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Available in 62 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-62" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">62 kalbų</span> </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/%D9%88%D8%AD%D8%AF%D8%A9_%D8%AA%D8%AE%D9%8A%D9%84%D9%8A%D8%A9" title="وحدة تخيلية – arabų" lang="ar" hreflang="ar" data-title="وحدة تخيلية" data-language-autonym="العربية" data-language-local-name="arabų" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0_%C9%99d%C9%99di" title="İ ədədi – azerbaidžaniečių" lang="az" hreflang="az" data-title="İ ədədi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidžaniečių" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%8F%D1%9E%D0%BD%D0%B0%D1%8F_%D0%B0%D0%B4%D0%B7%D1%96%D0%BD%D0%BA%D0%B0" title="Уяўная адзінка – baltarusių" lang="be" hreflang="be" data-title="Уяўная адзінка" data-language-autonym="Беларуская" data-language-local-name="baltarusių" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A3%D1%8F%D1%9E%D0%BD%D0%B0%D1%8F_%D0%B0%D0%B4%D0%B7%D1%96%D0%BD%D0%BA%D0%B0" title="Уяўная адзінка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Уяўная адзінка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BC%D0%B0%D0%B3%D0%B8%D0%BD%D0%B5%D1%80%D0%BD%D0%B0_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B0" title="Имагинерна единица – bulgarų" lang="bg" hreflang="bg" data-title="Имагинерна единица" data-language-autonym="Български" data-language-local-name="bulgarų" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Imaginarna_jedinica" title="Imaginarna jedinica – bosnių" lang="bs" hreflang="bs" data-title="Imaginarna jedinica" data-language-autonym="Bosanski" data-language-local-name="bosnių" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Unitat_imagin%C3%A0ria" title="Unitat imaginària – katalonų" lang="ca" hreflang="ca" data-title="Unitat imaginària" data-language-autonym="Català" data-language-local-name="katalonų" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DB%8C%DB%95%DA%A9%DB%95%DB%8C_%D8%AE%DB%95%DB%8C%D8%A7%DA%B5%DB%8C" title="یەکەی خەیاڵی – soranių kurdų" lang="ckb" hreflang="ckb" data-title="یەکەی خەیاڵی" data-language-autonym="کوردی" data-language-local-name="soranių kurdų" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Imagin%C3%A1rn%C3%AD_jednotka" title="Imaginární jednotka – čekų" lang="cs" hreflang="cs" data-title="Imaginární jednotka" data-language-autonym="Čeština" data-language-local-name="čekų" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AB%D1%82%D0%B0%D1%80%D0%BB%C4%83_%D0%BF%C4%95%D1%80%D1%87%C4%95" title="Ытарлă пĕрчĕ – čiuvašų" lang="cv" hreflang="cv" data-title="Ытарлă пĕрчĕ" data-language-autonym="Чӑвашла" data-language-local-name="čiuvašų" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Imagin%C3%A6re_enhed" title="Imaginære enhed – danų" lang="da" hreflang="da" data-title="Imaginære enhed" data-language-autonym="Dansk" data-language-local-name="danų" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Imagin%C3%A4re_Einheit" title="Imaginäre Einheit – vokiečių" lang="de" hreflang="de" data-title="Imaginäre Einheit" data-language-autonym="Deutsch" data-language-local-name="vokiečių" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit – anglų" lang="en" hreflang="en" data-title="Imaginary unit" data-language-autonym="English" data-language-local-name="anglų" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Imaginara_unuo" title="Imaginara unuo – esperanto" lang="eo" hreflang="eo" data-title="Imaginara unuo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Unidad_imaginaria" title="Unidad imaginaria – ispanų" lang="es" hreflang="es" data-title="Unidad imaginaria" data-language-autonym="Español" data-language-local-name="ispanų" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Imaginaar%C3%BChik" title="Imaginaarühik – estų" lang="et" hreflang="et" data-title="Imaginaarühik" data-language-autonym="Eesti" data-language-local-name="estų" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Unitate_irudikari" title="Unitate irudikari – baskų" lang="eu" hreflang="eu" data-title="Unitate irudikari" data-language-autonym="Euskara" data-language-local-name="baskų" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%8C%DA%A9%D9%87_%D9%85%D9%88%D9%87%D9%88%D9%85%DB%8C" title="یکه موهومی – persų" lang="fa" hreflang="fa" data-title="یکه موهومی" data-language-autonym="فارسی" data-language-local-name="persų" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Imaginaariyksikk%C3%B6" title="Imaginaariyksikkö – suomių" lang="fi" hreflang="fi" data-title="Imaginaariyksikkö" data-language-autonym="Suomi" data-language-local-name="suomių" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Unit%C3%A9_imaginaire" title="Unité imaginaire – prancūzų" lang="fr" hreflang="fr" data-title="Unité imaginaire" data-language-autonym="Français" data-language-local-name="prancūzų" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Unidade_imaxinaria" title="Unidade imaxinaria – galisų" lang="gl" hreflang="gl" data-title="Unidade imaxinaria" data-language-autonym="Galego" data-language-local-name="galisų" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/I_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="I (מספר) – hebrajų" lang="he" hreflang="he" data-title="I (מספר)" data-language-autonym="עברית" data-language-local-name="hebrajų" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A7%E0%A4%BF%E0%A4%95%E0%A4%B2%E0%A5%8D%E0%A4%AA%E0%A4%BF%E0%A4%A4_%E0%A4%8F%E0%A4%95%E0%A4%95" title="अधिकल्पित एकक – hindi" lang="hi" hreflang="hi" data-title="अधिकल्पित एकक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Imaginarna_jedinica" title="Imaginarna jedinica – kroatų" lang="hr" hreflang="hr" data-title="Imaginarna jedinica" data-language-autonym="Hrvatski" data-language-local-name="kroatų" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Imagin%C3%A1rius_egys%C3%A9g" title="Imaginárius egység – vengrų" lang="hu" hreflang="hu" data-title="Imaginárius egység" data-language-autonym="Magyar" data-language-local-name="vengrų" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A5%D5%B2%D5%AE_%D5%B4%D5%AB%D5%A1%D5%BE%D5%B8%D6%80" title="Կեղծ միավոր – armėnų" lang="hy" hreflang="hy" data-title="Կեղծ միավոր" data-language-autonym="Հայերեն" data-language-local-name="armėnų" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Unitate_imaginari" title="Unitate imaginari – tarpinė" lang="ia" hreflang="ia" data-title="Unitate imaginari" data-language-autonym="Interlingua" data-language-local-name="tarpinė" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Unit_imajiner" title="Unit imajiner – indoneziečių" lang="id" hreflang="id" data-title="Unit imajiner" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziečių" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Imaginara_unajo" title="Imaginara unajo – ido" lang="io" hreflang="io" data-title="Imaginara unajo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Unit%C3%A0_immaginaria" title="Unità immaginaria – italų" lang="it" hreflang="it" data-title="Unità immaginaria" data-language-autonym="Italiano" data-language-local-name="italų" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%99%9A%E6%95%B0%E5%8D%98%E4%BD%8D" title="虚数単位 – japonų" lang="ja" hreflang="ja" data-title="虚数単位" data-language-autonym="日本語" data-language-local-name="japonų" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/ka%27o" title="ka'o – loiban" lang="jbo" hreflang="jbo" data-title="ka'o" data-language-autonym="La .lojban." data-language-local-name="loiban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%97%88%EC%88%98_%EB%8B%A8%EC%9C%84" title="허수 단위 – korėjiečių" lang="ko" hreflang="ko" data-title="허수 단위" data-language-autonym="한국어" data-language-local-name="korėjiečių" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/I_(getal)" title="I (getal) – limburgiečių" lang="li" hreflang="li" data-title="I (getal)" data-language-autonym="Limburgs" data-language-local-name="limburgiečių" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Unit%C3%A0_immaginaria" title="Unità immaginaria – lombardų" lang="lmo" hreflang="lmo" data-title="Unità immaginaria" data-language-autonym="Lombard" data-language-local-name="lombardų" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Imagin%C4%81r%C4%81_vien%C4%ABba" title="Imaginārā vienība – latvių" lang="lv" hreflang="lv" data-title="Imaginārā vienība" data-language-autonym="Latviešu" data-language-local-name="latvių" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BC%D0%B0%D0%B3%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B0" title="Имагинарна единица – makedonų" lang="mk" hreflang="mk" data-title="Имагинарна единица" data-language-autonym="Македонски" data-language-local-name="makedonų" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Imaginaire_eenheid" title="Imaginaire eenheid – olandų" lang="nl" hreflang="nl" data-title="Imaginaire eenheid" data-language-autonym="Nederlands" data-language-local-name="olandų" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Imagin%C3%A6r_enhet" title="Imaginær enhet – norvegų bukmolas" lang="nb" hreflang="nb" data-title="Imaginær enhet" data-language-autonym="Norsk bokmål" data-language-local-name="norvegų bukmolas" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Jednostka_urojona" title="Jednostka urojona – lenkų" lang="pl" hreflang="pl" data-title="Jednostka urojona" data-language-autonym="Polski" data-language-local-name="lenkų" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Unidade_imagin%C3%A1ria" title="Unidade imaginária – portugalų" lang="pt" hreflang="pt" data-title="Unidade imaginária" data-language-autonym="Português" data-language-local-name="portugalų" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Unitatea_imaginar%C4%83" title="Unitatea imaginară – rumunų" lang="ro" hreflang="ro" data-title="Unitatea imaginară" data-language-autonym="Română" data-language-local-name="rumunų" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B8%D0%BC%D0%B0%D1%8F_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B0" title="Мнимая единица – rusų" lang="ru" hreflang="ru" data-title="Мнимая единица" data-language-autonym="Русский" data-language-local-name="rusų" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Imaginarna_jedinica" title="Imaginarna jedinica – serbų-kroatų" lang="sh" hreflang="sh" data-title="Imaginarna jedinica" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbų-kroatų" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%AD%E0%B7%8F%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B6%92%E0%B6%9A%E0%B6%9A%E0%B6%BA" title="අතාත්වික ඒකකය – sinhalų" lang="si" hreflang="si" data-title="අතාත්වික ඒකකය" data-language-autonym="සිංහල" data-language-local-name="sinhalų" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit – Simple English" lang="en-simple" hreflang="en-simple" data-title="Imaginary unit" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Imaginarna_enota" title="Imaginarna enota – slovėnų" lang="sl" hreflang="sl" data-title="Imaginarna enota" data-language-autonym="Slovenščina" data-language-local-name="slovėnų" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Nj%C3%ABsia_imagjinare" title="Njësia imagjinare – albanų" lang="sq" hreflang="sq" data-title="Njësia imagjinare" data-language-autonym="Shqip" data-language-local-name="albanų" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BC%D0%B0%D0%B3%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D1%98%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B0" title="Имагинарна јединица – serbų" lang="sr" hreflang="sr" data-title="Имагинарна јединица" data-language-autonym="Српски / srpski" data-language-local-name="serbų" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Imagin%C3%A4ra_enheten" title="Imaginära enheten – švedų" lang="sv" hreflang="sv" data-title="Imaginära enheten" data-language-autonym="Svenska" data-language-local-name="švedų" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%A9%E0%AF%88_%E0%AE%85%E0%AE%B2%E0%AE%95%E0%AF%81" title="கற்பனை அலகு – tamilų" lang="ta" hreflang="ta" data-title="கற்பனை அலகு" data-language-autonym="தமிழ்" data-language-local-name="tamilų" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%92%D0%BE%D2%B3%D0%B8%D0%B4%D0%B8_%D0%BC%D0%B0%D0%B2%D2%B3%D1%83%D0%BC" title="Воҳиди мавҳум – tadžikų" lang="tg" hreflang="tg" data-title="Воҳиди мавҳум" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžikų" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AB%E0%B8%99%E0%B9%88%E0%B8%A7%E0%B8%A2%E0%B8%88%E0%B8%B4%E0%B8%99%E0%B8%95%E0%B8%A0%E0%B8%B2%E0%B8%9E" title="หน่วยจินตภาพ – tajų" lang="th" hreflang="th" data-title="หน่วยจินตภาพ" data-language-autonym="ไทย" data-language-local-name="tajų" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0_say%C4%B1s%C4%B1" title="İ sayısı – turkų" lang="tr" hreflang="tr" data-title="İ sayısı" data-language-autonym="Türkçe" data-language-local-name="turkų" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A3%D0%B9%D0%BB%D0%B0%D0%BD%D0%BC%D0%B0_%D0%B1%D0%B5%D1%80%D0%BB%D0%B5" title="Уйланма берле – totorių" lang="tt" hreflang="tt" data-title="Уйланма берле" data-language-autonym="Татарча / tatarça" data-language-local-name="totorių" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A3%D1%8F%D0%B2%D0%BD%D0%B0_%D0%BE%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8F" title="Уявна одиниця – ukrainiečių" lang="uk" hreflang="uk" data-title="Уявна одиниця" data-language-autonym="Українська" data-language-local-name="ukrainiečių" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%AE%DB%8C%D9%84%DB%8C_%D8%A7%DA%A9%D8%A7%D8%A6%DB%8C" title="تخیلی اکائی – urdų" lang="ur" hreflang="ur" data-title="تخیلی اکائی" data-language-autonym="اردو" data-language-local-name="urdų" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%A1n_v%E1%BB%8B_%E1%BA%A3o" title="Đơn vị ảo – vietnamiečių" lang="vi" hreflang="vi" data-title="Đơn vị ảo" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamiečių" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%99%9A%E6%95%B0%E5%8D%95%E4%BD%8D" title="虚数单位 – kinų kalbos vu tarmė" lang="wuu" hreflang="wuu" data-title="虚数单位" data-language-autonym="吴语" data-language-local-name="kinų kalbos vu tarmė" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%E1%BA%B8y%E1%BB%8D_t%C3%ADk%C3%B2s%C3%AD" title="Ẹyọ tíkòsí – jorubų" lang="yo" hreflang="yo" data-title="Ẹyọ tíkòsí" data-language-autonym="Yorùbá" data-language-local-name="jorubų" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%99%9B%E6%95%B8%E5%96%AE%E4%BD%8D" title="虛數單位 – kinų" lang="zh" hreflang="zh" data-title="虛數單位" data-language-autonym="中文" data-language-local-name="kinų" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%99%9B%E6%95%B8%E5%96%AE%E4%BD%8D" title="虛數單位 – kinų kalbos Kantono tarmė" lang="yue" hreflang="yue" data-title="虛數單位" data-language-autonym="粵語" data-language-local-name="kinų kalbos Kantono tarmė" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q193796#sitelinks-wikipedia" title="Keisti tarpkalbines nuorodas" class="wbc-editpage">Keisti nuorodas</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Vardų sritys"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Menamasis_vienetas" title="Rodyti puslapio turinį [c]" accesskey="c"><span>Straipsnis</span></a></li><li id="ca-talk" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/220px-ImaginaryUnit5.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/330px-ImaginaryUnit5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/440px-ImaginaryUnit5.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>Skaičius <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span></b> kompleksinėje arba <a href="/wiki/Dekarto_koordina%C4%8Di%C5%B3_sistema" title="Dekarto koordinačių sistema">Dekarto plokštumoje</a>. Realieji skaičiai yra horizontalioje ašyje, o menamieji – vertikalioje.</figcaption></figure> <p><b>Menamasis vienetas</b> (arba <b>tariamasis vienetas</b>) – skaičius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> (kartais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>), praplečiantis <a href="/wiki/Realusis_skai%C4%8Dius" title="Realusis skaičius">realiųjų skaičių</a> aibę <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> iki <a href="/wiki/Kompleksinis_skai%C4%8Dius" title="Kompleksinis skaičius">kompleksinių skaičių</a> aibės <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Tariamasis vienetas įvestas daugiausia todėl, jog ne kiekviena <a href="/wiki/Polinomas" title="Polinomas">polinominė lygybė</a> <i>f</i>(<i>x</i>) = 0 turi sprendimą realiųjų skaičių aibėje. Pavyzdžiui, lygybė <i>x</i>² + 1 = 0 neturi realaus sprendinio. Praplėtus realiųjų skaičių aibę menamuoju (tariamuoju vienetu), kiekviena tokia lygybė turi sprendinį naujoje kompleksinių skaičių aibėje. Žinomi sprendiniai neretai padeda lengviau dirbti su tokiomis lygtimis. </p><p>Dažnai sakoma, kad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}}"></span> tačiau tai ne visai tikslu nes yra dvi reikšmės kurias pakėlę kvadratu gauname -1: <i>i</i> ir <i>-i</i>. </p><p>Skaičių <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}"></span> vadinti menamuoju vienetu XVII a. pasiūlė prancūzų matematikas, fizikas ir filosofas <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Renė Dekartas</a>, o XVIII a. <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonardas Oileris</a> įvedė šio skaičiaus įprastą žymėjimą <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, siejamą su prancūzų kalbos žodžui <i>imaginaire</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Apibrėžimas"><span id="Apibr.C4.97.C5.BEimas"></span>Apibrėžimas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Menamasis_vienetas&veaction=edit&section=1" title="Redaguoti skyrelį: Apibrėžimas" class="mw-editsection-visualeditor"><span>redaguoti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Menamasis_vienetas&action=edit&section=1" title="Edit section's source code: Apibrėžimas"><span>redaguoti vikitekstą</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pagal apibrėžimą menamasis vienetas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> yra vienas iš <a href="/wiki/Kvadratin%C4%97_lygtis" title="Kvadratinė lygtis">kvadratinės lygties</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30946aecca20c29fbabc4efb123abe335529f477" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.228ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0\ }"></span></dd></dl> <p>arba </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=-1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=-1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30893d9dacf84f83aa65155717cb9cf360046adf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.034ex; height:2.843ex;" alt="{\displaystyle x^{2}=-1\ }"></span></dd></dl> <p>sprendinių. </p><p>Kadangi nėra realiojo skaičiaus, kuris pakeltas kvadratu duotų neigiamą realų skaičių, mes galime įsivaizduoti jį (tarti jį egzistuojant – iš čia ir pavadinimas menamasis arba tariamasis vienetas) egzistuojant ir priskirti jam simbolį <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>. Tačiau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> yra tokia pat lygiavertė matematinė abstrakcija, kaip ir realusis skaičius, nors, aišku, jį sunkiau intuityviai suvokti. </p> <div class="mw-heading mw-heading2"><h2 id="i_ir_−i"><span id="i_ir_.E2.88.92i"></span><i>i</i> ir −<i>i</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Menamasis_vienetas&veaction=edit&section=2" title="Redaguoti skyrelį: i ir −i" class="mw-editsection-visualeditor"><span>redaguoti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Menamasis_vienetas&action=edit&section=2" title="Edit section's source code: i ir −i"><span>redaguoti vikitekstą</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kadangi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d687ce8b19e0ef0d8f872a4e0bfdcf3a6665a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.453ex; height:2.843ex;" alt="{\displaystyle x^{2}=-1}"></span> tai antros eilės polinomas, lygtis turi du skirtingus sprendinius: vienas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, kitas −<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> ≠ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>. Kadangi kvadratinė lygtis yra vienintelis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> apibrėžimas, atrodo, kad jis nevienareikšmis. Tačiau jokių dviprasmybių nelieka, jei pasirenkamas vienas iš sprendinių ir deklaruojamas kaip „teigiamas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>“. Tai yra dėl to kad nors −<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> ir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> nėra kiekybiškai vienodi (vienas neigiamas, kitas teigiamas), tačiau kokybiškai jie nesiskiria (tačiau to negalima pasakyti apie −1 ir +1): abu menamieji skaičiai turi vienodas teises būti −1 kvadratu. Jei visose matematinėse knygose apie kompleksinius skaičius pakeisti +<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> į −<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, visi faktai ir teoremos išliks teisingomis. Taigi, nė viena vertė nėra svarbesnė už kitą, o pažymėjimas vieną „teigiama“ tėra tik užrašymo rudimentas. </p><p>Ši problema matematine prasme gana subtili. Nors kompleksinių skaičių laukas apibrėžtas kaip <b>R</b>[<i>X</i>]/ (<i>X</i>² + 1), iš tiesų yra du <b>R</b>[<i>X</i>]/ (<i>X</i>² + 1) <a href="/wiki/Automorfizmas" class="mw-redirect" title="Automorfizmas">automorfizmai</a>, pats <i>X</i> ir automorfizmas, atvaizduojantis <i>X</i> į −<i>X</i>. </p><p>Panaši problema atsiranda, kai kompleksiniai skaičiai yra vaizduojami kaip 2 × 2 realiosios <a href="/wiki/Matrica_(matematika)" title="Matrica (matematika)">matricos</a>, kadangi tiek </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5b479b5edc6bd9c5094c8d0ffd578e905a4fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.707ex; height:6.176ex;" alt="{\displaystyle X={\begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix}}}"></span></dd></dl> <p>tiek ir </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\begin{pmatrix}0&1\\-1&\;\;0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\begin{pmatrix}0&1\\-1&\;\;0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a5bb563ffe691ad0391a70b3a766836e7da86f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.997ex; height:6.176ex;" alt="{\displaystyle X={\begin{pmatrix}0&1\\-1&\;\;0\end{pmatrix}}}"></span></dd></dl> <p>yra matricinės lygties sprendiniai: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}=-I\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>I</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}=-I\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8eded1e0d559fa85a862ed44ba1a0285afe1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.71ex; height:2.843ex;" alt="{\displaystyle X^{2}=-I\ }"></span> .</dd></dl> <p>Šiuo atveju nevienareikšmiškumas atsiranda dėl geometrinio teigiamos krypties apskritime pasirinkimo. Matematiškai išsireiškus, <a href="/wiki/Grup%C4%97_(algebra)" title="Grupė (algebra)">grupės</a> SO (2, <b>R</b>) automorfizmas turi du elementus – vienetą ir automorfizmą, kuris sukeičia vietomis pasukimus kryptimis pagal laikrodžio rodyklę ir prieš laikrodžio rodyklę. </p><p>Visų šių nevienareikšmiškumų yra išvengiama, jei įvedama griežtesnis <a href="/wiki/Kompleksinis_skai%C4%8Dius#Kompleksinių_skaičių_laukas" title="Kompleksinis skaičius">kompleksinio skaičiaus apibrėžimas</a>, kuomet išreikštai pasirenkamas vienas iš sprendinių kaip menamasis vienetas. </p><p>Menamasis vienetas kartais yra užrašomas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}"></span>, tačiau kompleksinių skaičių aibėje <a href="/wiki/%C5%A0aknis_(matematika)" class="mw-redirect" title="Šaknis (matematika)">šaknies</a> operaciją reikia naudoti labai atidžiai. Pagrindinė perspėjimo priežastis yra ta, kad šaknies operacija nėra vienareikšmis atvaizdavimas: pagal Muavro formulę yra lygiai <i>n</i> skirtingų <i>n</i>-tojo laipsnio šaknų iš kompleksinio skaičiaus. Visos šios šaknys sudaro aibę iš <i>n</i> elementų. Taigi ir <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}"></span> formaliai yra aibė, sudaryta iš dviejų elementų (<i>i</i> ir -<i>i</i>), o užrašas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}}"></span> reikštų, kad skaičius yra lygus aibei. </p> <div class="mw-heading mw-heading2"><h2 id="Šaltiniai"><span id=".C5.A0altiniai"></span>Šaltiniai</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Menamasis_vienetas&veaction=edit&section=3" title="Redaguoti skyrelį: Šaltiniai" class="mw-editsection-visualeditor"><span>redaguoti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Menamasis_vienetas&action=edit&section=3" title="Edit section's source code: Šaltiniai"><span>redaguoti vikitekstą</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Eric W. Weisstein: Imaginary Number. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ImaginaryNumber.html">MathWorld</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Algirdas Matulis. Kompleksiniai skaičiai ir funkcijos. – Vilnius: Ciklonas, 2003. – 2 p. <a href="/wiki/Specialus:Knyg%C5%B3_%C5%A1altiniai/9955497289" class="internal mw-magiclink-isbn">ISBN 9955-497-28-9</a></span> </li> </ol> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Rodomas puslapis "<a dir="ltr" href="https://lt.wikipedia.org/w/index.php?title=Menamasis_vienetas&oldid=7096174">https://lt.wikipedia.org/w/index.php?title=Menamasis_vienetas&oldid=7096174</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Specialus:Kategorijos" title="Specialus:Kategorijos">Kategorijos</a>: <ul><li><a href="/wiki/Kategorija:Skai%C4%8Di%C5%B3_teorija" title="Kategorija:Skaičių teorija">Skaičių teorija</a></li><li><a href="/wiki/Kategorija:Kompleksin%C4%97_analiz%C4%97" title="Kategorija:Kompleksinė analizė">Kompleksinė analizė</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Paslėpta kategorija: <ul><li><a href="/wiki/Kategorija:Puslapiai,_naudojantys_ISBN_magi%C5%A1kas_nuorodas" title="Kategorija:Puslapiai, naudojantys ISBN magiškas nuorodas">Puslapiai, naudojantys ISBN magiškas nuorodas</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Šis puslapis paskutinį kartą keistas 6 lapkričio 2023 11:56.</li> <li id="footer-info-copyright">Tekstas pateikiamas pagal <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike Licenciją</a>; gali būti taikomos papildomos sąlygos. 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