to conserve its meaning by use of <a href="/wiki/Vyakarana" class="mw-redirect" title="Vyakarana">vyākaraṇa</a> (<a href="/wiki/Grammar" title="Grammar">grammar</a>) and <a href="/wiki/Nirukta" title="Nirukta">nirukta</a> (<a href="/wiki/Etymology" title="Etymology">etymology</a>); and to correctly perform the rites at the correct time by the use of <a href="/wiki/Kalpa_(aeon)" class="mw-redirect" title="Kalpa (aeon)">kalpa</a> (<a href="/wiki/Ritual" title="Ritual">ritual</a>) and <a href="/wiki/Jyotisha" class="mw-redirect" title="Jyotisha">jyotiṣa</a> (<a href="/wiki/Astrology" title="Astrology">astrology</a>), gave rise to the six disciplines of the Vedāṇgas.<a href="#cite_note-filliozat2004-p140-141-46">[46]</a> Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in a highly compressed mnemonic form, the <a href="/wiki/Sutra" title="Sutra">sūtra</a> (literally, "thread"): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.<a href="#cite_note-filliozat2004-p140-141-46">[46]</a></blockquote> Extreme brevity was achieved through multiple means, which included using <a href="/wiki/Ellipsis" title="Ellipsis">ellipsis</a> "beyond the tolerance of natural language,"<a href="#cite_note-filliozat2004-p140-141-46">[46]</a> using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables.<a href="#cite_note-filliozat2004-p140-141-46">[46]</a> The sūtras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called <a href="/wiki/Guru-shishya_tradition" class="mw-redirect" title="Guru-shishya tradition">Guru-shishya parampara</a>, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret.<a href="#cite_note-47">[47]</a> The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE). <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Domestic_fire_altar.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Domestic_fire_altar.jpg/300px-Domestic_fire_altar.jpg" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Domestic_fire_altar.jpg/450px-Domestic_fire_altar.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/70/Domestic_fire_altar.jpg 2x" data-file-width="560" data-file-height="420" /></a><figcaption>The design of the domestic fire altar in the Śulba Sūtra</figcaption></figure> The domestic fire-altar in the <a href="/wiki/Vedic_period" title="Vedic period">Vedic period</a> was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely.<a href="#cite_note-filliozat2004-p143-144-48">[48]</a> The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra, this procedure is described in the following words: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three. II.65. In another layer one places the [bricks] North-pointing.<a href="#cite_note-filliozat2004-p143-144-48">[48]</a></blockquote> According to Filliozat,<a href="#cite_note-49">[49]</a> the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju, f.), two pegs (Sanskrit, śanku, m.), and clay to make the bricks (Sanskrit, iṣṭakā, f.). Concision is achieved in the sūtra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the east–west direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory.<a href="#cite_note-filliozat2004-p143-144-48">[48]</a> <div class="mw-heading mw-heading2"><h2 id="The_written_tradition:_prose_commentary">The written tradition: prose commentary</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=10" title="Edit section: The written tradition: prose commentary">edit</a>]</div> With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally.<a href="#cite_note-pingree1988b-50">[50]</a></blockquote> The earliest mathematical prose commentary was that on the work, <a href="/wiki/Aryabhatiya" title="Aryabhatiya">Āryabhaṭīya</a> (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs.<a href="#cite_note-hayashi03-p122-123-51">[51]</a> However, according to Hayashi,<a href="#cite_note-52">[52]</a> "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of <a href="/wiki/Bhaskara_I" class="mw-redirect" title="Bhaskara I">Bhaskara I</a> (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:<a href="#cite_note-hayashi03-p122-123-51">[51]</a> <ul><li>Rule ('sūtra') in verse by <a href="/wiki/Aryabhata" title="Aryabhata">Āryabhaṭa</a></li> <li>Commentary by Bhāskara I, consisting of: <ul><li>Elucidation of rule (derivations were still rare then, but became more common later)</li> <li>Example (uddeśaka) usually in verse.</li> <li>Setting (nyāsa/sthāpanā) of the numerical data.</li> <li>Working (karana) of the solution.</li> <li>Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then.<a href="#cite_note-hayashi03-p122-123-51">[51]</a></li></ul></li></ul> Typically, for any mathematical topic, students in ancient India first memorised the sūtras, which, as explained earlier, were "deliberately inadequate"<a href="#cite_note-pingree1988b-50">[50]</a> in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> (<a href="/wiki/Floruit" title="Floruit">fl.</a> 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: dhulikarman).<a href="#cite_note-hayashi2003-p119-53">[53]</a> <div class="mw-heading mw-heading2"><h2 id="Numerals_and_the_decimal_number_system">Numerals and the decimal number system</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=11" title="Edit section: Numerals and the decimal number system">edit</a>]</div> It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe.<a href="#cite_note-plofker2007-p395-54">[54]</a> The Syrian bishop <a href="/wiki/Severus_Sebokht" title="Severus Sebokht">Severus Sebokht</a> wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.<a href="#cite_note-plofker2007-p395-54">[54]</a> However, how, when, and where the first decimal place value system was invented is not so clear.<a href="#cite_note-55">[55]</a> The earliest extant <a href="/wiki/Writing_system" title="Writing system">script</a> used in India was the <a href="/wiki/Kharo%E1%B9%A3%E1%B9%ADh%C4%AB" class="mw-redirect" title="Kharoṣṭhī">Kharoṣṭhī</a> script used in the <a href="/wiki/Gandhara" title="Gandhara">Gandhara</a> culture of the north-west. It is thought to be of <a href="/wiki/Aramaic" title="Aramaic">Aramaic</a> origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the <a href="/wiki/Br%C4%81hm%C4%AB_script" class="mw-redirect" title="Brāhmī script">Brāhmī script</a>, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.<a href="#cite_note-hayashi2005-p366-56">[56]</a> The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.<a href="#cite_note-plofker2009-p45-57">[57]</a> A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate.<a href="#cite_note-plofker2009-p45-57">[57]</a> Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.<a href="#cite_note-plofker2009-p45-57">[57]</a> There are older textual sources, although the extant manuscript copies of these texts are from much later dates.<a href="#cite_note-plofker2009-p46-58">[58]</a> Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE.<a href="#cite_note-plofker2009-p46-58">[58]</a> Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred."<a href="#cite_note-plofker2009-p46-58">[58]</a> Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."<a href="#cite_note-plofker2009-p46-58">[58]</a> A third decimal representation was employed in a verse composition technique, later labelled <a href="/wiki/Bhuta-sankhya" class="mw-redirect" title="Bhuta-sankhya">Bhuta-sankhya</a> (literally, "object numbers") used by early Sanskrit authors of technical books.<a href="#cite_note-plofker2009-p47-59">[59]</a> Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier.<a href="#cite_note-plofker2009-p47-59">[59]</a> According to Plofker,<a href="#cite_note-Plofker_2009-60">[60]</a> the number 4, for example, could be represented by the word "<a href="/wiki/Veda" class="mw-redirect" title="Veda">Veda</a>" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon).<a href="#cite_note-plofker2009-p47-59">[59]</a> So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left.<a href="#cite_note-plofker2009-p47-59">[59]</a> The earliest reference employing object numbers is a c. 269 CE Sanskrit text, <a href="/wiki/Yavanajataka" title="Yavanajataka">Yavanajātaka</a> (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (c. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.<a href="#cite_note-61">[61]</a> Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.<a href="#cite_note-plofker2009-p47-59">[59]</a> It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE.<a href="#cite_note-plofker2009-p48-62">[62]</a> According to Plofker,<a href="#cite_note-Plofker_2009-60">[60]</a> <blockquote>These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."<a href="#cite_note-plofker2009-p48-62">[62]</a></blockquote> <div class="mw-heading mw-heading2"><h2 id="Bakhshali_Manuscript">Bakhshali Manuscript</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=12" title="Edit section: Bakhshali Manuscript">edit</a>]</div> The oldest extant mathematical manuscript in India is the <a href="/wiki/Bakhshali_Manuscript" class="mw-redirect" title="Bakhshali Manuscript">Bakhshali Manuscript</a>, a birch bark manuscript written in "Buddhist hybrid Sanskrit"<a href="#cite_note-plofker-brit6-12">[12]</a> in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE.<a href="#cite_note-hayashi2005-371-63">[63]</a> The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near <a href="/wiki/Peshawar" title="Peshawar">Peshawar</a> (then in <a href="/wiki/British_India" class="mw-redirect" title="British India">British India</a> and now in <a href="/wiki/Pakistan" title="Pakistan">Pakistan</a>). Of unknown authorship and now preserved in the <a href="/wiki/Bodleian_Library" title="Bodleian Library">Bodleian Library</a> in the <a href="/wiki/University_of_Oxford" title="University of Oxford">University of Oxford</a>, the manuscript has been dated recently as 224 AD- 383 AD.<a href="#cite_note-64">[64]</a> The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples.<a href="#cite_note-hayashi2005-371-63">[63]</a> The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the <a href="/wiki/Rule_of_three_(mathematics)" class="mw-redirect" title="Rule of three (mathematics)">rule of three</a>, and <a href="/wiki/Regula_falsi" title="Regula falsi">regula falsi</a>) and algebra (simultaneous linear equations and <a href="/wiki/Quadratic_equations" class="mw-redirect" title="Quadratic equations">quadratic equations</a>), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<a href="#cite_note-hayashi2005-371-63">[63]</a> Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant.<a href="#cite_note-anton-65">[65]</a></blockquote> The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.<a href="#cite_note-anton-65">[65]</a> In 2017, three samples from the manuscript were shown by <a href="/wiki/Radiocarbon_dating" title="Radiocarbon dating">radiocarbon dating</a> to come from three different centuries: from 224 to 383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.<a href="#cite_note-Devlin-66">[66]</a><a href="#cite_note-Mason-67">[67]</a><a href="#cite_note-Bodleian_Library-68">[68]</a> <div class="mw-heading mw-heading2"><h2 id="Classical_period_(400–1300)">Classical period (400–1300)</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=13" title="Edit section: Classical period (400–1300)">edit</a>]</div> This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, <a href="/wiki/Varahamihira" class="mw-redirect" title="Varahamihira">Varahamihira</a>, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>, <a href="/wiki/Bhaskara_I" class="mw-redirect" title="Bhaskara I">Bhaskara I</a>, <a href="/wiki/Mahavira_(mathematician)" class="mw-redirect" title="Mahavira (mathematician)">Mahavira</a>, <a href="/wiki/Bhaskara_II" class="mw-redirect" title="Bhaskara II">Bhaskara II</a>, <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> and <a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a> give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā).<a href="#cite_note-hayashi2003-p119-53">[53]</a> This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika<a href="#cite_note-69">[69]</a> (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 <a href="/wiki/Common_Era" title="Common Era">CE</a>)—of five earlier works, <a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a>, <a href="/wiki/Romaka_Siddhanta" title="Romaka Siddhanta">Romaka Siddhanta</a>, <a href="/wiki/Paulisa_Siddhanta" title="Paulisa Siddhanta">Paulisa Siddhanta</a>, <a href="/wiki/Vasishtha_Siddhanta" title="Vasishtha Siddhanta">Vasishtha Siddhanta</a> and <a href="/wiki/Paitamaha_Siddhanta" class="mw-redirect" title="Paitamaha Siddhanta">Paitamaha Siddhanta</a>, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.<a href="#cite_note-hayashi2003-p119-53">[53]</a> <div class="mw-heading mw-heading3"><h3 id="Fourth_to_sixth_centuries">Fourth to sixth centuries</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=14" title="Edit section: Fourth to sixth centuries">edit</a>]</div> <dl><dt>Surya Siddhanta</dt></dl> Though its authorship is unknown, the <a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a> (c. 400) contains the roots of modern <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Because it contains many words of foreign origin, some authors consider that it was written under the influence of <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Mesopotamia</a> and Greece.<a href="#cite_note-Origins_of_Sulva_Sutras_and_Siddhanta-70">[70]</a>[<a href="/wiki/Wikipedia:NOTRS" class="mw-redirect" title="Wikipedia:NOTRS">better source needed</a>] This ancient text uses the following as trigonometric functions for the first time:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <ul><li>Sine (<a href="/wiki/Jya" class="mw-redirect" title="Jya">Jya</a>).</li> <li>Cosine (<a href="/wiki/Kojya" class="mw-redirect" title="Kojya">Kojya</a>).</li> <li><a href="/wiki/Inverse_sine" class="mw-redirect" title="Inverse sine">Inverse sine</a> (Otkram jya).</li></ul> Later Indian mathematicians such as Aryabhata made references to this text, while later <a href="/wiki/Arabic" title="Arabic">Arabic</a> and <a href="/wiki/Latin" title="Latin">Latin</a> translations were very influential in Europe and the Middle East. <dl><dt>Chhedi calendar</dt></dl> This Chhedi calendar (594) contains an early use of the modern <a href="/wiki/Place-value" class="mw-redirect" title="Place-value">place-value</a> <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> now used universally. <dl><dt>Aryabhata I</dt></dl> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 <a href="/wiki/Shlokas" class="mw-redirect" title="Shlokas">shlokas</a>. The treatise contained: <ul><li><a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equations</a></li> <li><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></li> <li>The value of <a href="/wiki/Pi" title="Pi">π</a>, correct to 4 decimal places.</li></ul> Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include: Trigonometry: (See also : <a href="/wiki/Aryabhata%27s_sine_table" class="mw-redirect" title="Aryabhata's sine table">Aryabhata's sine table</a>) <ul><li>Introduced the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>.</li> <li>Defined the sine (<a href="/wiki/Jya" class="mw-redirect" title="Jya">jya</a>) as the modern relationship between half an angle and half a chord.</li> <li>Defined the cosine (<a href="/wiki/Kojya" class="mw-redirect" title="Kojya">kojya</a>).</li> <li>Defined the <a href="/wiki/Versine" title="Versine">versine</a> (<a href="/wiki/Utkrama-jya" class="mw-redirect" title="Utkrama-jya">utkrama-jya</a>).</li> <li>Defined the inverse sine (otkram jya).</li> <li>Gave methods of calculating their approximate numerical values.</li> <li>Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.</li> <li>Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.</li> <li><a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">Spherical trigonometry</a>.</li></ul> Arithmetic: <ul><li><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a>.</li></ul> Algebra: <ul><li>Solutions of simultaneous quadratic equations.</li> <li>Whole number solutions of <a href="/wiki/Linear_equations" class="mw-redirect" title="Linear equations">linear equations</a> by a method equivalent to the modern method.</li> <li>General solution of the indeterminate linear equation .</li></ul> Mathematical astronomy: <ul><li>Accurate calculations for astronomical constants, such as the: <ul><li><a href="/wiki/Solar_eclipse" title="Solar eclipse">Solar eclipse</a>.</li> <li><a href="/wiki/Lunar_eclipse" title="Lunar eclipse">Lunar eclipse</a>.</li> <li>The formula for the sum of the <a href="/wiki/Cube_(algebra)" title="Cube (algebra)">cubes</a>, which was an important step in the development of integral calculus.<a href="#cite_note-katz-71">[71]</a></li></ul></li></ul> <dl><dt>Varahamihira</dt></dl> <a href="/wiki/Varahamihira" class="mw-redirect" title="Varahamihira">Varahamihira</a> (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> functions: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890c54228f31bf6d711a0fcb0b7e130ebf58fa51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.454ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d801ebcf28716d0e509afa5743fe2e963906c247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.318ex; height:4.843ex;" alt="{\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9ed0ecaf0ffec7c0644f52d0968274fcdb4f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.399ex; height:5.676ex;" alt="{\displaystyle {\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)}" /></li></ul> <div class="mw-heading mw-heading3"><h3 id="Seventh_and_eighth_centuries">Seventh and eighth centuries</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=15" title="Edit section: Seventh and eighth centuries">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Brahmaguptra%27s_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Brahmaguptra%27s_theorem.svg/200px-Brahmaguptra%27s_theorem.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Brahmaguptra%27s_theorem.svg/300px-Brahmaguptra%27s_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Brahmaguptra%27s_theorem.svg/400px-Brahmaguptra%27s_theorem.svg.png 2x" data-file-width="234" data-file-height="234" /></a><figcaption><a href="/wiki/Brahmagupta%27s_theorem" class="mw-redirect" title="Brahmagupta's theorem">Brahmagupta's theorem</a> states that AF = FD.</figcaption></figure> In the 7th century, two separate fields, <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> (which included <a href="/wiki/Measurement" title="Measurement">measurement</a>) and <a href="/wiki/Algebra" title="Algebra">algebra</a>, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations).<a href="#cite_note-hayashi2005-p369-72">[72]</a> <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>, in his astronomical work <a href="/wiki/Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta">Brāhma Sphuṭa Siddhānta</a> (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<a href="#cite_note-hayashi2003-p121-122-73">[73]</a> In the latter section, he stated his famous theorem on the diagonals of a <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a>:<a href="#cite_note-hayashi2003-p121-122-73">[73]</a> Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a>), as well as a complete description of <a href="/wiki/Rational_triangle" class="mw-redirect" title="Rational triangle">rational triangles</a> (i.e. triangles with rational sides and rational areas). Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e283116b6c89d65af7214a4d2ed55dbd421841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.963ex; height:4.843ex;" alt="{\displaystyle A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,}" /></dd></dl> where s, the <a href="/wiki/Semiperimeter" title="Semiperimeter">semiperimeter</a>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {a+b+c+d}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {a+b+c+d}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c00b5b950826ed62c4a7d026fc6a118ea62ffbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.643ex; height:5.343ex;" alt="{\displaystyle s={\frac {a+b+c+d}{2}}.}" /> Brahmagupta's Theorem on rational triangles: A triangle with rational sides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" /> and rational area is of the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {u^{2}}{v}}+v,\ \ b={\frac {u^{2}}{w}}+w,\ \ c={\frac {u^{2}}{v}}+{\frac {u^{2}}{w}}-(v+w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> </mfrac> </mrow> <mo>+</mo> <mi>v</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>w</mi> </mfrac> </mrow> <mo>+</mo> <mi>w</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>w</mi> </mfrac> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {u^{2}}{v}}+v,\ \ b={\frac {u^{2}}{w}}+w,\ \ c={\frac {u^{2}}{v}}+{\frac {u^{2}}{w}}-(v+w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bdc80a0589adca6e57a155cd71dca6379b6b05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.395ex; height:5.676ex;" alt="{\displaystyle a={\frac {u^{2}}{v}}+v,\ \ b={\frac {u^{2}}{w}}+w,\ \ c={\frac {u^{2}}{v}}+{\frac {u^{2}}{w}}-(v+w)}" /></dd></dl> for some rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb98c62121b5b3c9c84158b9c64d7d57c4eb3c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.138ex; height:2.009ex;" alt="{\displaystyle u,v,}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" />.<a href="#cite_note-74">[74]</a> Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers<a href="#cite_note-hayashi2003-p121-122-73">[73]</a> and is considered the first systematic treatment of the subject. The rules (which included <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=\ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=\ a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33ae5811b8adee3281ed20336f0954afe203c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.142ex; height:2.343ex;" alt="{\displaystyle a+0=\ a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times 0=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c086c199718699c3a64124ef56a1fec50cc071c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.494ex; height:2.176ex;" alt="{\displaystyle a\times 0=0}" />) were all correct, with one exception: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {0}{0}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <mn>0</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0}{0}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824c2ad35c100948390673d0c9175e409544e280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.26ex; height:5.176ex;" alt="{\displaystyle {\frac {0}{0}}=0}" />.<a href="#cite_note-hayashi2003-p121-122-73">[73]</a> Later in the chapter, he gave the first explicit (although still not completely general) solution of the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ ax^{2}+bx=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ax^{2}+bx=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615c04bc20de8046e0425269d2d963ffd4d260cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.467ex; height:2.843ex;" alt="{\displaystyle \ ax^{2}+bx=c}" /></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.<a href="#cite_note-stillwell2004-p87-75">[75]</a></blockquote> This is equivalent to: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mi>a</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc34ee5d75442197c093aef961a8a0cf085feb86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.717ex; height:6.176ex;" alt="{\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}" /></dd></dl> Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a>,<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}-Ny^{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}-Ny^{2}=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5927fd6715685d5bda4aa12f76fe709d5f69a0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.991ex; height:3.009ex;" alt="{\displaystyle \ x^{2}-Ny^{2}=1,}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> is a nonsquare integer. He did this by discovering the following identity:<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> Brahmagupta's Identity: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>N</mi> <mi>y</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e6cf62e1e08a707b6acb42fde1466eea3782b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.273ex; height:3.176ex;" alt="{\displaystyle \ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}}" /> which was a generalisation of an earlier identity of <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a>:<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> Brahmagupta used his identity to prove the following lemma:<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> Lemma (Brahmagupta): If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1},\ \ y=y_{1}\ \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1},\ \ y=y_{1}\ \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f44811adf4a53885c03242faf9c46a3236d554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.616ex; height:2.009ex;" alt="{\displaystyle x=x_{1},\ \ y=y_{1}\ \ }" /> is a solution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \ x^{2}-Ny^{2}=k_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \ x^{2}-Ny^{2}=k_{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc9ffea5caa0d09938673862cfdcf2cb57c4b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.675ex; height:3.009ex;" alt="{\displaystyle \ \ x^{2}-Ny^{2}=k_{1},}" /> and, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{2},\ \ y=y_{2}\ \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{2},\ \ y=y_{2}\ \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15bdb07dca9bf4ba84571431a34e7bed5bd2bb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.616ex; height:2.009ex;" alt="{\displaystyle x=x_{2},\ \ y=y_{2}\ \ }" /> is a solution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \ x^{2}-Ny^{2}=k_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \ x^{2}-Ny^{2}=k_{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0711ddd2665ac39bb73c785b0524af46a0153170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.675ex; height:3.009ex;" alt="{\displaystyle \ \ x^{2}-Ny^{2}=k_{2},}" />, then: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>N</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>y</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d658a15a6d0bb70ec7e755aa5311df6558c9c106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.093ex; height:2.509ex;" alt="{\displaystyle x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \ }" /> is a solution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}-Ny^{2}=k_{1}k_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}-Ny^{2}=k_{1}k_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7f7a8e6c836636109d9ca5d26fa77dacc2d4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.713ex; height:3.009ex;" alt="{\displaystyle \ x^{2}-Ny^{2}=k_{1}k_{2}}" /></dd></dl> He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem: Theorem (Brahmagupta): If the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}-Ny^{2}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}-Ny^{2}=k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb899df296f0cbd7c8ca6219e38d82080f78cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.393ex; height:3.009ex;" alt="{\displaystyle \ x^{2}-Ny^{2}=k}" /> has an integer solution for any one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k=\pm 4,\pm 2,-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mo>±</mo> <mn>4</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k=\pm 4,\pm 2,-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490107f763b9334d287b8d42f8ed7f337671e7e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.87ex; height:2.509ex;" alt="{\displaystyle \ k=\pm 4,\pm 2,-1}" /> then Pell's equation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}-Ny^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>N</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}-Ny^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32c9d593ecc3ebbcd518dcd72a1e1135772ebb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.344ex; height:3.009ex;" alt="{\displaystyle \ x^{2}-Ny^{2}=1}" /></dd></dl> also has an integer solution.<a href="#cite_note-stillwell2004-p74-76-77">[77]</a> Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> Example (Brahmagupta): Find integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x,\ y\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x,\ y\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720767bb78dd07ec8359d10bbfeaf51a43637715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.261ex; height:2.009ex;" alt="{\displaystyle \ x,\ y\ }" /> such that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x^{2}-92y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>92</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{2}-92y^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c42b19bf8e2aebb20884e57eda74f55b0041e1f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.606ex; height:3.009ex;" alt="{\displaystyle \ x^{2}-92y^{2}=1}" /></dd></dl> In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."<a href="#cite_note-stillwell2004-p72-73-76">[76]</a> The solution he provided was: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x=1151,\ y=120}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>1151</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>=</mo> <mn>120</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x=1151,\ y=120}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b573e8fedf27bddbd801f4559bfb7e54363a8318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.015ex; height:2.509ex;" alt="{\displaystyle \ x=1151,\ y=120}" /></dd></dl> <dl><dt>Bhaskara I</dt></dl> <a href="/wiki/Bhaskara_I" class="mw-redirect" title="Bhaskara I">Bhaskara I</a> (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced: <ul><li>Solutions of indeterminate equations.</li> <li>A rational approximation of the <a href="/wiki/Sine_function" class="mw-redirect" title="Sine function">sine function</a>.</li> <li>A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Ninth_to_twelfth_centuries">Ninth to twelfth centuries</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=16" title="Edit section: Ninth to twelfth centuries">edit</a>]</div> <dl><dt>Virasena</dt></dl> <a href="/wiki/Virasena" title="Virasena">Virasena</a> (8th century) was a Jain mathematician in the court of <a href="/wiki/Rashtrakuta" class="mw-redirect" title="Rashtrakuta">Rashtrakuta</a> King <a href="/wiki/Amoghavarsha" title="Amoghavarsha">Amoghavarsha</a> of <a href="/wiki/Manyakheta" class="mw-redirect" title="Manyakheta">Manyakheta</a>, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which: <ul><li>Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a> when applied to <a href="/wiki/Power_of_two" title="Power of two">powers of two</a>,<a href="#cite_note-78">[78]</a><a href="#cite_note-Dhavala-79">[79]</a> but differs on other numbers, more closely resembling the <a href="/wiki/P-adic_order" class="mw-redirect" title="P-adic order">2-adic order</a>.</li></ul> Virasena also gave: <ul><li>The derivation of the <a href="/wiki/Volume" title="Volume">volume</a> of a <a href="/wiki/Frustum" title="Frustum">frustum</a> by a sort of infinite procedure.</li></ul> It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.<a href="#cite_note-Dhavala-79">[79]</a> <dl><dt>Mahavira</dt></dl> <a href="/wiki/Mahavira_(mathematician)" class="mw-redirect" title="Mahavira (mathematician)">Mahavira Acharya</a> (c. 800–870) from <a href="/wiki/Karnataka" title="Karnataka">Karnataka</a>, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of: <ul><li><a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">Zero</a></li> <li><a href="/wiki/Square_(algebra)" title="Square (algebra)">Squares</a></li> <li><a href="/wiki/Cube_(arithmetic)" class="mw-redirect" title="Cube (arithmetic)">Cubes</a></li> <li><a href="/wiki/Square_root" title="Square root">square roots</a>, <a href="/wiki/Cube_root" title="Cube root">cube roots</a>, and the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> extending beyond these</li> <li>Plane geometry</li> <li><a href="/wiki/Solid_geometry" title="Solid geometry">Solid geometry</a></li> <li>Problems relating to the casting of <a href="/wiki/Shadows" class="mw-redirect" title="Shadows">shadows</a></li> <li>Formulae derived to calculate the area of an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> and <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> inside a <a href="/wiki/Circle" title="Circle">circle</a>.</li></ul> Mahavira also: <ul><li>Asserted that the square root of a <a href="/wiki/Negative_number" title="Negative number">negative number</a> did not exist</li> <li>Gave the sum of a series whose terms are <a href="/wiki/Square_(algebra)" title="Square (algebra)">squares</a> of an <a href="/wiki/Arithmetical_progression" class="mw-redirect" title="Arithmetical progression">arithmetical progression</a>, and gave empirical rules for area and <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> of an ellipse.</li> <li>Solved cubic equations.</li> <li>Solved quartic equations.</li> <li>Solved some <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equations</a> and higher-order <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>.</li> <li>Gave the general solutions of the higher order polynomial equations: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ ax^{n}=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ ax^{n}=q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87d3c9f88b3fe6b21571d19c4991be0bf4da0359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.527ex; height:2.676ex;" alt="{\displaystyle \ ax^{n}=q}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {x^{n}-1}{x-1}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {x^{n}-1}{x-1}}=p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ed34d74c12a8cdb09fc8e74f650e949fbd63c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.885ex; height:5.343ex;" alt="{\displaystyle a{\frac {x^{n}-1}{x-1}}=p}" /></li></ul></li> <li>Solved indeterminate quadratic equations.</li> <li>Solved indeterminate cubic equations.</li> <li>Solved indeterminate higher order equations.</li></ul> <dl><dt>Shridhara</dt></dl> <a href="/wiki/Shridhara" class="mw-redirect" title="Shridhara">Shridhara</a> (c. 870–930), who lived in <a href="/wiki/Bengal" title="Bengal">Bengal</a>, wrote the books titled Nav Shatika, Tri Shatika and Pati Ganita. He gave: <ul><li>A good rule for finding the volume of a <a href="/wiki/Sphere" title="Sphere">sphere</a>.</li> <li>The formula for solving <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a>.</li></ul> The Pati Ganita is a work on arithmetic and <a href="/wiki/Measurement" title="Measurement">measurement</a>. It deals with various operations, including: <ul><li>Elementary operations</li> <li>Extracting square and cube roots.</li> <li>Fractions.</li> <li>Eight rules given for operations involving zero.</li> <li>Methods of <a href="/wiki/Summation" title="Summation">summation</a> of different arithmetic and geometric series, which were to become standard references in later works.</li></ul> <dl><dt>Manjula</dt></dl> Aryabhata's equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression<a href="#cite_note-Joseph-298-300-80">[80]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sin w'-\sin w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sin w'-\sin w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf932af1ef27956f10f338d799d4274ddd3d5222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.307ex; height:2.676ex;" alt="{\displaystyle \ \sin w'-\sin w}" /></dd></dl> could be approximately expressed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (w'-w)\cos w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (w'-w)\cos w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6848ddbc5320c8c18a5212f2508f0ecb6f3da222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.793ex; height:3.009ex;" alt="{\displaystyle \ (w'-w)\cos w}" /></dd></dl> This was elaborated by his later successor Bhaskara ii thereby finding the derivative of sine.<a href="#cite_note-Joseph-298-300-80">[80]</a> <dl><dt>Aryabhata II</dt></dl> <a href="/wiki/Aryabhata_II" title="Aryabhata II">Aryabhata II</a> (c. 920–1000) wrote a commentary on Shridhara, and an astronomical treatise <a href="/wiki/Maha-Siddhanta" class="mw-redirect" title="Maha-Siddhanta">Maha-Siddhanta</a>. The Maha-Siddhanta has 18 chapters, and discusses: <ul><li>Numerical mathematics (Ank Ganit).</li> <li>Algebra.</li> <li>Solutions of indeterminate equations (kuttaka).</li></ul> <dl><dt>Shripati</dt></dl> <a href="/wiki/Sripati" class="mw-redirect" title="Sripati">Shripati Mishra</a> (1019–1066) wrote the books Siddhanta Shekhara, a major work on astronomy in 19 chapters, and Ganit Tilaka, an incomplete <a href="/wiki/Arithmetic" title="Arithmetic">arithmetical</a> treatise in 125 verses based on a work by Shridhara. He worked mainly on: <ul><li><a href="/wiki/Permutation" title="Permutation">Permutations and combinations</a>.</li> <li>General solution of the simultaneous indeterminate linear equation.</li></ul> He was also the author of Dhikotidakarana, a work of twenty verses on: <ul><li><a href="/wiki/Solar_eclipse" title="Solar eclipse">Solar eclipse</a>.</li> <li><a href="/wiki/Lunar_eclipse" title="Lunar eclipse">Lunar eclipse</a>.</li></ul> The Dhruvamanasa is a work of 105 verses on: <ul><li>Calculating planetary <a href="/wiki/Longitude" title="Longitude">longitudes</a></li> <li><a href="/wiki/Eclipse" title="Eclipse">eclipses</a>.</li> <li>planetary <a href="/wiki/Astronomical_transit" title="Astronomical transit">transits</a>.</li></ul> <dl><dt>Nemichandra Siddhanta Chakravati</dt></dl> Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled Gome-mat Saar. <dl><dt>Bhaskara II</dt></dl> <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a> (1114–1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, <a href="/wiki/Lilavati" class="mw-redirect" title="Lilavati">Lilavati</a>, <a href="/wiki/Bijaganita" title="Bijaganita">Bijaganita</a>, Gola Addhaya, Griha Ganitam and Karan Kautoohal. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include: Arithmetic: <ul><li>Interest computation</li> <li>Arithmetical and geometrical progressions</li> <li>Plane geometry</li> <li>Solid geometry</li> <li>The shadow of the <a href="/wiki/Gnomon" title="Gnomon">gnomon</a></li> <li>Solutions of <a href="/wiki/Combinations" class="mw-redirect" title="Combinations">combinations</a></li> <li>Gave a proof for division by zero being <a href="/wiki/Infinity" title="Infinity">infinity</a>.</li></ul> Algebra: <ul><li>The recognition of a positive number having two square roots.</li> <li><a href="/wiki/Nth_root" title="Nth root">Surds</a>.</li> <li>Operations with products of several unknowns.</li> <li>The solutions of: <ul><li>Quadratic equations.</li> <li>Cubic equations.</li> <li>Quartic equations.</li> <li>Equations with more than one unknown.</li> <li>Quadratic equations with more than one unknown.</li> <li>The general form of <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a> using the <a href="/wiki/Chakravala_method" title="Chakravala method">chakravala method</a>.</li> <li>The general indeterminate quadratic equation using the chakravala method.</li> <li>Indeterminate cubic equations.</li> <li>Indeterminate quartic equations.</li> <li>Indeterminate higher-order polynomial equations.</li></ul></li></ul> Geometry: <ul><li>Gave a proof of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.</li></ul> Calculus: <ul><li>Preliminary concept of differentiation</li> <li>Discovered the <a href="/wiki/Differential_coefficient" title="Differential coefficient">differential coefficient</a>.</li> <li>Stated early form of <a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a>, a special case of the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a> (one of the most important theorems of calculus and analysis).</li> <li>Derived the differential of the sine function although didn't perceive the notion of derivative.</li> <li>Computed <a href="/wiki/Pi" title="Pi">π</a>, correct to five decimal places.</li> <li>Calculated the length of the Earth's revolution around the Sun to 9 decimal places.<a href="#cite_note-81">[81]</a></li></ul> Trigonometry: <ul><li>Developments of <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a></li> <li>The trigonometric formulas: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b84d43083a6fad1e2d502e96ea8ee398f6b4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.039ex; height:2.843ex;" alt="{\displaystyle \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8cb20365b6d4e6122d1b4452badeaf6f34c7b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.039ex; height:2.843ex;" alt="{\displaystyle \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)}" /></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Medieval_and_early_modern_mathematics_(1300–1800)">Medieval and early modern mathematics (1300–1800)</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=17" title="Edit section: Medieval and early modern mathematics (1300–1800)">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Navya-Nyaya">Navya-Nyaya</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=18" title="Edit section: Navya-Nyaya">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Navya-Ny%C4%81ya" title="Navya-Nyāya">Navya-Nyāya</a></div> The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century by the philosopher <a href="/wiki/Gangesha_Upadhyaya" class="mw-redirect" title="Gangesha Upadhyaya">Gangesha Upadhyaya</a> of <a href="/wiki/Mithila_(India)" class="mw-redirect" title="Mithila (India)">Mithila</a>.<a href="#cite_note-82">[82]</a> It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers <a href="/wiki/V%C4%81caspati_Mi%C5%9Bra" class="mw-redirect" title="Vācaspati Miśra">Vācaspati Miśra</a> (900–980 CE) and <a href="/wiki/Udayana" title="Udayana">Udayana</a> (late 10th century). Gangeśa's book <a href="/wiki/Tattvacint%C4%81ma%E1%B9%87i" class="mw-redirect" title="Tattvacintāmaṇi">Tattvacintāmaṇi</a> ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of <a href="/wiki/Advaita_Vedanta" title="Advaita Vedanta">Advaita Vedānta</a>, which had offered a set of thorough criticisms of Nyāya theories of thought and language.<a href="#cite_note-83">[83]</a> Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyze, and solve problems in logic and epistemology. It involves naming each object to be analyzed, identifying a distinguishing characteristic for the named object, and verifying the appropriateness of the defining characteristic using pramanas.<a href="#cite_note-84">[84]</a> <div class="mw-heading mw-heading3"><h3 id="Kerala_School">Kerala School</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=19" title="Edit section: Kerala School">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kerala_school_chain_of_teachers.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Kerala_school_chain_of_teachers.jpg/250px-Kerala_school_chain_of_teachers.jpg" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Kerala_school_chain_of_teachers.jpg/330px-Kerala_school_chain_of_teachers.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Kerala_school_chain_of_teachers.jpg/500px-Kerala_school_chain_of_teachers.jpg 2x" data-file-width="2193" data-file-height="1668" /></a><figcaption>Chain of teachers of <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Pages_from_Yuktibhasa.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Pages_from_Yuktibhasa.jpg/220px-Pages_from_Yuktibhasa.jpg" decoding="async" width="220" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Pages_from_Yuktibhasa.jpg/330px-Pages_from_Yuktibhasa.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Pages_from_Yuktibhasa.jpg/440px-Pages_from_Yuktibhasa.jpg 2x" data-file-width="3750" data-file-height="1214" /></a><figcaption>Pages from the <a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhasa</a> c.1530</figcaption></figure> The <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a> was founded by <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> in Kerala, <a href="/wiki/South_India" title="South India">South India</a> and included among its members: <a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a>, <a href="/wiki/Neelakanta_Somayaji" class="mw-redirect" title="Neelakanta Somayaji">Neelakanta Somayaji</a>, <a href="/wiki/Jyeshtadeva" class="mw-redirect" title="Jyeshtadeva">Jyeshtadeva</a>, <a href="/wiki/Achyuta_Pisharati" class="mw-redirect" title="Achyuta Pisharati">Achyuta Pisharati</a>, <a href="/wiki/Melpathur_Narayana_Bhattathiri" title="Melpathur Narayana Bhattathiri">Melpathur Narayana Bhattathiri</a> and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematics concepts. The most important results, series expansion for <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>, were given in <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> verse in a book by Neelakanta called Tantrasangraha and a commentary on this work called Tantrasangraha-vakhya of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work <a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a> (c.1500–c.1610), written in <a href="/wiki/Malayalam" title="Malayalam">Malayalam</a>, by <a href="/wiki/Jyesthadeva" class="mw-redirect" title="Jyesthadeva">Jyesthadeva</a>.<a href="#cite_note-roy-85">[85]</a> Their discovery of these three important series expansions of <a href="/wiki/Calculus" title="Calculus">calculus</a>—several centuries before calculus was developed in Europe by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>—was an achievement. However, the Kerala School did not invent calculus,<a href="#cite_note-bressoud-86">[86]</a> because, while they were able to develop <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansions for the important <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, they developed neither a theory of <a href="/wiki/Derivative" title="Derivative">differentiation</a> or <a href="/wiki/Integral" title="Integral">integration</a>, nor the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>.<a href="#cite_note-katz-71">[71]</a> The results obtained by the Kerala school include: <ul><li>The (infinite) <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+x^{4}+\cdots {\text{ for }}|x|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+x^{4}+\cdots {\text{ for }}|x|<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0adcb6bbc05ef1826f956b47d08395648b1e74e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.055ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+x^{4}+\cdots {\text{ for }}|x|<1}" /><a href="#cite_note-singh-87">[87]</a></li> <li>A semi-rigorous proof (see "induction" remark below) of the result: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54eca5f4e177d583418540fc66581d0421b0a33d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.248ex; height:6.176ex;" alt="{\displaystyle 1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}}" /> for large n.<a href="#cite_note-roy-85">[85]</a></li> <li>Intuitive use of <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>, however, the <a href="/wiki/Mathematical_induction#Description" title="Mathematical induction">inductive hypothesis</a> was not formulated or employed in proofs.<a href="#cite_note-roy-85">[85]</a></li> <li>Applications of ideas from (what was to become) differential and integral calculus to obtain <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">(Taylor–Maclaurin) infinite series</a> for sin x, cos x, and arctan x.<a href="#cite_note-bressoud-86">[86]</a> The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:<a href="#cite_note-roy-85">[85]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>y</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>≤</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aac0095704595975d16be37eaa662098def9974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:66.543ex; height:6.009ex;" alt="{\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef425a8c3e8a783197747f28eb15952d135c52e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.923ex; height:6.676ex;" alt="{\displaystyle r\sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a5a87f9944ef59edd46f6e538f5765247384aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:56.254ex; height:6.676ex;" alt="{\displaystyle r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,}" /></dd></dl></dd> <dd>where, for r = 1, the series reduces to the standard power series for these trigonometric functions, for example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a9027c77b90e40215c01281b39d37730e7e537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.746ex; height:5.843ex;" alt="{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }" /></dd></dl></dd> <dd>and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ec2d1dd64d61a18f031ed242dd310840bcbbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.834ex; height:5.843ex;" alt="{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }" /></dd></dl></dd></dl> <ul><li>Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature, i.e. computation of area under the arc of the circle, was not used.)<a href="#cite_note-roy-85">[85]</a></li> <li>Use of the series expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7315ab90c047f16ec0972c15e14988ed737c1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.183ex; height:2.009ex;" alt="{\displaystyle \arctan x}" /> to obtain the <a href="/wiki/Leibniz_formula_for_%CF%80" title="Leibniz formula for π">Leibniz formula for π</a>:<a href="#cite_note-roy-85">[85]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ac5f27203f1869a2c740c98202f11a0e78e30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.51ex; height:5.343ex;" alt="{\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }" /></dd></dl></dd></dl> <ul><li>A rational approximation of error for the finite sum of their series of interest. For example, the error, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}(n+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0c8921836dbb46968f58a4ccb43b50bfb7b38e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.146ex; height:2.843ex;" alt="{\displaystyle f_{i}(n+1)}" />, (for n odd, and i = 1, 2, 3) for the series:</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots +(-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>≈</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots +(-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6d2748ac039769ab8aa95cc91e20166de30b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.772ex; height:5.176ex;" alt="{\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots +(-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{where }}f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{where }}f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57e7b86173924c707fe9bd2f3b9e535f11f7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.39ex; height:6.676ex;" alt="{\displaystyle {\text{where }}f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}" /></dd></dl></dd></dl> <ul><li>Manipulation of error term to derive a faster converging series for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" />:<a href="#cite_note-roy-85">[85]</a></li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>7</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ef8b29347050c41378497b9957830d2c09f620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.517ex; height:5.843ex;" alt="{\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots }" /></dd></dl></dd></dl> <ul><li>Using the improved series to derive a rational expression,<a href="#cite_note-roy-85">[85]</a> 104348/33215 for π correct up to nine decimal places, i.e. 3.141592653.</li> <li>Use of an intuitive notion of limit to compute these results.<a href="#cite_note-roy-85">[85]</a></li> <li>A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.<a href="#cite_note-katz-71">[71]</a> However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functions.</li></ul> The works of the Kerala school were first written up for the Western world by Englishman <a href="/wiki/C.M._Whish" class="mw-redirect" title="C.M. Whish">C.M. Whish</a> in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."<a href="#cite_note-whish-88">[88]</a> However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhāṣā given in two papers,<a href="#cite_note-89">[89]</a><a href="#cite_note-90">[90]</a> a commentary on the Yuktibhāṣā's proof of the sine and cosine series<a href="#cite_note-91">[91]</a> and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).<a href="#cite_note-92">[92]</a><a href="#cite_note-93">[93]</a> Parameshvara (c. 1370–1460) wrote commentaries on the works of <a href="/wiki/Bhaskara_I" class="mw-redirect" title="Bhaskara I">Bhaskara I</a>, <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his important discoveries: a version of the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>. <a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a> (1444–1544) composed the Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501). He elaborated and extended the contributions of Madhava. <a href="/wiki/Citrabhanu" class="mw-redirect" title="Citrabhanu">Citrabhanu</a> (c. 1530) was a 16th-century mathematician from Kerala who gave integer solutions to 21 types of systems of two <a href="/wiki/Simultaneous_equation" class="mw-redirect" title="Simultaneous equation">simultaneous</a> algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,\\[8pt]&x^{2}-y^{2}=e,\ x^{3}+y^{3}=f,\ x^{3}-y^{3}=g\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>c</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>g</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,\\[8pt]&x^{2}-y^{2}=e,\ x^{3}+y^{3}=f,\ x^{3}-y^{3}=g\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7c2147f7c241cb7c7eaf65c879c86be5dfb2ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:43.081ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}&x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,\\[8pt]&x^{2}-y^{2}=e,\ x^{3}+y^{3}=f,\ x^{3}-y^{3}=g\end{aligned}}}" /></dd></dl> For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. <a href="/wiki/Jyesthadeva" class="mw-redirect" title="Jyesthadeva">Jyesthadeva</a> (c. 1500–1575) was another member of the Kerala School. His key work was the Yukti-bhāṣā (written in Malayalam, a regional language of Kerala). Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians. <div class="mw-heading mw-heading3"><h3 id="Others">Others</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=20" title="Edit section: Others">edit</a>]</div> <a href="/wiki/Narayana_Pandit" class="mw-redirect" title="Narayana Pandit">Narayana Pandit</a> was a 14th century mathematician who composed two important mathematical works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa. Ganita Kaumudi is one of the most revolutionary works in the field of combinatorics with developing a method for <a href="/wiki/Permutation#Generation_in_lexicographic_order" title="Permutation">systematic generation of all permutations</a> of a given sequence. In his Ganita Kaumudi Narayana proposed the following problem on a herd of cows and calves: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote">A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after 20 years?</blockquote> Translated into the modern mathematical language of <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence sequences</a>: <dl><dd>Nn = Nn-1 + Nn-3 for n > 2,</dd></dl> with initial values <dl><dd>N0 = N1 = N2 = 1.</dd></dl> The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... (sequence <a href="//oeis.org/A000930" class="extiw" title="oeis:A000930">A000930</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The limit ratio between consecutive terms is the <a href="/wiki/Supergolden_ratio" title="Supergolden ratio">supergolden ratio</a>. . Narayana is also thought to be the author of an elaborate commentary of <a href="/wiki/Bhaskara_II" class="mw-redirect" title="Bhaskara II">Bhaskara II</a>'s <a href="/wiki/Lilavati" class="mw-redirect" title="Lilavati">Lilavati</a>, titled <a href="/wiki/Ganita_Kaumudi" title="Ganita Kaumudi">Ganita Kaumudia</a>(or Karma-Paddhati).<a href="#cite_note-94">[94]</a> <div class="mw-heading mw-heading2"><h2 id="Charges_of_Eurocentrism">Charges of Eurocentrism</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=21" title="Edit section: Charges of Eurocentrism">edit</a>]</div> It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by <a href="/wiki/Indian_mathematicians" class="mw-redirect" title="Indian mathematicians">Indian mathematicians</a> are presently culturally attributed to their <a href="/wiki/Western_world" title="Western world">Western</a> counterparts, as a result of <a href="/wiki/Eurocentrism" title="Eurocentrism">Eurocentrism</a>. According to G. G. Joseph's take on "<a href="/wiki/Ethnomathematics" title="Ethnomathematics">Ethnomathematics</a>": <blockquote>[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"<a href="#cite_note-95">[95]</a></blockquote> Historian of mathematics <a href="/wiki/Florian_Cajori" title="Florian Cajori">Florian Cajori</a> wrote that he and others "suspect that <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> got his first glimpse of algebraic knowledge from India".<a href="#cite_note-96">[96]</a> He also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".<a href="#cite_note-97">[97]</a> More recently, as discussed in the above section, the infinite series of <a href="/wiki/Calculus" title="Calculus">calculus</a> for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described in India, by mathematicians of the <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school</a>, some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from <a href="/wiki/Kerala" title="Kerala">Kerala</a> by traders and <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> missionaries.<a href="#cite_note-almeida-98">[98]</a> Kerala was in continuous contact with China and <a href="/wiki/Arabia" class="mw-redirect" title="Arabia">Arabia</a>, and, from around 1500, with Europe. The fact that the communication routes existed and the chronology is suitable certainly make such transmission a possibility. However, no evidence of transmission has been found.<a href="#cite_note-almeida-98">[98]</a> According to <a href="/wiki/David_Bressoud" title="David Bressoud">David Bressoud</a>, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century".<a href="#cite_note-bressoud-86">[86]</a><a href="#cite_note-gold-99">[99]</a> Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<a href="#cite_note-katz-71">[71]</a> However, they did not (as <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> did) "combine many differing ideas under the two unifying themes of the <a href="/wiki/Derivative" title="Derivative">derivative</a> and the <a href="/wiki/Integral" title="Integral">integral</a>, show the connection between the two, and turn calculus into the great problem-solving tool we have today".<a href="#cite_note-katz-71">[71]</a> The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;<a href="#cite_note-katz-71">[71]</a> however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, [may have] learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."<a href="#cite_note-katz-71">[71]</a> This is an area of current research, especially in the manuscript collections of Spain and <a href="/wiki/Maghreb" title="Maghreb">Maghreb</a>, and is being pursued, among other places, at the <a href="/wiki/CNRS" class="mw-redirect" title="CNRS">CNRS</a>.<a href="#cite_note-katz-71">[71]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=22" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 27em;"> <ul><li><a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a></li> <li><a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a></li> <li><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a></li> <li><a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a></li> <li><a href="/wiki/List_of_Indian_mathematicians" title="List of Indian mathematicians">List of Indian mathematicians</a></li> <li><a href="/wiki/Indian_science_and_technology" class="mw-redirect" title="Indian science and technology">Indian science and technology</a></li> <li><a href="/wiki/Indian_logic" title="Indian logic">Indian logic</a></li> <li><a href="/wiki/Indian_astronomy" title="Indian astronomy">Indian astronomy</a></li> <li><a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a></li> <li><a href="/wiki/List_of_numbers_in_Hindu_scriptures" title="List of numbers in Hindu scriptures">List of numbers in Hindu scriptures</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=23" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-plofker-1">^ <a href="#cite_ref-plofker_1-0">a</a> <a href="#cite_ref-plofker_1-1">b</a> (<a href="#CITEREFKim_Plofker2007">Kim Plofker 2007</a>, p. 1) </li> <li id="cite_note-hayashi2005-p360-361-2">^ <a href="#cite_ref-hayashi2005-p360-361_2-0">a</a> <a href="#cite_ref-hayashi2005-p360-361_2-1">b</a> <a href="#cite_ref-hayashi2005-p360-361_2-2">c</a> <a href="#cite_ref-hayashi2005-p360-361_2-3">d</a> (<a href="#CITEREFHayashi2005">Hayashi 2005</a>, pp. 360–361) </li> <li id="cite_note-irfah346-3"><a href="#cite_ref-irfah346_3-0">^</a> (<a href="#CITEREFIfrah2000">Ifrah 2000</a>, p. 346): "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own." </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> (<a href="#CITEREFPlofker2009">Plofker 2009</a>, pp. 44–47) </li> <li id="cite_note-bourbaki46-5"><a href="#cite_ref-bourbaki46_5-0">^</a> (<a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, p. 46): "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus." </li> <li id="cite_note-bourbaki49-6"><a href="#cite_ref-bourbaki49_6-0">^</a> (<a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, p. 49): Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. <a href="/wiki/Leonardo_of_Pisa" class="mw-redirect" title="Leonardo of Pisa">Leonardo of Pisa</a> wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> during 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic." </li> <li id="cite_note-concise-britannica-7">^ <a href="#cite_ref-concise-britannica_7-0">a</a> <a href="#cite_ref-concise-britannica_7-1">b</a> "algebra" 2007. <a rel="nofollow" class="external text" href="https://www.britannica.com/ebc/article-231064">Britannica Concise Encyclopedia</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070929134632/http://www.britannica.com/ebc/article-231064">Archived</a> 29 September 2007 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra." </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> (<a href="#CITEREFPingree2003">Pingree 2003</a>, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> (<a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (<a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a>, <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a>, <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta <\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta <\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053c2f5cb3d9460d5fe43fcc48920484fb2ce4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta <\pi }" /> on a circle of radius r, in other words the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2r\sin \left(\theta /2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r\sin \left(\theta /2\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca756adcc10a403716e769f6214640ed6802c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:2.843ex;" alt="{\displaystyle 2r\sin \left(\theta /2\right)}" />; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)." </li> <li id="cite_note-filliozat-p140to143-10"><a href="#cite_ref-filliozat-p140to143_10-0">^</a> (<a href="#CITEREFFilliozat2004">Filliozat 2004</a>, pp. 140–143) </li> <li id="cite_note-hayashi95-11"><a href="#cite_ref-hayashi95_11-0">^</a> (<a href="#CITEREFHayashi1995">Hayashi 1995</a>) </li> <li id="cite_note-plofker-brit6-12">^ <a href="#cite_ref-plofker-brit6_12-0">a</a> <a href="#cite_ref-plofker-brit6_12-1">b</a> (<a href="#CITEREFKim_Plofker2007">Kim Plofker 2007</a>, p. 6) </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> (<a href="#CITEREFStillwell2004">Stillwell 2004</a>, p. 173) </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> (<a href="#CITEREFBressoud2002">Bressoud 2002</a>, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use." </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> (<a href="#CITEREFPlofker2001">Plofker 2001</a>, p. 293) Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here" </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> (<a href="#CITEREFPingree1992">Pingree 1992</a>, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by <a href="/wiki/C.M._Whish" class="mw-redirect" title="C.M. Whish">Charles Matthew Whish</a>, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution." </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> (<a href="#CITEREFKatz1995">Katz 1995</a>, pp. 173–174) Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today." </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSergent1997" class="citation cs2 cs1-prop-foreign-lang-source">Sergent, Bernard (1997), Genèse de l'Inde (in French), Paris: Payot, p. 113, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-228-89116-5" title="Special:BookSources/978-2-228-89116-5"><bdi>978-2-228-89116-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gen%C3%A8se+de+l%27Inde&rft.place=Paris&rft.pages=113&rft.pub=Payot&rft.date=1997&rft.isbn=978-2-228-89116-5&rft.aulast=Sergent&rft.aufirst=Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoppa2006" class="citation cs2">Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Natur.440..755C">2006Natur.440..755C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F440755a">10.1038/440755a</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16598247">16598247</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6787162">6787162</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Early+Neolithic+tradition+of+dentistry%3A+Flint+tips+were+surprisingly+effective+for+drilling+tooth+enamel+in+a+prehistoric+population&rft.volume=440&rft.issue=7085&rft.pages=%3Cspan+class%3D%22nowrap%22%3E755-%3C%2Fspan%3E6&rft.date=2006-04-06&rft_id=info%3Adoi%2F10.1038%2F440755a&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6787162%23id-name%3DS2CID&rft_id=info%3Apmid%2F16598247&rft_id=info%3Abibcode%2F2006Natur.440..755C&rft.aulast=Coppa&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBisht1982" class="citation cs2">Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113–124</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Excavations+at+Banawali%3A+1974%E2%80%9377&rft.btitle=Harappan+Civilisation%3A+A+Contemporary+Perspective&rft.place=New+Delhi&rft.pages=%3Cspan+class%3D%22nowrap%22%3E113-%3C%2Fspan%3E124&rft.pub=Oxford+and+IBH+Publishing+Co.&rft.date=1982&rft.aulast=Bisht&rft.aufirst=R.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRao1992" class="citation journal cs1">Rao, S. R. (July 1992). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170808011822/http://drs.nio.org/drs/bitstream/handle/2264/3082/J_Mar_Archaeol_3_61.pdf?sequence=2">"A Navigational Instrument of the Harappan Sailors"</a> (PDF). Marine Archaeology. 3: 61–62. Archived from <a rel="nofollow" class="external text" href="http://drs.nio.org/drs/bitstream/handle/2264/3082/J_Mar_Archaeol_3_61.pdf?sequence=2">the original</a> (PDF) on 8 August 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Marine+Archaeology&rft.atitle=A+Navigational+Instrument+of+the+Harappan+Sailors&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E61-%3C%2Fspan%3E62&rft.date=1992-07&rft.aulast=Rao&rft.aufirst=S.+R.&rft_id=http%3A%2F%2Fdrs.nio.org%2Fdrs%2Fbitstream%2Fhandle%2F2264%2F3082%2FJ_Mar_Archaeol_3_61.pdf%3Fsequence%3D2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> A. Seidenberg, 1978. The origin of mathematics. Archive for History of Exact Sciences, vol 18. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> (<a href="#CITEREFStaal1999">Staal 1999</a>) </li> <li id="cite_note-hayashi2003-p118-24">^ <a href="#cite_ref-hayashi2003-p118_24-0">a</a> <a href="#cite_ref-hayashi2003-p118_24-1">b</a> (<a href="#CITEREFHayashi2003">Hayashi 2003</a>, p. 118) </li> <li id="cite_note-hayashi2005-p363-25">^ <a href="#cite_ref-hayashi2005-p363_25-0">a</a> <a href="#cite_ref-hayashi2005-p363_25-1">b</a> (<a href="#CITEREFHayashi2005">Hayashi 2005</a>, p. 363) </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> Pythagorean triples are triples of integers (a, b, c) with the property: a2+b2 = c2. Thus, 32+42 = 52, 82+152 = 172, 122+352 = 372, etc. </li> <li id="cite_note-cooke198-27"><a href="#cite_ref-cooke198_27-0">^</a> (<a href="#CITEREFCooke2005">Cooke 2005</a>, p. 198): "The arithmetic content of the Śulva Sūtras consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others." </li> <li id="cite_note-cooke199-200-28"><a href="#cite_ref-cooke199-200_28-0">^</a> (<a href="#CITEREFCooke2005">Cooke 2005</a>, pp. 199–200): "The requirement of three altars of equal areas but different shapes would explain the interest in transformation of areas. Among other transformation of area problems the Hindus considered in particular the problem of squaring the circle. The Bodhayana Sutra states the converse problem of constructing a circle equal to a given square. The following approximate construction is given as the solution.... this result is only approximate. The authors, however, made no distinction between the two results. In terms that we can appreciate, this construction gives a value for π of 18 (3 − 2√2), which is about 3.088." </li> <li id="cite_note-joseph229-29">^ <a href="#cite_ref-joseph229_29-0">a</a> <a href="#cite_ref-joseph229_29-1">b</a> <a href="#cite_ref-joseph229_29-2">c</a> (<a href="#CITEREFJoseph2000">Joseph 2000</a>, p. 229) </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.theintellibrain.com/vedicmaths/">"Vedic Maths Complete Detail"</a>. ALLEN IntelliBrain. Retrieved 22 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ALLEN+IntelliBrain&rft.atitle=Vedic+Maths+Complete+Detail&rft_id=https%3A%2F%2Fwww.theintellibrain.com%2Fvedicmaths%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-cooke200-31">^ <a href="#cite_ref-cooke200_31-0">a</a> <a href="#cite_ref-cooke200_31-1">b</a> (<a href="#CITEREFCooke2005">Cooke 2005</a>, p. 200) </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> The value of this approximation, 577/408, is the seventh in a sequence of increasingly accurate approximations 3/2, 7/5, 17/12, ... to √2, the numerators and denominators of which were known as "side and diameter numbers" to the ancient Greeks, and in modern mathematics are called the <a href="/wiki/Pell_numbers" class="mw-redirect" title="Pell numbers">Pell numbers</a>. If x/y is one term in this sequence of approximations, the next is (x + 2y)/(x + y). These approximations may also be derived by truncating the <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> representation of √2. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Neugebauer, O. and A. Sachs. 1945. Mathematical Cuneiform Texts, New Haven, CT, Yale University Press. p. 45. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> Mathematics Department, University of British Columbia, <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html">The Babylonian tabled Plimpton 322</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200617151320/http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html">Archived</a> 17 June 2020 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> Three positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}" /> form a primitive Pythagorean triple if c2 = a2+b2 and if the highest common factor of a, b, c is 1. In the particular Plimpton322 example, this means that 135002+127092 = 185412 and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details. </li> <li id="cite_note-dani-36">^ <a href="#cite_ref-dani_36-0">a</a> <a href="#cite_ref-dani_36-1">b</a> (<a href="#CITEREFDani2003">Dani 2003</a>) </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIngerman1967" class="citation journal cs1">Ingerman, Peter Zilahy (1 March 1967). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363162.363165">""Pānini-Backus Form" suggested"</a>. Communications of the ACM. 10 (3): 137. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363162.363165">10.1145/363162.363165</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:52817672">52817672</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=%22P%C4%81nini-Backus+Form%22+suggested&rft.volume=10&rft.issue=3&rft.pages=137&rft.date=1967-03-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52817672%23id-name%3DS2CID&rft.issn=0001-0782&rft_id=info%3Adoi%2F10.1145%2F363162.363165&rft.aulast=Ingerman&rft.aufirst=Peter+Zilahy&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F363162.363165&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-fowler96-38"><a href="#cite_ref-fowler96_38-0">^</a> (<a href="#CITEREFFowler1996">Fowler 1996</a>, p. 11) </li> <li id="cite_note-singh36-39"><a href="#cite_ref-singh36_39-0">^</a> (<a href="#CITEREFSingh1936">Singh 1936</a>, pp. 623–624) </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="display:inline;"><ul style="display:inline;"><li style="margin-bottom:.5em; display:block;;display:inline; margin:0;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHarper2011" class="citation encyclopaedia cs1">Harper, Douglas (2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170703014638/http://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none">"Zero"</a>. Etymonline Etymology Dictionary. Archived from <a rel="nofollow" class="external text" href="https://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none">the original</a> on 3 July 2017. <q>figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", <abbr title="circa">c.</abbr> 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zero&rft.btitle=Etymonline+Etymology+Dictionary&rft.date=2011&rft.aulast=Harper&rft.aufirst=Douglas&rft_id=https%3A%2F%2Fwww.etymonline.com%2Findex.php%3Fallowed_in_frame%3D0%26search%3Dzero%26searchmode%3Dnone&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li><li style="margin-bottom:.5em; display:block;;margin-top:.5em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMenninger1992" class="citation book cs1">Menninger, Karl (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BFJHzSIj2u0C">Number Words and Number Symbols: A cultural history of numbers</a>. Courier Dover Publications. pp. 399–404. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-27096-8" title="Special:BookSources/978-0-486-27096-8"><bdi>978-0-486-27096-8</bdi></a>. Retrieved 5 January 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Words+and+Number+Symbols%3A+A+cultural+history+of+numbers&rft.pages=%3Cspan+class%3D%22nowrap%22%3E399-%3C%2Fspan%3E404&rft.pub=Courier+Dover+Publications&rft.date=1992&rft.isbn=978-0-486-27096-8&rft.aulast=Menninger&rft.aufirst=Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBFJHzSIj2u0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li><li style="margin-bottom:.5em; display:block;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false">"zero, n."</a> <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">OED</a> Online. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. December 2011. <a rel="nofollow" class="external text" href="https://www.webcitation.org/65yd7ur9u?url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false">Archived</a> from the original on 7 March 2012. Retrieved 4 March 2012. <q>French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=OED+Online&rft.atitle=zero%2C+n.&rft.date=2011-12&rft_id=http%3A%2F%2Fwww.oed.com%2Fview%2FEntry%2F232803%3Frskey%3DzGcSoq%26result%3D1%26isAdvanced%3Dfalse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li></ul></div> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDattaSingh2019" class="citation book cs1">Datta, Bibhutibhusan; Singh, Awadhesh Narayan (2019). "Use of permutations and combinations in India". In Kolachana, Aditya; Mahesh, K.; Ramasubramanian, K. (eds.). Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla. Sources and Studies in the History of Mathematics and Physical Sciences. Springer Singapore. pp. 356–376. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-13-7326-8_18">10.1007/978-981-13-7326-8_18</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-13-7325-1" title="Special:BookSources/978-981-13-7325-1"><bdi>978-981-13-7325-1</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:191141516">191141516</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Use+of+permutations+and+combinations+in+India&rft.btitle=Studies+in+Indian+Mathematics+and+Astronomy%3A+Selected+Articles+of+Kripa+Shankar+Shukla&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pages=%3Cspan+class%3D%22nowrap%22%3E356-%3C%2Fspan%3E376&rft.pub=Springer+Singapore&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A191141516%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-981-13-7326-8_18&rft.isbn=978-981-13-7325-1&rft.aulast=Datta&rft.aufirst=Bibhutibhusan&rft.au=Singh%2C+Awadhesh+Narayan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">. Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=MR1189487">MR1189487</a>. 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Delhi: Motilal Banarsidass. pp. 405–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788120805651" title="Special:BookSources/9788120805651"><bdi>9788120805651</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Indian+Logic%3A+Ancient%2C+Mediaeval+and+Modern+Schools&rft.place=Delhi&rft.pages=%3Cspan+class%3D%22nowrap%22%3E405-%3C%2Fspan%3E6&rft.pub=Motilal+Banarsidass&rft.date=1920&rft.isbn=9788120805651&rft.aulast=Vidyabhusana&rft.aufirst=Satis+Chandra&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0lG85RD9YZoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSatis_Chandra_Vidyabhusana1920" class="citation book cs1">Satis Chandra Vidyabhusana (1920). <a rel="nofollow" class="external text" href="https://archive.org/details/historyindianlog00vidy">A History of Indian Logic: Ancient, Mediaeval and Modern Schools</a>. 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(1977), "On an untapped source of medieval Keralese mathematics", Archive for History of Exact Sciences, 18 (2): 89–102, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00348142">10.1007/BF00348142</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51861422">51861422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=On+an+untapped+source+of+medieval+Keralese+mathematics&rft.volume=18&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E89-%3C%2Fspan%3E102&rft.date=1977&rft_id=info%3Adoi%2F10.1007%2FBF00348142&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51861422%23id-name%3DS2CID&rft.aulast=Rajagopal&rft.aufirst=C.&rft.au=Rangachari%2C+M.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"> </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRajagopalRangachari1986" class="citation cs2">Rajagopal, C.; Rangachari, M. 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(1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2691411">10.2307/2691411</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2691411">2691411</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Ideas+of+Calculus+in+Islam+and+India&rft.volume=68&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E163-%3C%2Fspan%3E174&rft.date=1995&rft_id=info%3Adoi%2F10.2307%2F2691411&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2691411%23id-name%3DJSTOR&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKatz2007" class="citation cs2">Katz, Victor J., ed. (2007), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, pp. 385–514, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India%2C+and+Islam%3A+A+Sourcebook&rft.place=Princeton%2C+NJ&rft.pages=%3Cspan+class%3D%22nowrap%22%3E385-%3C%2Fspan%3E514&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeller2005" class="citation cs2">Keller, Agathe (2005), <a rel="nofollow" class="external text" href="https://halshs.archives-ouvertes.fr/halshs-00000445/file/Diags.pdf">"Making diagrams speak, in Bhāskara I's commentary on the Aryabhaṭīya"</a> (PDF), Historia Mathematica, 32 (3): 275–302, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.hm.2004.09.001">10.1016/j.hm.2004.09.001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Making+diagrams+speak%2C+in+Bh%C4%81skara+I%27s+commentary+on+the+Aryabha%E1%B9%AD%C4%ABya&rft.volume=32&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E275-%3C%2Fspan%3E302&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2Fj.hm.2004.09.001&rft.aulast=Keller&rft.aufirst=Agathe&rft_id=https%3A%2F%2Fhalshs.archives-ouvertes.fr%2Fhalshs-00000445%2Ffile%2FDiags.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKichenassamy2006" class="citation cs2">Kichenassamy, Satynad (2006), "Baudhāyana's rule for the quadrature of the circle", Historia Mathematica, 33 (2): 149–183, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.hm.2005.05.001">10.1016/j.hm.2005.05.001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Baudh%C4%81yana%27s+rule+for+the+quadrature+of+the+circle&rft.volume=33&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E149-%3C%2Fspan%3E183&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.hm.2005.05.001&rft.aulast=Kichenassamy&rft.aufirst=Satynad&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNeugebauerPingree1970" class="citation cs2"><a href="/wiki/Otto_Neugebauer" class="mw-redirect" title="Otto Neugebauer">Neugebauer, Otto</a>; <a href="/wiki/David_Pingree" title="David Pingree">Pingree, David</a>, eds. 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(1978), The Yavanajātaka of Sphujidhvaja, <a href="/wiki/Harvard_Oriental_Series" title="Harvard Oriental Series">Harvard Oriental Series</a> 48 (2 vols.), Edited, translated and commented by D. 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href="https://doi.org/10.1007%2F978-1-4684-9281-1">10.1007/978-1-4684-9281-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95336-6" title="Special:BookSources/978-0-387-95336-6"><bdi>978-0-387-95336-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+its+History&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=2&rft.pub=Springer%2C+Berlin+and+New+York%2C+568+pages&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2F978-1-4684-9281-1&rft.isbn=978-0-387-95336-6&rft.aulast=Stillwell&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThibaut1984" class="citation cs2"><a href="/wiki/George_Thibaut" title="George Thibaut">Thibaut, George</a> (1984) [1875], Mathematics in the Making in Ancient India: reprints of 'On the Sulvasutras' and 'Baudhyayana Sulva-sutra', Calcutta and Delhi: K. P. Bagchi and Company (orig. Journal of the Asiatic Society of Bengal), 133 pages</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+the+Making+in+Ancient+India%3A+reprints+of+%27On+the+Sulvasutras%27+and+%27Baudhyayana+Sulva-sutra%27&rft.pub=Calcutta+and+Delhi%3A+K.+P.+Bagchi+and+Company+%28orig.+Journal+of+the+Asiatic+Society+of+Bengal%29%2C+133+pages&rft.date=1984&rft.aulast=Thibaut&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_der_Waerden1983" class="citation cs2"><a href="/wiki/B._L._van_der_Waerden" class="mw-redirect" title="B. L. van der Waerden">van der Waerden, B. L.</a> (1983), <a rel="nofollow" class="external text" href="https://archive.org/details/geometryalgebrai0000waer">Geometry and Algebra in Ancient Civilisations</a>, Berlin and New York: Springer, 223 pages, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-12159-8" title="Special:BookSources/978-0-387-12159-8"><bdi>978-0-387-12159-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+and+Algebra+in+Ancient+Civilisations&rft.pub=Berlin+and+New+York%3A+Springer%2C+223+pages&rft.date=1983&rft.isbn=978-0-387-12159-8&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryalgebrai0000waer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_der_Waerden1988" class="citation cs2"><a href="/wiki/B._L._van_der_Waerden" class="mw-redirect" title="B. L. van der Waerden">van der Waerden, B. L.</a> (1988), "On the Romaka-Siddhānta", Archive for History of Exact Sciences, 38 (1): 1–11, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00329976">10.1007/BF00329976</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:189788738">189788738</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=On+the+Romaka-Siddh%C4%81nta&rft.volume=38&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E11&rft.date=1988&rft_id=info%3Adoi%2F10.1007%2FBF00329976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189788738%23id-name%3DS2CID&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_der_Waerden1988" class="citation cs2"><a href="/wiki/B._L._van_der_Waerden" class="mw-redirect" title="B. L. van der Waerden">van der Waerden, B. L.</a> (1988), "Reconstruction of a Greek table of chords", Archive for History of Exact Sciences, 38 (1): 23–38, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988AHES...38...23V">1988AHES...38...23V</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00329978">10.1007/BF00329978</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:189793547">189793547</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Reconstruction+of+a+Greek+table+of+chords&rft.volume=38&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E23-%3C%2Fspan%3E38&rft.date=1988&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189793547%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00329978&rft_id=info%3Abibcode%2F1988AHES...38...23V&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVan_Nooten1993" class="citation cs2">Van Nooten, B. (1993), "Binary numbers in Indian antiquity", Journal of Indian Philosophy, 21 (1), Springer Netherlands: 31–50, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01092744">10.1007/BF01092744</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:171039636">171039636</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Indian+Philosophy&rft.atitle=Binary+numbers+in+Indian+antiquity&rft.volume=21&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E31-%3C%2Fspan%3E50&rft.date=1993&rft_id=info%3Adoi%2F10.1007%2FBF01092744&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A171039636%23id-name%3DS2CID&rft.aulast=Van+Nooten&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWhish1835" class="citation cs2"><a href="/wiki/C.M._Whish" class="mw-redirect" title="C.M. Whish">Whish, Charles</a> (1835), <a rel="nofollow" class="external text" href="https://zenodo.org/record/2223599">"On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála"</a>, Transactions of the Royal Asiatic Society of Great Britain and Ireland, 3 (3): 509–523, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0950473700001221">10.1017/S0950473700001221</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25581775">25581775</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Royal+Asiatic+Society+of+Great+Britain+and+Ireland&rft.atitle=On+the+Hind%C3%BA+Quadrature+of+the+Circle%2C+and+the+infinite+Series+of+the+proportion+of+the+circumference+to+the+diameter+exhibited+in+the+four+S%27%C3%A1stras%2C+the+Tantra+Sangraham%2C+Yucti+Bh%C3%A1sh%C3%A1%2C+Carana+Padhati%2C+and+Sadratnam%C3%A1la&rft.volume=3&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E509-%3C%2Fspan%3E523&rft.date=1835&rft_id=info%3Adoi%2F10.1017%2FS0950473700001221&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25581775%23id-name%3DJSTOR&rft.aulast=Whish&rft.aufirst=Charles&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2223599&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYano2006" class="citation cs2">Yano, Michio (2006), "Oral and Written Transmission of the Exact Sciences in Sanskrit", Journal of Indian Philosophy, 34 (1–2), Springer Netherlands: 143–160, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10781-005-8175-6">10.1007/s10781-005-8175-6</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170679879">170679879</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Indian+Philosophy&rft.atitle=Oral+and+Written+Transmission+of+the+Exact+Sciences+in+Sanskrit&rft.volume=34&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E143-%3C%2Fspan%3E160&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2Fs10781-005-8175-6&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170679879%23id-name%3DS2CID&rft.aulast=Yano&rft.aufirst=Michio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=25" title="Edit section: Further reading">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Source_books_in_Sanskrit">Source books in Sanskrit</h3>[<a href="/w/index.php?title=Indian_mathematics&action=edit&section=26" title="Edit section: Source books in Sanskrit">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeller2006" class="citation cs2">Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-7291-0" title="Special:BookSources/978-3-7643-7291-0"><bdi>978-3-7643-7291-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Expounding+the+Mathematical+Seed.+Vol.+1%3A+The+Translation%3A+A+Translation+of+Bhaskara+I+on+the+Mathematical+Chapter+of+the+Aryabhatiya&rft.pub=Basel%2C+Boston%2C+and+Berlin%3A+Birkh%C3%A4user+Verlag%2C+172+pages&rft.date=2006&rft.isbn=978-3-7643-7291-0&rft.aulast=Keller&rft.aufirst=Agathe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKeller2006" class="citation cs2">Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-7292-7" title="Special:BookSources/978-3-7643-7292-7"><bdi>978-3-7643-7292-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Expounding+the+Mathematical+Seed.+Vol.+2%3A+The+Supplements%3A+A+Translation+of+Bhaskara+I+on+the+Mathematical+Chapter+of+the+Aryabhatiya&rft.pub=Basel%2C+Boston%2C+and+Berlin%3A+Birkh%C3%A4user+Verlag%2C+206+pages&rft.date=2006&rft.isbn=978-3-7643-7292-7&rft.aulast=Keller&rft.aufirst=Agathe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSarma1976" class="citation cs2"><a href="/wiki/K._V._Sarma" title="K. V. Sarma">Sarma, K. V.</a>, ed. (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, critically edited with Introduction and Appendices, New Delhi: Indian National Science Academy</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%3Cspan+title%3D%22International+Alphabet+of+Sanskrit+transliteration%22%3E%3Ci+lang%3D%22sa-Latn%22%3E%C4%80ryabha%E1%B9%AD%C4%ABya%3C%2Fi%3E%3C%2Fspan%3E+of+%3Cspan+title%3D%22International+Alphabet+of+Sanskrit+transliteration%22%3E%3Ci+lang%3D%22sa-Latn%22%3E%C4%80ryabha%E1%B9%ADa%3C%2Fi%3E%3C%2Fspan%3E+with+the+commentary+of+S%C5%ABryadeva+Yajvan&rft.pub=critically+edited+with+Introduction+and+Appendices%2C+New+Delhi%3A+Indian+National+Science+Academy&rft.date=1976&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSenBag1983" class="citation cs2">Sen, S. N.; Bag, A. K., eds. (1983), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science Academy</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+%C5%9Aulbas%C5%ABtras+of+Baudh%C4%81yana%2C+%C4%80pastamba%2C+K%C4%81ty%C4%81yana+and+M%C4%81nava&rft.pub=with+Text%2C+English+Translation+and+Commentary%2C+New+Delhi%3A+Indian+National+Science+Academy&rft.date=1983&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShukla1976" class="citation cs2">Shukla, K. S., ed. (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara, critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science Academy</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%3Cspan+title%3D%22International+Alphabet+of+Sanskrit+transliteration%22%3E%3Ci+lang%3D%22sa-Latn%22%3E%C4%80ryabha%E1%B9%AD%C4%ABya%3C%2Fi%3E%3C%2Fspan%3E+of+%3Cspan+title%3D%22International+Alphabet+of+Sanskrit+transliteration%22%3E%3Ci+lang%3D%22sa-Latn%22%3E%C4%80ryabha%E1%B9%ADa%3C%2Fi%3E%3C%2Fspan%3E+with+the+commentary+of+Bh%C4%81skara+I+and+Some%C5%9Bvara&rft.pub=critically+edited+with+Introduction%2C+English+Translation%2C+Notes%2C+Comments+and+Indexes%2C+New+Delhi%3A+Indian+National+Science+Academy&rft.date=1976&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIndian+mathematics" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShukla1988" class="citation cs2">Shukla, K. S., ed. (1988), Āryabhaṭīya of Āryabhaṭa, critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with <a href="/wiki/K.V._Sarma" class="mw-redirect" title="K.V. Sarma">K.V. 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Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.</li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p0038xb0">Indian Mathematics</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)">In Our Time</a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><a rel="nofollow" class="external text" href="http://cs.annauniv.edu/insight/index.htm">InSIGHT 2009</a>, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.</li> <li><a rel="nofollow" class="external text" href="https://drive.google.com/file/d/0BxearzN4q-babEVJZ2g0VnZCQzA/view?usp=sharing">Mathematics in ancient India by R. 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navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Geology_of_India" title="Category:Geology of India">Geology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fossil_Parks_of_India" class="mw-redirect" title="Fossil Parks of India">Fossil Parks</a></li> <li><a href="/wiki/Geology_of_India" title="Geology of India">Geology of India</a></li> <li><a href="/wiki/Indian_Plate" class="mw-redirect" title="Indian Plate">Indian Plate</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Heritage</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/National_Geological_Monuments_of_India" title="National Geological Monuments of India">National Geological Monuments of India</a></li> <li><a href="/wiki/Sacred_groves_of_India" title="Sacred groves of India">Sacred groves of India</a></li> <li><a href="/wiki/Sacred_mountains#India" title="Sacred mountains">Sacred mountains of India</a></li> <li><a href="/wiki/River#Sacred" title="River">Sacred rivers of India</a></li> <li><a href="/wiki/Stones_of_India" title="Stones of India">Stones of India</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Environment_of_India" title="Environment of India">Environment</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biogeographic_classification_of_India" title="Biogeographic classification of India">Biogeographic classification</a></li> <li><a href="/wiki/Biosphere_reserves_of_India" title="Biosphere reserves of India">Biosphere reserves</a></li> <li><a href="/wiki/Climate_of_India" title="Climate of India">Climate</a></li> <li><a href="/wiki/Climate_change_in_India" title="Climate change in India">Climate change</a></li> <li><a href="/wiki/List_of_earthquakes_in_India" title="List of earthquakes in India">Earthquakes</a></li> <li><a href="/wiki/List_of_ecoregions_in_India" title="List of ecoregions in India">Ecoregions</a></li> <li><a href="/wiki/Environmental_issues_in_India" title="Environmental issues in India">Environmental issues</a></li> <li><a href="/wiki/Fauna_of_India" title="Fauna of India">Fauna</a></li> <li><a href="/wiki/List_of_forests_in_India" title="List of forests in India">Forests</a></li> <li><a href="/wiki/Flora_of_India" title="Flora of India">Flora</a></li> <li><a href="/wiki/Geology_of_India" title="Geology of India">Geology</a></li> <li><a href="/wiki/List_of_national_parks_of_India" title="List of national parks of India">National parks</a></li> <li><a href="/wiki/Protected_areas_of_India" title="Protected areas of India">Protected areas</a></li> <li><a href="/wiki/Wildlife_of_India" title="Wildlife of India">Wildlife</a></li> <li><a href="/wiki/List_of_wildlife_sanctuaries_of_India" title="List of wildlife sanctuaries of India">sanctuaries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Landforms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_beaches_in_India" title="List of beaches in India">Beaches</a></li> <li><a href="/wiki/List_of_canals_in_India" title="List of canals in India">Canals</a></li> <li><a href="/wiki/Coastal_India" title="Coastal India">Coasts</a></li> <li><a href="/wiki/Thar_Desert" title="Thar Desert">Desert</a></li> <li><a href="/wiki/Exclusive_economic_zone_of_India" title="Exclusive economic zone of India">ECZ</a></li> <li><a href="/wiki/List_of_extreme_points_of_India" title="List of extreme points of India">Extreme points</a></li> <li><a href="/wiki/List_of_glaciers_of_India" class="mw-redirect" title="List of glaciers of India">Glaciers</a></li> <li><a href="/wiki/List_of_Indian_states_and_union_territories_by_highest_point" title="List of Indian states and union territories by highest point">Highest point by states</a></li> <li><a href="/wiki/List_of_islands_of_India" title="List of islands of India">Islands</a></li> <li><a href="/wiki/List_of_lakes_of_India" title="List of lakes of India">Lakes</a></li> <li><a href="/wiki/List_of_mountains_in_India" title="List of mountains in India">Mountains</a></li> <li><a href="/wiki/List_of_mountain_passes_of_India" title="List of mountain passes of India">Mountain passes</a></li> <li>Plains <ul><li><a href="/wiki/Indo-Gangetic_Plain" title="Indo-Gangetic Plain">Indo-Gangetic</a></li> <li><a href="/wiki/Eastern_Coastal_Plains" title="Eastern Coastal Plains">Eastern coastal</a></li> <li><a href="/wiki/Western_Coastal_Plains" title="Western Coastal Plains">Western coastal</a></li></ul></li> <li><a href="/wiki/List_of_rivers_of_India" title="List of rivers of India">Rivers</a></li> <li><a href="/wiki/List_of_valleys_in_India" title="List of valleys in India">Valleys</a></li> <li><a href="/wiki/List_of_volcanoes_in_India" title="List of volcanoes in India">Volcanoes</a></li> <li><a href="/wiki/List_of_waterfalls_of_India" class="mw-redirect" title="List of waterfalls of India">Waterfalls</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Administrative_divisions_of_India" title="Administrative divisions of India">Regions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_India" title="Central India">Central</a></li> <li><a href="/wiki/East_India" title="East India">East</a></li> <li><a href="/wiki/North_India" title="North India">North</a> <ul><li><a href="/wiki/Northwest_India" title="Northwest India">Northwest</a></li></ul></li> <li><a href="/wiki/Northeast_India" title="Northeast India">Northeast</a></li> <li><a href="/wiki/South_India" title="South India">South</a> <ul><li><a href="/wiki/Southwest_India" class="mw-redirect" title="Southwest India">Southwest</a></li> <li><a href="/wiki/Southeast_India" class="mw-redirect" title="Southeast India">Southeast</a></li></ul></li> <li><a href="/wiki/Western_India" title="Western India">West</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Administrative_divisions_of_India" title="Administrative divisions of India">Subdivisions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autonomous_administrative_divisions_of_India" title="Autonomous administrative divisions of India">Autonomous administrative divisions</a></li> <li><a href="/wiki/Template:Borders_of_India" title="Template:Borders of India">Borders</a></li> <li><a href="/wiki/List_of_towns_in_India_by_population" title="List of towns in India by population">Towns</a></li> <li><a href="/wiki/List_of_cities_in_India_by_population" title="List of cities in India by population">Cities</a></li> <li><a href="/wiki/List_of_districts_in_India" title="List of districts in India">Districts</a></li> <li><a href="/wiki/Municipal_governance_in_India" title="Municipal governance in India">Municipalities</a></li> <li><a href="/wiki/States_and_union_territories_of_India" title="States and union territories of India">States and union territories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monuments_of_National_Importance_of_India" class="mw-redirect" title="Monuments of National Importance of India">National monuments of India</a></li> <li><a href="/wiki/List_of_national_parks_of_India" title="List of national parks of India">National parks of India</a></li> <li><a href="/wiki/Nature_worship" title="Nature worship">Nature worship</a> in <a href="/wiki/Indian_religions" title="Indian religions">Indian-origin religions</a></li> <li><a href="/wiki/World_Heritage_Sites_in_India" class="mw-redirect" title="World Heritage Sites in India">World Heritage Sites in India</a></li> <li><a href="/wiki/Culture_of_India" title="Culture of India">Culture of India</a></li> <li><a href="/wiki/History_of_India" title="History of India">History of India</a></li> <li><a href="/wiki/Tourism_in_India" title="Tourism in India">Tourism in India</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Politics37" style="font-size:114%;margin:0 4em"><a href="/wiki/Politics_of_India" title="Politics of India">Politics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Government_of_India" title="Government of India">Government</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_agencies_of_the_government_of_India" title="List of agencies of the government of India">Agencies</a></li> <li><a href="/wiki/Energy_policy_of_India" title="Energy policy of India">Energy policy</a></li> <li><a href="/wiki/Foreign_relations_of_India" title="Foreign relations of India">Foreign relations</a></li> <li><a href="/wiki/Parliament_of_India" title="Parliament of India">Parliament</a> <ul><li><a href="/wiki/Lok_Sabha" title="Lok Sabha">Lok Sabha</a></li> <li><a href="/wiki/Rajya_Sabha" title="Rajya Sabha">Rajya Sabha</a></li></ul></li> <li><a href="/wiki/President_of_India" title="President of India">President</a></li> <li><a href="/wiki/Vice_President_of_India" title="Vice President of India">Vice President</a></li> <li><a href="/wiki/Prime_Minister_of_India" title="Prime Minister of India">Prime Minister</a></li> <li><a href="/wiki/Union_Council_of_Ministers" title="Union Council of Ministers">Union Council of Ministers</a></li> <li><a href="/wiki/Civil_Services_of_India" title="Civil Services of India">Civil Services</a></li> <li><a href="/wiki/Cabinet_Secretary_(India)" title="Cabinet Secretary (India)">Cabinet Secretary</a></li> <li><a href="/wiki/State_governments_of_India" title="State governments of India">State governments</a> <ul><li><a href="/wiki/State_legislative_assemblies_of_India" title="State legislative assemblies of India">State legislative assemblies</a></li> <li><a href="/wiki/State_legislative_councils_of_India" title="State legislative councils of India">State legislative councils</a></li></ul></li> <li><a href="/wiki/Governor_(India)" title="Governor (India)">Governors, Lieutenant Governors and Administrators</a></li> <li><a href="/wiki/Chief_minister_(India)" title="Chief minister (India)">Chief Ministers</a></li> <li><a href="/wiki/Chief_secretary_(India)" title="Chief secretary (India)">Chief Secretaries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Law_of_India" title="Law of India">Law</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constitution_of_India" title="Constitution of India">Constitution</a></li> <li><a href="/wiki/Indian_Penal_Code" title="Indian Penal Code">Penal Code</a></li> <li><a href="/wiki/Fundamental_Rights,_Directive_Principles_and_Fundamental_Duties_of_India" title="Fundamental Rights, Directive Principles and Fundamental Duties of India">Fundamental rights, principles and duties</a></li> <li><a href="/wiki/Human_rights_in_India" title="Human rights in India">Human rights</a> <ul><li><a href="/wiki/LGBT_rights_in_India" class="mw-redirect" title="LGBT rights in India">LGBT</a></li></ul></li> <li><a href="/wiki/Supreme_Court_of_India" title="Supreme Court of India">Supreme Court</a></li> <li><a href="/wiki/Chief_Justice_of_India" title="Chief Justice of India">Chief Justice</a></li> <li><a href="/wiki/High_courts_of_India" title="High courts of India">High Courts</a></li> <li><a href="/wiki/District_courts_of_India" title="District courts of India">District Courts</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0.5em;padding-right:0.5em;font-weight:normal;"><a href="/wiki/Law_enforcement_in_India" title="Law enforcement in India">Enforcement</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0.5em;padding-right:0.5em;font-weight:normal;">Federal</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Border_Security_Force" title="Border Security Force">Border Security Force (BSF)</a></li> <li><a href="/wiki/Central_Industrial_Security_Force" title="Central Industrial Security Force">Central Industrial Security Force (CISF)</a></li> <li><a href="/wiki/Central_Reserve_Police_Force" title="Central Reserve Police Force">Central Reserve Police Force (CRPF)</a></li> <li><a href="/wiki/Indo-Tibetan_Border_Police" title="Indo-Tibetan Border Police">Indo-Tibetan Border Police (ITBP)</a></li> <li><a href="/wiki/National_Security_Guard" title="National Security Guard">National Security Guard (NSG)</a></li> <li><a href="/wiki/Railway_Protection_Force" title="Railway Protection Force">Railway Protection Force (RPF)</a></li> <li><a href="/wiki/Sashastra_Seema_Bal" title="Sashastra Seema Bal">Sashastra Seema Bal (SSB)</a></li> <li><a href="/wiki/Special_Protection_Group" title="Special Protection Group">Special Protection Group (SPG)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0.5em;padding-right:0.5em;font-weight:normal;"><a href="/wiki/List_of_Indian_intelligence_agencies" title="List of Indian intelligence agencies">Intelligence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bureau_of_Police_Research_and_Development" title="Bureau of Police Research and Development">Bureau of Police Research and Development (BPR&D)</a></li> <li><a href="/wiki/Central_Bureau_of_Investigation" title="Central Bureau of Investigation">Central Bureau of Investigation (CBI)</a></li> <li><a href="/wiki/Directorate_of_Revenue_Intelligence" title="Directorate of Revenue Intelligence">Directorate of Revenue Intelligence (DRI)</a></li> <li><a href="/wiki/Enforcement_Directorate" title="Enforcement Directorate">Enforcement Directorate (ED)</a></li> <li><a href="/wiki/Intelligence_Bureau_(India)" title="Intelligence Bureau (India)">Intelligence Bureau (IB)</a></li> <li><a href="/wiki/National_Security_Council_(India)" title="National Security Council (India)">Joint Intelligence Committee (JIC)</a></li> <li><a href="/wiki/Narcotics_Control_Bureau" title="Narcotics Control Bureau">Narcotics Control Bureau (NCB)</a></li> <li><a href="/wiki/National_Investigation_Agency" title="National Investigation Agency">National Investigation Agency (NIA)</a></li> <li><a href="/wiki/Research_and_Analysis_Wing" title="Research and Analysis Wing">Research and Analysis Wing (R&AW)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Indian_Armed_Forces" title="Indian Armed Forces">Military</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indian_Army" title="Indian Army">Army</a></li> <li><a href="/wiki/Indian_Navy" title="Indian Navy">Navy</a></li> <li><a href="/wiki/Indian_Air_Force" title="Indian Air Force">Air Force</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Politics_of_India" title="Politics of India">Politics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Censorship_in_India" title="Censorship in India">Censorship</a></li> <li><a href="/wiki/Indian_nationality_law" title="Indian nationality law">Citizenship</a></li> <li><a href="/wiki/Elections_in_India" title="Elections in India">Elections</a></li> <li><a href="/wiki/Democracy_in_India" title="Democracy in India">Democracy</a></li> <li><a href="/wiki/Indian_nationalism" title="Indian nationalism">Nationalism</a></li> <li><a href="/wiki/List_of_political_parties_in_India" title="List of political parties in India">Political parties</a></li> <li><a href="/wiki/Reservation_in_India" title="Reservation in India">Reservations</a></li> <li><a href="/wiki/List_of_scandals_in_India" title="List of scandals in India">Scandals</a></li> <li><a href="/wiki/Scheduled_Castes_and_Scheduled_Tribes" title="Scheduled Castes and Scheduled Tribes">Scheduled groups</a></li> <li><a href="/wiki/Secularism_in_India" title="Secularism in India">Secularism</a></li> <li><a href="/wiki/Women%27s_political_participation_in_India" title="Women's political participation in India">Women in politics</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Economy37" style="font-size:114%;margin:0 4em"><a href="/wiki/Economy_of_India" title="Economy of India">Economy</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_companies_of_India" title="List of companies of India">Companies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BSE_SENSEX" title="BSE SENSEX">BSE SENSEX</a></li> <li><a href="/wiki/NIFTY_50" title="NIFTY 50">NIFTY 50</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Governance</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ministry_of_Finance_(India)" title="Ministry of Finance (India)">Ministry of Finance</a> <ul><li><a href="/wiki/Minister_of_Finance_(India)" title="Minister of Finance (India)">Finance ministers</a></li></ul></li> <li><a href="/wiki/Ministry_of_Commerce_and_Industry_(India)" title="Ministry of Commerce and Industry (India)">Ministry of Commerce and Industry</a></li> <li><a href="/wiki/Finance_Commission" title="Finance Commission">Finance Commission</a></li> <li><a href="/wiki/Economic_Advisory_Council" class="mw-redirect" title="Economic Advisory Council">Economic Advisory Council</a></li> <li><a href="/wiki/Central_Statistics_Office_(India)" title="Central Statistics Office (India)">Central Statistical Office</a></li> <li><a href="/wiki/Securities_and_Exchange_Board_of_India" title="Securities and Exchange Board of India">Securities and Exchange Board of India</a></li> <li><a href="/wiki/Enforcement_Directorate" title="Enforcement Directorate">Enforcement Directorate</a></li> <li><a href="/wiki/External_debt_of_India" title="External debt of India">External debt</a></li> <li><a href="/wiki/Foreign_trade_of_India" title="Foreign trade of India">Foreign trade</a></li> <li><a href="/wiki/Foreign_direct_investment_in_India" title="Foreign direct investment in India">Foreign direct investment</a></li> <li><a href="/wiki/Foreign-exchange_reserves_of_India" title="Foreign-exchange reserves of India">Foreign exchange reserves</a></li> <li><a href="/wiki/Remittances_to_India" title="Remittances to India">Remittances</a></li> <li><a href="/wiki/Taxation_in_India" title="Taxation in India">Taxation</a></li> <li><a href="/wiki/Subsidies_in_India" title="Subsidies in India">Subsidies</a></li> <li><a href="/wiki/Industrial_licensing_in_India" title="Industrial licensing in India">Industrial licensing</a></li> <li><a href="/wiki/National_Voluntary_Guidelines_on_Social,_Environmental_and_Economic_Responsibilities_of_Business" title="National Voluntary Guidelines on Social, Environmental and Economic Responsibilities of Business">Voluntary guidelines</a></li> <li><a href="/wiki/NITI_Aayog" title="NITI Aayog">NITI Aayog</a></li> <li><a href="/wiki/Make_in_India" title="Make in India">Make in India</a></li> <li><a href="/wiki/Atmanirbhar_Bharat" title="Atmanirbhar Bharat">Atmanirbhar Bharat</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Currency</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indian_rupee" title="Indian rupee">Indian rupee</a> <ul><li><a href="/wiki/Indian_rupee_sign" title="Indian rupee sign">Sign</a></li> <li><a href="/wiki/History_of_the_rupee" title="History of the rupee">History</a></li> <li><a href="/wiki/Exchange_rate_history_of_the_Indian_rupee" title="Exchange rate history of the Indian rupee">Historical Forex</a></li> <li><a href="/wiki/Digital_rupee" title="Digital rupee">Digital rupee</a></li> <li><a href="/wiki/Coinage_of_India" title="Coinage of India">Coinage</a></li> <li><a href="/wiki/Indian_paisa" title="Indian paisa">Paisa</a></li></ul></li> <li><a href="/wiki/Reserve_Bank_of_India" title="Reserve Bank of India">Reserve Bank of India</a> <ul><li><a href="/wiki/List_of_governors_of_the_Reserve_Bank_of_India" title="List of governors of the Reserve Bank of India">Governor</a></li> <li><a href="/wiki/India_Government_Mint" title="India Government Mint">Mint</a></li></ul></li> <li><a href="/wiki/Inflation_in_India" title="Inflation in India">Inflation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Financial services</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banking_in_India" title="Banking in India">Banking</a></li> <li><a href="/wiki/Insurance_in_India" title="Insurance in India">Insurance</a></li> <li><a href="/wiki/Multi_Commodity_Exchange" title="Multi Commodity Exchange">Multi Commodity Exchange</a></li> <li><a href="/wiki/The_Gold_(Control)_Act,_1968" title="The Gold (Control) Act, 1968">Bullion</a></li> <li><a href="/wiki/Indian_black_money" title="Indian black money">Black money</a> <ul><li><a href="/wiki/Bombay_Stock_Exchange" title="Bombay Stock Exchange">Bombay</a></li> <li><a href="/wiki/National_Stock_Exchange_of_India" title="National Stock Exchange of India">National</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economic_history_of_India" title="Economic history of India">History</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Economic_impact_of_the_COVID-19_pandemic_in_India" title="Economic impact of the COVID-19 pandemic in India">COVID-19 impact</a></li> <li><a href="/wiki/Economic_development_in_India" title="Economic development in India">Economic development</a></li> <li><a href="/wiki/Economic_liberalisation_in_India" title="Economic liberalisation in India">Liberalisation</a></li> <li><a href="/wiki/Licence_Raj" title="Licence Raj">Licence Raj</a></li> <li><a href="/wiki/Green_Revolution_in_India" title="Green Revolution in India">Green revolution</a></li> <li><a href="/wiki/List_of_schemes_of_the_government_of_India" title="List of schemes of the government of India">Government initiatives</a></li> <li><a href="/wiki/Indian_numbering_system" title="Indian numbering system">Numbering system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_Indians_by_net_worth" title="List of Indians by net worth">By net worth</a></li> <li><a href="/wiki/Demographics_of_India" title="Demographics of India">Demography</a></li> <li><a href="/wiki/Income_in_India" title="Income in India">Income</a> <ul><li><a href="/wiki/Poverty_in_India" title="Poverty in India">Poverty</a></li></ul></li> <li><a href="/wiki/Indian_labour_law" title="Indian labour law">Labour law</a></li> <li><a href="/wiki/Pensions_in_India" title="Pensions in India">Pensions</a> <ul><li><a href="/wiki/Employees%27_Provident_Fund_Organisation" title="Employees' Provident Fund Organisation">EPFO</a></li> <li><a href="/wiki/National_Pension_Scheme" class="mw-redirect" title="National Pension Scheme">NPS</a></li> <li><a href="/wiki/Public_Provident_Fund_(India)" title="Public Provident Fund (India)">PPF</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">States</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Economy_of_Andhra_Pradesh" title="Economy of Andhra Pradesh">Andhra Pradesh</a></li> <li><a href="/wiki/Economy_of_Assam" title="Economy of Assam">Assam</a></li> <li><a href="/wiki/Economy_of_Bihar" title="Economy of Bihar">Bihar</a></li> <li><a href="/wiki/Economy_of_Delhi" title="Economy of Delhi">Delhi</a></li> <li><a href="/wiki/Economy_of_Goa" class="mw-redirect" title="Economy of Goa">Goa</a></li> <li><a href="/wiki/Economy_of_Gujarat" title="Economy of Gujarat">Gujarat</a></li> <li><a href="/wiki/Economy_of_Haryana" title="Economy of Haryana">Haryana</a></li> <li><a href="/wiki/Economy_of_Himachal_Pradesh" title="Economy of Himachal Pradesh">Himachal Pradesh</a></li> <li><a href="/wiki/Economy_of_Jammu_and_Kashmir" class="mw-redirect" title="Economy of Jammu and Kashmir">Jammu and Kashmir</a></li> <li><a href="/wiki/Economy_of_Jharkhand" class="mw-redirect" title="Economy of Jharkhand">Jharkhand</a></li> <li><a href="/wiki/Economy_of_Karnataka" title="Economy of Karnataka">Karnataka</a></li> <li><a href="/wiki/Economy_of_Kerala" title="Economy of Kerala">Kerala</a></li> <li><a href="/wiki/Economy_of_Ladakh" class="mw-redirect" title="Economy of Ladakh">Ladakh</a></li> <li><a href="/wiki/Economy_of_Madhya_Pradesh" title="Economy of Madhya Pradesh">Madhya Pradesh</a></li> <li><a href="/wiki/Economy_of_Maharashtra" title="Economy of Maharashtra">Maharashtra</a></li> <li><a href="/wiki/Economy_of_Mizoram" title="Economy of Mizoram">Mizoram</a></li> <li><a href="/wiki/Economy_of_Odisha" title="Economy of Odisha">Odisha</a></li> <li><a href="/wiki/Economy_of_Punjab,_India" title="Economy of Punjab, India">Punjab</a></li> <li><a href="/wiki/Economy_of_Rajasthan" title="Economy of Rajasthan">Rajasthan</a></li> <li><a href="/wiki/Economy_of_Tamil_Nadu" title="Economy of Tamil Nadu">Tamil Nadu</a></li> <li><a href="/wiki/Economy_of_Telangana" title="Economy of Telangana">Telangana</a></li> <li><a href="/wiki/Economy_of_Uttarakhand" title="Economy of Uttarakhand">Uttarakhand</a></li> <li><a href="/wiki/Economy_of_Uttar_Pradesh" title="Economy of Uttar Pradesh">Uttar Pradesh</a></li> <li><a href="/wiki/Economy_of_West_Bengal" title="Economy of West Bengal">West Bengal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Agriculture_in_India" title="Agriculture in India">Agriculture</a> <ul><li><a href="/wiki/Animal_husbandry_in_India" title="Animal husbandry in India">Livestock</a></li> <li><a href="/wiki/Fishing_in_India" title="Fishing in India">Fishing</a></li></ul></li> <li><a href="/wiki/Automotive_industry_in_India" title="Automotive industry in India">Automotive</a></li> <li><a href="/wiki/Chemical_industry_in_India" title="Chemical industry in India">Chemical</a></li> <li><a href="/wiki/Construction_industry_of_India" title="Construction industry of India">Construction</a></li> <li><a href="/wiki/Defence_industry_of_India" title="Defence industry of India">Defence</a></li> <li><a href="/wiki/Education_in_India" title="Education in India">Education</a></li> <li><a href="/wiki/Energy_in_India" title="Energy in India">Energy</a> <ul><li><a href="/wiki/Electricity_sector_in_India" title="Electricity sector in India">Electricity</a> <ul><li><a href="/wiki/Nuclear_power_in_India" title="Nuclear power in India">Nuclear</a></li> <li><a href="/wiki/Oil_and_gas_industry_in_India" title="Oil and gas industry in India">Oil and gas</a></li> <li><a href="/wiki/Solar_power_in_India" title="Solar power in India">Solar</a></li> <li><a href="/wiki/Wind_power_in_India" title="Wind power in India">Wind</a></li></ul></li></ul></li> <li><a href="/wiki/Electronics_and_semiconductor_manufacturing_industry_in_India" title="Electronics and semiconductor manufacturing industry in India">Electronics and semiconductor</a></li> <li><a href="/wiki/Entertainment_industry_in_India" class="mw-redirect" title="Entertainment industry in India">Entertainment</a></li> <li><a href="/wiki/Forestry_in_India" title="Forestry in India">Forestry</a></li> <li><a href="/wiki/Gambling_in_India" title="Gambling in India">Gambling</a></li> <li><a href="/wiki/Healthcare_in_India" title="Healthcare in India">Healthcare</a> <ul><li><a href="/wiki/List_of_hospitals_in_India" title="List of hospitals in India">Hospitals</a></li></ul></li> <li><a href="/wiki/Information_technology_in_India" title="Information technology in India">Information technology</a></li> <li><a href="/wiki/Media_of_India" class="mw-redirect" title="Media of India">Media</a> <ul><li><a href="/wiki/Cinema_of_India" title="Cinema of India">Cinema</a></li> <li><a href="/wiki/FM_broadcasting_in_India" title="FM broadcasting in India">FM Radio</a></li> <li><a href="/wiki/Television_in_India" title="Television in India">Television</a></li> <li><a href="/wiki/Printing_industry_in_India" title="Printing industry in India">Printing</a></li></ul></li> <li><a href="/wiki/Mining_in_India" title="Mining in India">Mining</a> <ul><li><a href="/wiki/Coal_in_India" title="Coal in India">Coal</a></li> <li><a href="/wiki/Iron_and_steel_industry_in_India" title="Iron and steel industry in India">Iron and Steel</a></li></ul></li> <li><a href="/wiki/Pharmaceutical_industry_in_India" title="Pharmaceutical industry in India">Pharmaceuticals</a></li> <li><a href="/wiki/Pulp_and_paper_industry_in_India" title="Pulp and paper industry in India">Pulp and paper</a></li> <li><a href="/wiki/Retailing_in_India" title="Retailing in India">Retail</a></li> <li><a href="/wiki/Science_and_technology_in_India" title="Science and technology in India">Science and technology</a> <ul><li><a href="/wiki/Biotechnology_in_India" title="Biotechnology in India">Biotechnology</a></li></ul></li> <li><a href="/wiki/Space_industry_of_India" title="Space industry of India">Space</a></li> <li><a href="/wiki/Telecommunications_in_India" title="Telecommunications in India">Telecommunications</a></li> <li><a href="/wiki/Textile_industry_in_India" title="Textile industry in India">Textiles</a></li> <li><a href="/wiki/Tourism_in_India" title="Tourism in India">Tourism</a></li> <li><a href="/wiki/Transport_in_India" title="Transport in India">Transport</a> <ul><li><a href="/wiki/Aviation_in_India" title="Aviation in India">Aviation</a> <ul><li><a href="/wiki/Civil_aviation_in_India" title="Civil aviation in India">Civil</a></li></ul></li> <li><a href="/wiki/List_of_ports_in_India" title="List of ports in India">Ports</a></li> <li><a href="/wiki/Rail_transport_in_India" title="Rail transport in India">Rail</a></li> <li><a href="/wiki/Roads_in_India" title="Roads in India">Roads</a></li> <li><a href="/wiki/List_of_electricity_organisations_in_India" title="List of electricity organisations in India">Electricity</a></li> <li><a href="/wiki/Water_supply_and_sanitation_in_India" title="Water supply and sanitation in India">Water</a></li> <li><a href="/wiki/Power_sector_in_India" class="mw-redirect" title="Power sector in India">Power</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Regulator</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insurance_Regulatory_and_Development_Authority" title="Insurance Regulatory and Development Authority">IRDAI</a></li> <li><a href="/wiki/Reserve_Bank_of_India" title="Reserve Bank of India">RBI</a></li> <li><a href="/wiki/SEBI" class="mw-redirect" title="SEBI">SEBI</a></li> <li><a href="/wiki/Insolvency_and_Bankruptcy_Board_of_India" title="Insolvency and Bankruptcy Board of India">IBBI</a></li> <li><a href="/wiki/Pension_Fund_Regulatory_and_Development_Authority" title="Pension Fund Regulatory and Development Authority">PFRDA</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/National_Company_Law_Tribunal" title="National Company Law Tribunal">NCLT</a> <ul><li><a href="/wiki/National_Company_Law_Appellate_Tribunal" title="National Company Law Appellate Tribunal">NCLAT</a></li></ul></li> <li><a href="/wiki/Board_for_Industrial_and_Financial_Reconstruction" title="Board for Industrial and Financial Reconstruction">BIFR</a></li> <li><a href="/wiki/Insolvency_and_Bankruptcy_Board_of_India" title="Insolvency and Bankruptcy Board of India">IBBI</a></li> <li><a href="/wiki/Insolvency_and_Bankruptcy_Code,_2016" title="Insolvency and Bankruptcy Code, 2016">IBC</a></li> <li><a href="/wiki/Securitisation_and_Reconstruction_of_Financial_Assets_and_Enforcement_of_Security_Interest_Act,_2002" title="Securitisation and Reconstruction of Financial Assets and Enforcement of Security Interest Act, 2002">SARFESI Act</a></li> <li><a href="/wiki/Income-tax_Act,_1961" class="mw-redirect" title="Income-tax Act, 1961">Income Tax Act</a></li> <li><a href="/wiki/Companies_Act,_2013" class="mw-redirect" title="Companies Act, 2013">Companies Act</a></li> <li><a href="/wiki/Banking_Regulation_Act,_1949" title="Banking Regulation Act, 1949">Banking Act</a></li> <li><a href="/wiki/Insurance_Act,_1938" title="Insurance Act, 1938">Insurance Act</a></li> <li><a href="/wiki/Foreign_Exchange_Management_Act" title="Foreign Exchange Management Act">FEMA</a></li> <li><a href="/wiki/Mumbai_Consensus" title="Mumbai Consensus">Mumbai Consensus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Society_and_culture37" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li>Society and <a href="/wiki/Culture_of_India" title="Culture of India">culture</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Society</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Caste_system_in_India" title="Caste system in India">Caste system</a></li> <li><a href="/wiki/Corruption_in_India" title="Corruption in India">Corruption</a></li> <li><a href="/wiki/Crime_in_India" title="Crime in India">Crime</a></li> <li><a href="/wiki/Demographics_of_India" title="Demographics of India">Demographics</a> <ul><li><a href="/wiki/Indian_people" title="Indian people">Indians</a></li></ul></li> <li><a href="/wiki/Education_in_India" title="Education in India">Education</a> <ul><li><a href="/wiki/List_of_universities_in_India" title="List of universities in India">Universities in India</a></li> <li><a href="/wiki/List_of_medical_colleges_in_India" title="List of medical colleges in India">Medical colleges in India</a></li> <li><a href="/wiki/List_of_law_schools_in_India" title="List of law schools in India">Law colleges in India</a></li> <li><a href="/wiki/Engineering_education_in_India" title="Engineering education in India">Engineering colleges in India</a></li></ul></li> <li><a href="/wiki/Ethnic_relations_in_India" title="Ethnic relations in India">Ethnic relations</a></li> <li><a href="/wiki/Health_in_India" title="Health in India">Health</a></li> <li><a href="/wiki/Languages_of_India" title="Languages of India">Languages</a></li> <li><a href="/wiki/List_of_Indian_states_by_life_expectancy_at_birth" title="List of Indian states by life expectancy at birth">Life expectancy</a></li> <li><a href="/wiki/Literacy_in_India" title="Literacy in India">Literacy</a></li> <li><a href="/wiki/Poverty_in_India" title="Poverty in India">Poverty</a></li> <li><a href="/wiki/Prisons_in_India" title="Prisons in India">Prisons</a></li> <li><a href="/wiki/Religion_in_India" title="Religion in India">Religion</a></li> <li><a href="/wiki/Social_issue#India" title="Social issue">Socio-economic issues</a></li> <li><a href="/wiki/Standard_of_living_in_India" title="Standard of living in India">Standard of living</a></li> <li><a href="/wiki/Water_supply_and_sanitation_in_India" title="Water supply and sanitation in India">Water supply and sanitation</a></li> <li><a href="/wiki/Women_in_India" title="Women in India">Women</a></li> <li><a href="/wiki/Sexuality_in_India" title="Sexuality in India">Sexuality</a></li> <li><a href="/wiki/Youth_in_India" title="Youth in India">Youth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Culture_of_India" title="Culture of India">Culture</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arts_and_entertainment_in_India" title="Arts and entertainment in India">Arts and entertainment</a></li> <li><a href="/wiki/Architecture_of_India" title="Architecture of India">Architecture</a></li> <li><a href="/wiki/Indian_blogosphere" title="Indian blogosphere">Blogging</a></li> <li><a href="/wiki/Cinema_of_India" title="Cinema of India">Cinema</a></li> <li><a href="/wiki/Indian_comics" title="Indian comics">Comics</a> <ul><li><a href="/wiki/Webcomics_in_India" title="Webcomics in India">Webcomics</a></li></ul></li> <li><a href="/wiki/Indian_cuisine" title="Indian cuisine">Cuisine</a> <ul><li><a href="/wiki/Indian_wine" title="Indian wine">wine</a></li></ul></li> <li><a href="/wiki/Indian_classical_dance" title="Indian classical dance">Dance</a></li> <li><a href="/wiki/Clothing_in_India" title="Clothing in India">Dress</a></li> <li><a href="/wiki/Folklore_of_India" title="Folklore of India">Folklore</a></li> <li><a href="/wiki/List_of_festivals_in_India" title="List of festivals in India">Festivals</a></li> <li><a href="/wiki/Indian_literature" title="Indian literature">Literature</a></li> <li><a href="/wiki/Mass_media_in_India" title="Mass media in India">Media</a> <ul><li><a href="/wiki/Television_in_India" title="Television in India">television</a></li></ul></li> <li><a href="/wiki/Indian_martial_arts" title="Indian martial arts">Martial arts</a></li> <li><a href="/wiki/Music_of_India" title="Music of India">Music</a></li> <li><a href="/wiki/Indian_painting" title="Indian painting">Painting</a></li> <li><a href="/wiki/Indian_physical_culture" title="Indian physical culture">Physical culture</a></li> <li><a href="/wiki/Public_holidays_in_India" title="Public holidays in India">Public holidays</a></li> <li><a href="/wiki/Sculpture_in_the_Indian_subcontinent" title="Sculpture in the Indian subcontinent">Sculpture</a> <ul><li><a href="/wiki/List_of_the_tallest_statues_in_India" title="List of the tallest statues in India">tallest</a></li></ul></li> <li><a href="/wiki/Sport_in_India" title="Sport in India">Sport</a> <ul><li><a href="/wiki/Traditional_games_of_India" title="Traditional games of India">Traditional</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="History_of_mathematics_(timeline)250" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_mathematics" title="Template:History of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_mathematics" title="Template talk:History of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_mathematics" title="Special:EditPage/Template:History of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_mathematics_(timeline)250" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a> (<a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By topic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a> <ul><li><a href="/wiki/Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li> <li>Algorithms <ul><li><a href="/wiki/Timeline_of_algorithms" title="Timeline of algorithms">timeline</a></li></ul></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">Arithmetic</a> <ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li> <li><a href="/wiki/History_of_Grandi%27s_series" title="History of Grandi's series">Grandi's series</a></li></ul></li> <li>Category theory <ul><li><a href="/wiki/Timeline_of_category_theory_and_related_mathematics" title="Timeline of category theory and related mathematics">timeline</a></li> <li><a href="/wiki/History_of_topos_theory" title="History of topos theory">Topos theory</a></li></ul></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">Functions</a> <ul><li><a href="/wiki/History_of_logarithms" title="History of logarithms">Logarithms</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a> <ul><li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li> <li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">Group theory</a></li> <li><a href="/wiki/History_of_information_theory" title="History of information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_logic" title="History of logic">Logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">Math notation</a></li> <li>Number theory <ul><li><a href="/wiki/Timeline_of_number_theory" title="Timeline of number theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a> <ul><li><a href="/wiki/Timeline_of_probability_and_statistics" title="Timeline of probability and statistics">timeline</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li></ul></li> <li>Topology <ul><li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">Manifolds</a> <ul><li><a href="/wiki/Timeline_of_manifolds" title="Timeline of manifolds">timeline</a></li></ul></li> <li><a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">Separation axioms</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeral systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">Prehistoric</a></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">Ancient</a></li> <li><a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">Hindu-Arabic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By ancient cultures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Mesopotamia</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egypt</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greece</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a class="mw-selflink selflink">India</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Medieval Islamic world</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Controversies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brouwer%E2%80%93Hilbert_controversy" title="Brouwer–Hilbert controversy">Brouwer–Hilbert</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Over Cantor's theory</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton</a></li> <li><a href="/wiki/Hobbes%E2%80%93Wallis_controversy" title="Hobbes–Wallis controversy">Hobbes–Wallis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Women in mathematics <ul><li><a href="/wiki/Timeline_of_women_in_mathematics" title="Timeline of women in mathematics">timeline</a></li></ul></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">timeline</a></li></ul></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future of mathematics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:History_of_mathematics" title="Category:History of mathematics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="History_of_science661" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_science" title="Template:History of science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_science" class="mw-redirect" title="Template talk:History of science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_science" title="Special:EditPage/Template:History of science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_science661" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_science" title="History of science">History of science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sociology_of_the_history_of_science" title="Sociology of the history of science">Theories and sociology</a></li> <li><a href="/wiki/Historiography_of_science" title="Historiography of science">Historiography</a></li> <li><a href="/wiki/History_of_pseudoscience" title="History of pseudoscience">Pseudoscience</a></li> <li><a href="/wiki/History_and_philosophy_of_science" title="History and philosophy of science">History and philosophy of science</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/80px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" decoding="async" width="80" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/120px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/160px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 2x" data-file-width="3992" data-file-height="5880" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By era</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_in_the_ancient_world" title="Science in the ancient world">Ancient world</a></li> <li><a href="/wiki/Science_in_classical_antiquity" title="Science in classical antiquity">Classical Antiquity</a></li> <li><a href="/wiki/European_science_in_the_Middle_Ages" title="European science in the Middle Ages">Medieval European</a></li> <li><a href="/wiki/History_of_science_in_the_Renaissance" class="mw-redirect" title="History of science in the Renaissance">Renaissance</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Science_in_the_Age_of_Enlightenment" title="Science in the Age of Enlightenment">Age of Enlightenment</a></li> <li><a href="/wiki/Romanticism_in_science" title="Romanticism in science">Romanticism</a></li> <li><a href="/wiki/19th_century_in_science" title="19th century in science">19th century in science</a></li> <li><a href="/wiki/20th_century_in_science" title="20th century in science">20th century in science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By culture</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_science_and_technology_in_Africa" title="History of science and technology in Africa">African</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Argentina" title="History of science and technology in Argentina">Argentine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Brazil" class="mw-redirect" title="History of science and technology in Brazil">Brazilian</a></li> <li><a href="/wiki/Byzantine_science" title="Byzantine science">Byzantine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_China" title="History of science and technology in China">Chinese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_France" class="mw-redirect" title="History of science and technology in France">French</a></li> <li><a href="/wiki/History_of_science_and_technology_in_the_Indian_subcontinent" class="mw-redirect" title="History of science and technology in the Indian subcontinent">Indian</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Japan" title="History of science and technology in Japan">Japanese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Korea" title="History of science and technology in Korea">Korean</a></li> <li><a href="/wiki/Science_in_the_medieval_Islamic_world" title="Science in the medieval Islamic world">Medieval Islamic</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Mexico" title="History of science and technology in Mexico">Mexican</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Russia" class="mw-redirect" title="History of science and technology in Russia">Russian</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Spain" title="History of science and technology in Spain">Spanish</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_natural_science" class="mw-redirect" title="History of natural science">Natural sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_astronomy" title="History of astronomy">Astronomy</a></li> <li><a href="/wiki/History_of_biology" title="History of biology">Biology</a></li> <li><a href="/wiki/History_of_chemistry" title="History of chemistry">Chemistry</a></li> <li><a href="/wiki/Outline_of_Earth_sciences#History_of_Earth_science" title="Outline of Earth sciences">Earth science</a></li> <li><a href="/wiki/History_of_physics" title="History of physics">Physics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_mathematics" title="History of mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">Logic</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_the_social_sciences" title="History of the social sciences">Social sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_anthropology" title="History of anthropology">Anthropology</a></li> <li><a href="/wiki/History_of_archaeology" title="History of archaeology">Archaeology</a></li> <li><a href="/wiki/History_of_economic_thought" title="History of economic thought">Economics</a></li> <li><a href="/wiki/History" title="History">History</a></li> <li><a href="/wiki/History_of_political_science" title="History of political science">Political science</a></li> <li><a href="/wiki/History_of_psychology" title="History of psychology">Psychology</a></li> <li><a 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class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-65585cc8dc-cq4nw","wgBackendResponseTime":298,"wgPageParseReport":{"limitreport":{"cputime":"1.433","walltime":"1.719","ppvisitednodes":{"value":12076,"limit":1000000},"postexpandincludesize":{"value":423768,"limit":2097152},"templateargumentsize":{"value":13615,"limit":2097152},"expansiondepth":{"value":18,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":352123,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1262.623 1 -total"," 32.24% 407.112 1 Template:Reflist"," 29.16% 368.187 73 Template:Citation"," 10.20% 128.730 39 Template:IAST"," 9.89% 124.919 39 Template:Transliteration"," 8.58% 108.331 61 Template:Harv"," 6.65% 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