Основні числові системи/Раціональні числа

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[o]" accesskey="o" class=""><span>Увійти</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Більше опцій" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Особисті інструменти" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Особисті інструменти</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Меню користувача" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_uk.wikibooks.org&uselang=uk"><span>Пожертвувати</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D1%82%D0%B2%D0%BE%D1%80%D0%B8%D1%82%D0%B8_%D0%BE%D0%B1%D0%BB%D1%96%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Пропонуємо створити обліковий запис і увійти в систему; однак, це не обов'язково"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Створити обліковий запис</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D1%85%D1%96%D0%B4&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Заохочуємо Вас увійти в систему, але це необов'язково. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Увійти</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Сторінки для редакторів, які не ввійшли в систему <a href="/wiki/%D0%94%D0%BE%D0%B2%D1%96%D0%B4%D0%BA%D0%B0:%D0%92%D1%81%D1%82%D1%83%D0%BF" aria-label="Дізнатися більше про редагування"><span>дізнатися більше</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D1%96%D0%B9_%D0%B2%D0%BD%D0%B5%D1%81%D0%BE%D0%BA" title="Список редагувань, зроблених з цієї IP-адреси [y]" accesskey="y"><span>Внесок</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D0%BE%D1%94_%D0%BE%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Обговорення редагувань з цієї IP-адреси [n]" accesskey="n"><span>Обговорення</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Зміст" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Зміст</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">сховати</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Вступ</div> </a> </li> <li id="toc-Дроби_та_їх_властивості" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Дроби_та_їх_властивості"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Дроби та їх властивості</span> </div> </a> <ul id="toc-Дроби_та_їх_властивості-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Масштаб" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Масштаб"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Масштаб</span> </div> </a> <ul id="toc-Масштаб-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Дії_над_дробами" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Дії_над_дробами"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Дії над дробами</span> </div> </a> <button aria-controls="toc-Дії_над_дробами-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Перемкнути підрозділ Дії над дробами</span> </button> <ul id="toc-Дії_над_дробами-sublist" class="vector-toc-list"> <li id="toc-Додавання_дробів_та_віднімання_дробів" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Додавання_дробів_та_віднімання_дробів"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Додавання дробів та віднімання дробів</span> </div> </a> <ul id="toc-Додавання_дробів_та_віднімання_дробів-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Множення_дробів" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Множення_дробів"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Множення дробів</span> </div> </a> <ul id="toc-Множення_дробів-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ділення_дробів" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ділення_дробів"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Ділення дробів</span> </div> </a> <ul id="toc-Ділення_дробів-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Раціональні_вирази" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Раціональні_вирази"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Раціональні вирази</span> </div> </a> <button aria-controls="toc-Раціональні_вирази-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Перемкнути підрозділ Раціональні вирази</span> </button> <ul id="toc-Раціональні_вирази-sublist" class="vector-toc-list"> <li id="toc-Дії_над_раціональними_виразами" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Дії_над_раціональними_виразами"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Дії над раціональними виразами</span> </div> </a> <ul id="toc-Дії_над_раціональними_виразами-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Цілі_раціональні_рівняння" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Цілі_раціональні_рівняння"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Цілі раціональні рівняння</span> </div> </a> <ul id="toc-Цілі_раціональні_рівняння-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="Зміст" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Сховати/показати зміст" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Сховати/показати зміст</span> </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"><span class="mw-page-title-main">Основні числові системи/Раціональні числа</span></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="Ця стаття існує лише цією мовою. Додайте статтю для інших мов" > <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-0" 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">Додати мови</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span><span class="wb-langlinks-add wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1244890#sitelinks-wikibooks" title="Додати міжмовні посилання" class="wbc-editpage">Додати посилання</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="Простори назв"> <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/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Вміст статті [c]" accesskey="c"><span>Сторінка</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F:%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&redlink=1" rel="discussion" class="new" title="Обговорення сторінки (такої сторінки не існує) [t]" accesskey="t"><span>Обговорення</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Змінити варіант мови" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">українська</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Перегляди"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Читати</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e"><span>Редагувати</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history" title="Журнал змін сторінки [h]" accesskey="h"><span>Переглянути історію</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Інструменти сторінки"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Інструменти" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Інструменти</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Інструменти</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">сховати</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Більше опцій" > <div class="vector-menu-heading"> Дії </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Читати</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e"><span>Редагувати</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history"><span>Переглянути історію</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Загальний </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%81%D0%B8%D0%BB%D0%B0%D0%BD%D0%BD%D1%8F_%D1%81%D1%8E%D0%B4%D0%B8/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Перелік усіх сторінок, які посилаються на цю сторінку [j]" accesskey="j"><span>Посилання сюди</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%27%D1%8F%D0%B7%D0%B0%D0%BD%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" rel="nofollow" title="Останні зміни на сторінках, на які посилається ця сторінка [k]" accesskey="k"><span>Пов'язані редагування</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=uk" title="Завантажити файли [u]" accesskey="u"><span>Завантажити файл</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8" title="Перелік спеціальних сторінок [q]" accesskey="q"><span>Спеціальні сторінки</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&oldid=37417" title="Постійне посилання на цю версію цієї сторінки"><span>Постійне посилання</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=info" title="Додаткові відомості про цю сторінку"><span>Інформація про сторінку</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A6%D0%B8%D1%82%D0%B0%D1%82%D0%B0&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&id=37417&wpFormIdentifier=titleform" title="Інформація про те, як цитувати цю сторінку"><span>Цитувати сторінку</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:UrlShortener&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%25A0%25D0%25B0%25D1%2586%25D1%2596%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0"><span>Отримати вкорочену URL-адресу</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%25A0%25D0%25B0%25D1%2586%25D1%2596%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0"><span>Завантажити QR-код</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Друк/експорт </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Створити книгу</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:DownloadAsPdf&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=show-download-screen"><span>Завантажити як PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&printable=yes" title="Версія цієї сторінки для друку [p]" accesskey="p"><span>Версія до друку</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> В інших проєктах </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Rational_numbers" hreflang="en"><span>Вікісховище</span></a></li><li class="wb-otherproject-link wb-otherproject-wikipedia mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" hreflang="uk"><span>Вікіпедія</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1244890" title="Посилання на пов’язаний елемент сховища даних [g]" accesskey="g"><span>Елемент Вікіданих</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Інструменти сторінки"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Зовнішній вигляд"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Зовнішній вигляд</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">сховати</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" 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">Матеріал з Вікіпідручника</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="subpages">< <bdi dir="ltr"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8" title="Основні числові системи">Основні числові системи</a></bdi></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="uk" dir="ltr"><div class="infobox sisterproject"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Wikipedia-logo-v2.svg" class="mw-file-description" title="Логотип Вікіпедії"><img alt="Логотип Вікіпедії" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/50px-Wikipedia-logo-v2.svg.png" decoding="async" width="50" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/75px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/100px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></a><figcaption>Логотип Вікіпедії</figcaption></figure> <a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F" class="mw-redirect" title="Вікіпедія">Вікіпедія</a> має пов'язану з цією темою інформацію на сторінці <i><b><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" class="extiw" title="w:Раціональне число">Раціональне число</a></b></i> </div> <p>У множині цілих чисел <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 {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> операція ділення не виконується. За розширення цілих чисел до раціональних <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> керуються такими самими правилами, як і за розширення натуральних до цілих, зокрема: </p> <ul><li>система цілих чисел <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 {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> повинна бути підсистемою системи раціональних <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>, тобто <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 {Z} \subset \mathbb {Q} ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \subset \mathbb {Q} ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30516c3e2a6f8227b4a1421157e016495ea9c85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.104ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} \subset \mathbb {Q} ;}"></span></li> <li>у системі <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> повинна виконуватися операція ділення, окрім ділення на нуль.</li></ul> <p>Раціональні числа - це множина цілих чисел та усі числа виду <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 {\frac {m}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd4bc206b907a993249aed8a4253595f61f2baf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.523ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}},}"></span> де <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 m\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad6b3baee714b61d6caa3a6edede2d2c6cf14f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.078ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {Z} ,}"></span> a <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 n\in \mathbb {N} \setminus \{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \setminus \{0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1409a5a07c72c56889b02cf70289378d19762efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.242ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {N} \setminus \{0\}.}"></span> Для раціональних чисел справджуються наступні твердження: </p> <ul><li>якщо <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 a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span> та <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 a_{1}\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd37e73f9689f624f70bcbaa1d5ead178666c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.192ex; height:2.676ex;" alt="{\displaystyle a_{1}\neq 0,}"></span> то <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 {\frac {b}{a}}={\frac {b_{1}}{a_{1}}}\Leftrightarrow ba_{1}=ab_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">⇔</mo> <mi>b</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b}{a}}={\frac {b_{1}}{a_{1}}}\Leftrightarrow ba_{1}=ab_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65e4ba2a9fce43a72a708a6a09fc3c0dd083d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.56ex; height:5.676ex;" alt="{\displaystyle {\frac {b}{a}}={\frac {b_{1}}{a_{1}}}\Leftrightarrow ba_{1}=ab_{1}}"></span>;</li> <li>для усіх <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 a,b,a_{1},b_{1}\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2fab8fbac70282efa9586b80bf365b58349000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.314ex; height:2.509ex;" alt="{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }"></span> справджується <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 a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {a}{b}}\pm {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}\pm ab_{1}}{aa_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>±</mo> <mi>a</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {a}{b}}\pm {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}\pm ab_{1}}{aa_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd337e1b9a063d5837bc308164fe43d995f9bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:39.597ex; height:5.676ex;" alt="{\displaystyle a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {a}{b}}\pm {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}\pm ab_{1}}{aa_{1}}}}"></span>;</li> <li>для усіх <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 a,b,a_{1},b_{1}\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2fab8fbac70282efa9586b80bf365b58349000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.314ex; height:2.509ex;" alt="{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }"></span> справджується <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 a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {b}{a}}\cdot {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{aa_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {b}{a}}\cdot {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{aa_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bec2b7f7bc3d14cdd887207f3de433fd5deeed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.546ex; height:5.676ex;" alt="{\displaystyle a\neq 0\land a_{1}\neq 0\Rightarrow {\frac {b}{a}}\cdot {\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{aa_{1}}}}"></span>;</li> <li>для усіх <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 a,b,a_{1},b_{1}\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2fab8fbac70282efa9586b80bf365b58349000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.314ex; height:2.509ex;" alt="{\displaystyle a,b,a_{1},b_{1}\in \mathbb {Q} }"></span> справджується <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 a\neq 0\land a_{1}\neq 0\land b_{1}\neq 0\Rightarrow {\frac {b}{a}}:{\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{ab_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>a</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0\land a_{1}\neq 0\land b_{1}\neq 0\Rightarrow {\frac {b}{a}}:{\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{ab_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433c1552e90c55f612d7b440a345c29c86faacfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.467ex; height:5.843ex;" alt="{\displaystyle a\neq 0\land a_{1}\neq 0\land b_{1}\neq 0\Rightarrow {\frac {b}{a}}:{\frac {b_{1}}{a_{1}}}={\frac {ba_{1}}{ab_{1}}}}"></span>;</li> <li>для усіх <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 a,b\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a00cbe702ec1d24ff0707e2a618290801a95856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.91ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {Q} }"></span> справджується <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 a\neq 0\Rightarrow {\frac {-b}{a}}=-{\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0\Rightarrow {\frac {-b}{a}}=-{\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9c9acc3ca627bcd3cef1d5b1f221be705276f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.719ex; height:5.343ex;" alt="{\displaystyle a\neq 0\Rightarrow {\frac {-b}{a}}=-{\frac {b}{a}}}"></span>;</li> <li>для усіх <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 a\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb03a87dc03fc305e7fe382cc673c048b6bd599e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.879ex; height:2.509ex;" alt="{\displaystyle a\in \mathbb {Q} }"></span> та усіх <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 m,n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f932c786e686277b8f060aae74134ce9fc3e3348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.86ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {Z} }"></span> справджується <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 a^{m}\cdot a^{n}=a^{mn}\land (a^{m})^{n}=a^{mn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>∧</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}\cdot a^{n}=a^{mn}\land (a^{m})^{n}=a^{mn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17421252cb275f3072589a76d9579288b85750b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.526ex; height:2.843ex;" alt="{\displaystyle a^{m}\cdot a^{n}=a^{mn}\land (a^{m})^{n}=a^{mn}}"></span></li></ul> <p>Систему <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> можна впорядкувати. Вважаймо, що число <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 r={\frac {k}{l}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>l</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {k}{l}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7b44354f82bfadb6942bebe371eeb550cd3dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.841ex; height:5.509ex;" alt="{\displaystyle r={\frac {k}{l}},}"></span> де <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 l\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e6e8495703048e241ef4ac28c06ff175a7012c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.601ex; height:2.676ex;" alt="{\displaystyle l\neq 0,}"></span> додатним тоді, коли обидва числа <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> та <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> одночасно додатні або від'ємні. Інакше кажучи, число <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 r={\frac {k}{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>l</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {k}{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4626eed99c3987a49ce5035d8e34171cfa499e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.194ex; height:5.509ex;" alt="{\displaystyle r={\frac {k}{l}}}"></span> будемо вважати додатним лише тоді, коли число <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 kl}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kl}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff70eab6b51d0630ed898a3b25b91fba03721b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.905ex; height:2.176ex;" alt="{\displaystyle kl}"></span> є додатним. </p><p>Нехай тепер раціональне число <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> є меншим від раціонального числа <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 r_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9d9537b1d764e5c4f986e3c63f3bfb362df6c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.009ex;" alt="{\displaystyle r_{1},}"></span> а раціональне число <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 r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> - більше від числа <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> (тобто <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 r<r_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<r_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c00db32f7d05ba41b8a90fbaa4184f9231726be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.897ex; height:2.176ex;" alt="{\displaystyle r<r_{1},}"></span> якщо різниця <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 r_{1}-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdffffde8118989b879e21ee1a19805626ee5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.992ex; height:2.343ex;" alt="{\displaystyle r_{1}-r}"></span> є додатним раціональним числом). Зрозуміло, що додатні числа є більшими від <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 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95547343453ea34a314dd174f8458012f5a39ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0,}"></span> а від'ємні - менше від <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 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}"></span> Визначене таким чином відношення <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 <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> ("менше") є відношенням лінійного порядку. Таким чином, система раціональних чисел <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> за допомогою відношення "менше" лінійно впорядковується. Визначене щойно відношення "<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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> менше <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 r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span>" задовільняє наступним вимогам: </p> <ul><li>для усіх <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 r,r_{1},r_{2}\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,r_{1},r_{2}\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05cbb369dfec98e5d8dbd58ac6ef81bb11ac5339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.971ex; height:2.509ex;" alt="{\displaystyle r,r_{1},r_{2}\in \mathbb {Q} }"></span> справджується <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 r<r_{1}\Rightarrow r+r_{2}<r_{1}+r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">⇒</mo> <mi>r</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<r_{1}\Rightarrow r+r_{2}<r_{1}+r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be456fbb47ba6fe3e75f91964161b482fd0a95a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.001ex; height:2.343ex;" alt="{\displaystyle r<r_{1}\Rightarrow r+r_{2}<r_{1}+r_{2}}"></span>;</li> <li>для усіх <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 r,r_{1},r_{2}\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,r_{1},r_{2}\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05cbb369dfec98e5d8dbd58ac6ef81bb11ac5339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.971ex; height:2.509ex;" alt="{\displaystyle r,r_{1},r_{2}\in \mathbb {Q} }"></span> справджується <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 r<r_{1}\land r_{2}>0\Rightarrow rr_{2}<r_{1}r_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<r_{1}\land r_{2}>0\Rightarrow rr_{2}<r_{1}r_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02ebb4d03fbe572bb121e62442a117d960226be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.913ex; height:2.509ex;" alt="{\displaystyle r<r_{1}\land r_{2}>0\Rightarrow rr_{2}<r_{1}r_{2}.}"></span></li></ul> <p>Справді, якщо <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 r<r_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<r_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c00db32f7d05ba41b8a90fbaa4184f9231726be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.897ex; height:2.176ex;" alt="{\displaystyle r<r_{1},}"></span> то <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 r_{1}-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdffffde8118989b879e21ee1a19805626ee5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.992ex; height:2.343ex;" alt="{\displaystyle r_{1}-r}"></span> дорівнює додатному числу. Тоді <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 (r_{1}+r_{2})-(r-r_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{1}+r_{2})-(r-r_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe4027e7fdccb2766bb521cde7b16dc735ec552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.497ex; height:2.843ex;" alt="{\displaystyle (r_{1}+r_{2})-(r-r_{2})}"></span> є додатним числом та, відповідно, <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 r+r_{2}<r_{1}+r_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+r_{2}<r_{1}+r_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58db57b4026b7c06704af3a202c1b8ade45bc6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.783ex; height:2.343ex;" alt="{\displaystyle r+r_{2}<r_{1}+r_{2}.}"></span> Якщо <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 r_{1}-r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}-r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a96735b4cb380091768d225be7487f25f50b5eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.253ex; height:2.509ex;" alt="{\displaystyle r_{1}-r>0}"></span> та <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 r_{2}>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713e5718811d39dc2daceeae65c61c573c8c0534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.011ex; height:2.509ex;" alt="{\displaystyle r_{2}>0,}"></span> то число <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 r_{1}r_{2}-rr_{2}=(r_{1}-r)r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}r_{2}-rr_{2}=(r_{1}-r)r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66822a6dd3cdda78ad49c197f41e88693b6c34e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.2ex; height:2.843ex;" alt="{\displaystyle r_{1}r_{2}-rr_{2}=(r_{1}-r)r_{2}}"></span> також є додатним та, відповідно, <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 rr_{2}<r_{1}r_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rr_{2}<r_{1}r_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d67257bb8305a820c29d6502b4519b694c0a8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.103ex; height:2.176ex;" alt="{\displaystyle rr_{2}<r_{1}r_{2}.}"></span> Таким чином, система <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 {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> із визначеним на ній відношенням "менше" є впорядкованою системою. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Дроби_та_їх_властивості"><span id=".D0.94.D1.80.D0.BE.D0.B1.D0.B8_.D1.82.D0.B0_.D1.97.D1.85_.D0.B2.D0.BB.D0.B0.D1.81.D1.82.D0.B8.D0.B2.D0.BE.D1.81.D1.82.D1.96"></span>Дроби та їх властивості</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=1" title="Редагувати розділ: Дроби та їх властивості"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Дробом є число виду <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 {\frac {a}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101faa653872fb89cf12cbea8c0a99b1c507943a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}},}"></span> де <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> називається чисельником даного дробу, а <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> - знаменником даного дробу.</p><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Cake_quarters.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/220px-Cake_quarters.svg.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/330px-Cake_quarters.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Cake_quarters.svg/440px-Cake_quarters.svg.png 2x" data-file-width="504" data-file-height="383" /></a><figcaption>Один торт ділиться на 4 частини - ілюстрація до дробу 1/4</figcaption></figure><p> Наприклад, дріб <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 {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"></span> із чисельником <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> та знаменником <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> означає, що число <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> ділиться на 2 частини (читається "одна друга"). Якщо у знаменнику покласти число <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 4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dc0881187d7c549d76cf6e7274b30039a73853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 4,}"></span> то отримаємо дріб <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 {\frac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2dfb63ee75ec084f2abb25d248bc151a2687508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}}"></span> (читається "одна четверта"). </p><p>Якщо чисельник і знаменник дробу помножити на одне й те саме число, то результуючий дріб буде рівний початковому. Зокрема, <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 {\frac {a\cdot n}{b\cdot n}}={\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>n</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a\cdot n}{b\cdot n}}={\frac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629e74f75270326f6a70160ced786c7d65cfc3df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.951ex; height:4.843ex;" alt="{\displaystyle {\frac {a\cdot n}{b\cdot n}}={\frac {a}{b}}.}"></span> Аналогічно, якщо чисельник і знаменник поділити на одне й те саме число, то отримаємо дріб, рівний початковому. Зокрема, <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 {\frac {a:n}{b:n}}={\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>:</mo> <mi>n</mi> </mrow> <mrow> <mi>b</mi> <mo>:</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a:n}{b:n}}={\frac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/523c8b9270f0236f8ce0a02f3f14752f81d91715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.209ex; height:4.843ex;" alt="{\displaystyle {\frac {a:n}{b:n}}={\frac {a}{b}}.}"></span> Відтак ми можемо вважати, що <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 {\frac {4}{8}}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>8</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{8}}={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0fff62c3bf297b90295963d74bfd3d8d52d330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.096ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{8}}={\frac {1}{2}}}"></span> чи <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 {\frac {6}{4}}={\frac {36}{24}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>36</mn> <mn>24</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6}{4}}={\frac {36}{24}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2812e0c58c0fc7281b72101ee5de2e0fc71f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.258ex; height:5.176ex;" alt="{\displaystyle {\frac {6}{4}}={\frac {36}{24}}}"></span> тощо. </p><p>Скоротити дріб - означає поділити чисельник і знаменник дробу на їх спільний дільник. Наприклад, дріб <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 {\frac {27}{25}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>27</mn> <mn>25</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {27}{25}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413c538278e179165778923a7fa2e67c776a2d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {27}{25}}}"></span> не можна скоротити, оскільки числа <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 27}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>27</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 27}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89455d53868bd9b952932246fe41aaf82e3f77e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 27}"></span> та <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 25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8acf6937f13156a1301ebb614c5364d16597e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 25}"></span> не мають спільних дільників. У такому випадку дріб називають нескоротним. Зокрема, якщо чисельник і знаменник дробу - взаємно прості числа, то такий дріб є нескоротним. Наприклад, дріб <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 {\frac {7}{13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>13</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{13}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c0e975b80ebfa7c23be54e13b655c8f7da6385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{13}}}"></span> є нескоротним. Взагалі, якщо дріб розділити на найбільший спільний дільник чисельника і знаменника, то отримаємо нескоротний дріб. Наприклад, <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 {\frac {4}{16}}={\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>16</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{16}}={\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8998fcfc8448cb1deab99ea256da997f4d2999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.905ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{16}}={\frac {1}{4}}.}"></span> Дріб <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 {\frac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2dfb63ee75ec084f2abb25d248bc151a2687508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}}"></span> не скорочується далі. </p><p><i>Зведення дробів до спільного знаменника</i>. Нехай дані дроби <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 {\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7572f1241ec7c2f9311985bc3dfb0b7d6f491e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{4}}}"></span> та <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 {\frac {5}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/986be6a57886d17a6f2514dbea17ccbfd10c6e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.645ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{6}}.}"></span> На які числа числа потрібно помножити кожний з цих дробів, щоб їхні знаменники були однаковими? В якості таких чисел можна обрати, наприклад, <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 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}"></span> та <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 8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}"></span> відповідно. Відтак, дроби <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 {\frac {3\cdot 12}{4\cdot 12}}={\frac {36}{48}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>12</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>36</mn> <mn>48</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\cdot 12}{4\cdot 12}}={\frac {36}{48}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd24da5d7be318baa297cdd4e373affd0031a0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.262ex; height:5.343ex;" alt="{\displaystyle {\frac {3\cdot 12}{4\cdot 12}}={\frac {36}{48}}}"></span> та <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 {\frac {5\cdot 8}{6\cdot 8}}={\frac {40}{48}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>8</mn> </mrow> <mrow> <mn>6</mn> <mo>⋅</mo> <mn>8</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>40</mn> <mn>48</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5\cdot 8}{6\cdot 8}}={\frac {40}{48}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5fceef6df7a25550c226e2b2f1a1cd3c6b02a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.1ex; height:5.343ex;" alt="{\displaystyle {\frac {5\cdot 8}{6\cdot 8}}={\frac {40}{48}}}"></span> мають однакові знаменники - число <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 48.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>48.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 48.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bac6a3abc07642aa6ff1fdf703337c6c51240eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 48.}"></span> Однак при зведенні дробів до спільного знаменника зручніше зводити їх до найменошого спільного знаменника. Зокрема, замість чисел <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 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}"></span> та <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 8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}"></span> ми могли б обрати числа <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 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"></span> та <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 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span> і отримали б найменший з можливих спільний знаменник для дробів <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 {\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7572f1241ec7c2f9311985bc3dfb0b7d6f491e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{4}}}"></span> та <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 {\frac {5}{6}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{6}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b18e0ba9f3b48adad5dc335d2d2d61415c422e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.645ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{6}},}"></span> зокрема <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 {\frac {18}{24}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>24</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {18}{24}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7dc552637497cb252261cf13b51ecbb64e7a5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {18}{24}}}"></span> та <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 {\frac {20}{24}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>20</mn> <mn>24</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {20}{24}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730426d2151961a748385ebca02d6bac92cd5b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.808ex; height:5.176ex;" alt="{\displaystyle {\frac {20}{24}}.}"></span> </p><p>Щб звести дроби до спільних знаменників можна користуватися наступною послідовністю дій: </p> <ul><li>знайти НСК знаменників даних дробів; <ul><li>віднайти додаткові множники для кожного з дробів окремо, поділивши спільний знаменник на знаменники даних дробів; <ul><li>помножити чисельник і знаменник кожного дробу на його додатковий множник.</li></ul></li></ul></li></ul> <p>Розгляньмо дроби <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 {\frac {5}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f3e79a6a2f8288b72472a4649e78fba98f9290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{4}}}"></span> та <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 {\frac {7}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748cc3b4c368ca89ec85cd87ca9e27a1b3ebb9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.645ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{6}}.}"></span> НСК(4; 6)<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 =12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =12.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3baf4042296140da8d8eff3c2b3b8165914637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.425ex; height:2.176ex;" alt="{\displaystyle =12.}"></span> Розділимо <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 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}"></span> на кожний із знаменників даних дробів: <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 12:4=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>:</mo> <mn>4</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12:4=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7158c7dee6ca81df2e0ee1aa372259c1014bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.685ex; height:2.176ex;" alt="{\displaystyle 12:4=3}"></span> та <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 12:6=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>:</mo> <mn>6</mn> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12:6=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca81419424d7aa55a1e5bdc9853025f668762246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.332ex; height:2.176ex;" alt="{\displaystyle 12:6=2.}"></span> Множником дробу <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 {\frac {5}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f3e79a6a2f8288b72472a4649e78fba98f9290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{4}}}"></span> є число <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 3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e86384bc2ac4ac6c2a68904ba067110f0876bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 3,}"></span> а множником дробу <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 {\frac {7}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa4d811472804c34c559cc72576eecd7e5e9bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{6}}}"></span> є число <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 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}"></span> Зведімо дані дроби до спільного знаменника: <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 {\frac {5\cdot 3}{4\cdot 3}}={\frac {15}{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5\cdot 3}{4\cdot 3}}={\frac {15}{12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0757b38bf6503a4a13be47180b25f44ee5a21b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.1ex; height:5.343ex;" alt="{\displaystyle {\frac {5\cdot 3}{4\cdot 3}}={\frac {15}{12}}}"></span> й <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 {\frac {7\cdot 2}{6\cdot 2}}={\frac {14}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>6</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7\cdot 2}{6\cdot 2}}={\frac {14}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c2caed2dc72ff1c39dbf964aa31ec182fa38e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.746ex; height:5.176ex;" alt="{\displaystyle {\frac {7\cdot 2}{6\cdot 2}}={\frac {14}{12}}.}"></span> </p><p>Щоб порівняти два дроби, їх спочатку потрібно звести до спільного знаменника, а потім порівняти їх числельники. Наприклад, нас може цікавити питання про те, який з дробів <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 {\frac {5}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f3e79a6a2f8288b72472a4649e78fba98f9290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{4}}}"></span> та <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 {\frac {7}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa4d811472804c34c559cc72576eecd7e5e9bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{6}}}"></span> є більшим. Зводячи їх до спільного знаменника й порівнюючи чисельники <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 15>14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>></mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15>14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351bf1df95aef289a7fb1e820ba6b062846e3af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.748ex; height:2.176ex;" alt="{\displaystyle 15>14}"></span> (або <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 14<15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>14</mn> <mo><</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14<15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76320f31caa190484c0dd6eca9b10c16f728ee69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.748ex; height:2.176ex;" alt="{\displaystyle 14<15}"></span>) можна переконатися у тому, що <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 {\frac {5}{4}}>{\frac {7}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{4}}>{\frac {7}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52070acc5fc135075f5d60bae48c920119042aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.096ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{4}}>{\frac {7}{6}}}"></span> чи, навпаки, <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 {\frac {7}{6}}<{\frac {5}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{6}}<{\frac {5}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d16332531ee6c26710126392c41c423f1507c2d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.742ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{6}}<{\frac {5}{4}}.}"></span> Зрозуміло також, що <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 -{\frac {32}{7}}<{\frac {2}{7}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>7</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>7</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {32}{7}}<{\frac {2}{7}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4bf49dd4b456c2aac52469a1ebb8f8cdbe2252a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.713ex; height:5.343ex;" alt="{\displaystyle -{\frac {32}{7}}<{\frac {2}{7}},}"></span> оскільки дріб <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 -{\frac {32}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {32}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c967f5f6d0d8ba43e18fc849a5b4be421034368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.969ex; height:5.343ex;" alt="{\displaystyle -{\frac {32}{7}}}"></span> є від'ємним. </p><p>Розгляньмо приклад: </p> <ul><li>Розмістимо наступні дроби у порядку зростання: <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 {\frac {1}{12}};\,{\frac {5}{12}};{\frac {12}{12}};\,{\frac {18}{24}};\,{\frac {5}{3}};\,{\frac {4}{12}};\,{\frac {10}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>12</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>24</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>12</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{12}};\,{\frac {5}{12}};{\frac {12}{12}};\,{\frac {18}{24}};\,{\frac {5}{3}};\,{\frac {4}{12}};\,{\frac {10}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa2351489bcfa99470313b1a5246678796198c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.751ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{12}};\,{\frac {5}{12}};{\frac {12}{12}};\,{\frac {18}{24}};\,{\frac {5}{3}};\,{\frac {4}{12}};\,{\frac {10}{12}}.}"></span> Звернімо увагу на те, що частина наведених дробів мають однакові знаменники; зведімо інші дроби до спільного знаменника <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 12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ee8bf1c252cecdcbd35ffeae377d0e64dd73e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 12.}"></span> Отримаємо: <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 {\frac {18}{24}}={\frac {18:2}{24:2}}={\frac {9}{12}};\,\,\,\,{\frac {5}{3}}={\frac {5\cdot 4}{3\cdot 4}}={\frac {20}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>24</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>18</mn> <mo>:</mo> <mn>2</mn> </mrow> <mrow> <mn>24</mn> <mo>:</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>12</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>4</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>20</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {18}{24}}={\frac {18:2}{24:2}}={\frac {9}{12}};\,\,\,\,{\frac {5}{3}}={\frac {5\cdot 4}{3\cdot 4}}={\frac {20}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82515ac03373d7d5a22acab702b30df21ba3eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.206ex; height:5.343ex;" alt="{\displaystyle {\frac {18}{24}}={\frac {18:2}{24:2}}={\frac {9}{12}};\,\,\,\,{\frac {5}{3}}={\frac {5\cdot 4}{3\cdot 4}}={\frac {20}{12}}.}"></span> Тепер розташуймо у ланцюг виділені дроби: <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 {\frac {1}{12}}<{\frac {4}{12}}<{\frac {9}{12}}<{\frac {10}{12}}<{\frac {12}{12}}<{\frac {20}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>12</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>12</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>12</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>12</mn> </mfrac> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>20</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{12}}<{\frac {4}{12}}<{\frac {9}{12}}<{\frac {10}{12}}<{\frac {12}{12}}<{\frac {20}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff751939bdd8ff7692427d8aa0fea8665dc332e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.105ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{12}}<{\frac {4}{12}}<{\frac {9}{12}}<{\frac {10}{12}}<{\frac {12}{12}}<{\frac {20}{12}}.}"></span> Ми отримали лінійно впорядковану множину, мінімальним елементом якої є дріб <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 {\frac {1}{12}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{12}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a70247da938b261f46a5fa0c2368f68f6bd2f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.808ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{12}},}"></span> а максимальним - дріб <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 {\frac {20}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>20</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {20}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5beacd25ee197cc6f60fb15779578569c09d392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.808ex; height:5.176ex;" alt="{\displaystyle {\frac {20}{12}}.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Масштаб"><span id=".D0.9C.D0.B0.D1.81.D1.88.D1.82.D0.B0.D0.B1"></span>Масштаб</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=2" title="Редагувати розділ: Масштаб"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Щоб по зображенню мати уяву про розміри й положення у просторі якого-небудь предмета, потрібно, щоб це зображення мало наступні властивості: </p> <ul><li>зворотність - властивість зображення, яке дозволяє по ньому однозначно встановити реальну форму, розміри предмета та його положення у просторі;</li> <li>наочність - властивість зображення, яке викликає в уяві спостерігача просторові уявлення про предмет; найкращою формою наочності є натуральність; зорове судження про предмет по його натуральному зображенню є близьким до того, яке виникає за розгляду самого предмету у натурі;</li> <li>єдиність умовностей, прийнятих при виконанні зображення; ці конвенції (домовленості) повинні бути такими, щоб кожний спеціаліст міг "прочитати" зображення, виконане іншою особою.</li></ul> <p>Карта є прикладом малюнку, який містить відомості про просторове розташування об'єктів. Щоб скласти карту навіть невеличкого села із перенесенням на неї усіх відстаней між нерухомими об'єктами з їх розмірами потрібно було б багато паперу. Тому при складанні карт малюють зменшене кратно зображення і при цьому обов'язково вказують масштаб, тобто у скільки разів було зменшене зображення із збереженням усіх відстаней, щоб за необхідності можна було здійснити виміри по цій карті. При цьому один сантиметр на малюнку може означати 100 метрів (або більше) у натурі. При цьому пишуть <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 {\frac {1}{10\,000}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>10</mn> <mspace width="thinmathspace" /> <mn>000</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10\,000}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421b67e7ae1fb85cc8c05a42730bb7a4711f9c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.036ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10\,000}}}"></span> або <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 1:10\,000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>10</mn> <mspace width="thinmathspace" /> <mn>000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:10\,000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d9c88e0fdfc5e72bbed4604c4db8814ad688d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.299ex; height:2.176ex;" alt="{\displaystyle 1:10\,000}"></span> см (один сантиметр до десяти тисяч сантиметрів). Відношення довжини лінії з малюнку до її справжньої довжини у натурі називається масштабом. Відтак, у випадку, якщо лінія на малюнку дорівнює 1 м, а у натурі 1 км (1000 метрів), то масштаб буде складати <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 1:1000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:1000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9272cd8174947051eb24acb1b6d8aa1c5528da1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.749ex; height:2.176ex;" alt="{\displaystyle 1:1000}"></span> м. Варто враховувати, що зображення може бути й більшим за реальний об'єкт, особливо, якщо цей об'єкт у натурі мікроскопічного розміру. </p><p>Масштаби можна порівнювати. Більш крупним є той масштаб, у якого знаменник менший. Наприклад, масштаб <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 1:10\,000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>10</mn> <mspace width="thinmathspace" /> <mn>000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:10\,000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d9c88e0fdfc5e72bbed4604c4db8814ad688d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.299ex; height:2.176ex;" alt="{\displaystyle 1:10\,000}"></span> крупніший, ніж масштаб <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 1:100\,000.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mn>100</mn> <mspace width="thinmathspace" /> <mn>000.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:100\,000.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc302c7934489304f4837dd50af49d6ab03b998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.108ex; height:2.176ex;" alt="{\displaystyle 1:100\,000.}"></span> Відтак, <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 {\frac {1}{10\,000}}>{\frac {1}{100\,000}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>10</mn> <mspace width="thinmathspace" /> <mn>000</mn> </mrow> </mfrac> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>100</mn> <mspace width="thinmathspace" /> <mn>000</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10\,000}}>{\frac {1}{100\,000}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaed4fae2685ea71c35bb5bb2e027392d85a16b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.979ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10\,000}}>{\frac {1}{100\,000}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Дії_над_дробами"><span id=".D0.94.D1.96.D1.97_.D0.BD.D0.B0.D0.B4_.D0.B4.D1.80.D0.BE.D0.B1.D0.B0.D0.BC.D0.B8"></span>Дії над дробами</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=3" title="Редагувати розділ: Дії над дробами"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>У дробі <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 {\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"></span> чисельник може бути або меншим від знаменника <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 a<b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f9e3247a4373970cad2b3b37920af5d23ec7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a<b,}"></span> або рівним знаменнику <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 a=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a87d9bccbbd750d94a977aa90d98d60210d0c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a=b,}"></span> або більшим від нього <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 a>b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f3dbc209b9e7a3f89d6b918f0f67a5fd7cc2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.973ex; height:2.176ex;" alt="{\displaystyle a>b.}"></span> В залежності від значень чисельників й знаменників дроби діляться на правильні й неправильні. Дріб, чисельник якого є меншим від знаменника, тобто <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 a<b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f9e3247a4373970cad2b3b37920af5d23ec7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a<b,}"></span> називається правильним; дріб, у якому чисельник більший від знаменника, тобто <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 a>b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d8ca5031f98a774d037e63bfc4296c44d41d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a>b,}"></span> або рівний знаменнику, <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 a=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a87d9bccbbd750d94a977aa90d98d60210d0c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a=b,}"></span> називається неправильним. Розгляньмо дві групи дробів: першу </p><p><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 {\frac {1}{3}},\quad \quad {\frac {3}{5}},\quad \quad {\frac {4}{7}},\quad \quad {\frac {2}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}},\quad \quad {\frac {3}{5}},\quad \quad {\frac {4}{7}},\quad \quad {\frac {2}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc86f9da32f8a322d0a7cd140f013c2a8c5c65d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.679ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{3}},\quad \quad {\frac {3}{5}},\quad \quad {\frac {4}{7}},\quad \quad {\frac {2}{3}},}"></span> </p><p>та другу </p><p><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 {\frac {7}{7}},\quad \quad -{\frac {24}{6}},\quad \quad {\frac {32}{28}},\quad \quad {\frac {12}{9}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>7</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>24</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>28</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>9</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{7}},\quad \quad -{\frac {24}{6}},\quad \quad {\frac {32}{28}},\quad \quad {\frac {12}{9}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdc7330b16c09f107f5594751ae8cc93775539f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.974ex; height:5.343ex;" alt="{\displaystyle {\frac {7}{7}},\quad \quad -{\frac {24}{6}},\quad \quad {\frac {32}{28}},\quad \quad {\frac {12}{9}}.}"></span> </p><p>Перша група складається з правильних дробів (чисельник є меншим від знаменника); друга група складається з неправильних дробів (чисельник більший від знаменника або дорівнює йому). Можна помітити, що серед дробів другої групи є такі неправильні дроби, що в результаті виконання дії діленя чисельника на знаменник ми отримаємо цілі числа. Зокрема, такими дробами є дроби <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 {\frac {7}{7}}=1,\,\,\,-{\frac {24}{6}}=(-24):6=-4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>24</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>24</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mn>6</mn> <mo>=</mo> <mo>−</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{7}}=1,\,\,\,-{\frac {24}{6}}=(-24):6=-4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a208519e43a20bb558a2c888ef993ea51f6a9d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.28ex; height:5.343ex;" alt="{\displaystyle {\frac {7}{7}}=1,\,\,\,-{\frac {24}{6}}=(-24):6=-4.}"></span> У випадку із іншими неправильними дробами, тобто <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 {\frac {32}{28}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>28</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {32}{28}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2788411490f5297fdc67af54a2f304e49482ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {32}{28}}}"></span> та <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 {\frac {12}{9}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>9</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12}{9}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ddac578910cf448034d07eec66a68cde15ad00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.808ex; height:5.176ex;" alt="{\displaystyle {\frac {12}{9}},}"></span> необхідно вдаватися до ділення із остачею. Зокрема, можна здійснити ділення <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 {\frac {32}{28}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>28</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {32}{28}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2788411490f5297fdc67af54a2f304e49482ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {32}{28}}}"></span> наступним чином: <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 {\frac {32-4}{28}}+{\frac {4}{28}}=1+{\frac {1}{7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>32</mn> <mo>−</mo> <mn>4</mn> </mrow> <mn>28</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>28</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {32-4}{28}}+{\frac {4}{28}}=1+{\frac {1}{7}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2268996a57ec90d66387a93832bd71b276224fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.912ex; height:5.343ex;" alt="{\displaystyle {\frac {32-4}{28}}+{\frac {4}{28}}=1+{\frac {1}{7}}.}"></span> Так само у випадку із неправильним дробом <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 {\frac {24}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>24</mn> <mn>9</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {24}{9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9e8fca6d6c06116496bc69041e02c775393da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {24}{9}}}"></span> маємо <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 {\frac {24-6}{9}}+{\frac {6}{9}}={\frac {18}{9}}+{\frac {6}{9}}=2+{\frac {2}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>24</mn> <mo>−</mo> <mn>6</mn> </mrow> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {24-6}{9}}+{\frac {6}{9}}={\frac {18}{9}}+{\frac {6}{9}}=2+{\frac {2}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b55fcf1009d7f635f6be0c1848c29ad07d39e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.848ex; height:5.176ex;" alt="{\displaystyle {\frac {24-6}{9}}+{\frac {6}{9}}={\frac {18}{9}}+{\frac {6}{9}}=2+{\frac {2}{3}}.}"></span> У цих обох прикладах ми виділили цілі частини <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> та <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> неправильного дробу - неповні частки. Дробові лишкові доданки <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 {\frac {1}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7488185acac9cec8180b1f6a1f3c13b72d59b395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.999ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{7}}}"></span> та <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 {\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19eee5d63f2cf9106dc531cdfdea8cfb8f34b2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}}"></span> - остачі від неповного ділення. В загальному, для дробу <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 {\frac {a}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101faa653872fb89cf12cbea8c0a99b1c507943a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}},}"></span> де <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 b\leq a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05af0ccc5a6e91bf711507150765f14a209b592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle b\leq a,}"></span> маємо неповну частку <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 k={\frac {a-r}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>r</mi> </mrow> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {a-r}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ecb53def77fa6eea1a2971cffd4dcf8347d479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.265ex; height:5.176ex;" alt="{\displaystyle k={\frac {a-r}{b}}}"></span> (ділення чисельника, зменшеного на остачу <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 r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250644a0f511e9078be6f89ba78a606a0e08c0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.695ex; height:2.009ex;" alt="{\displaystyle r,}"></span> на знаменник) та лишковий доданок <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 {\frac {r}{b}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r}{b}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608bc79c8416792664133a260ca39105daa61de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.532ex; height:4.843ex;" alt="{\displaystyle {\frac {r}{b}},}"></span> тобто <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 k+{\frac {r}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k+{\frac {r}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7173264b090a2e26732d8408510a4fa005c1f128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.583ex; height:4.843ex;" alt="{\displaystyle k+{\frac {r}{b}}.}"></span> Зворотне перетворення здійснюється наступним чином: <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 {\frac {k\cdot b+r}{b}}={\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>r</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k\cdot b+r}{b}}={\frac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e1c2552a86f0104a688b0be525898da09eb9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.424ex; height:5.509ex;" alt="{\displaystyle {\frac {k\cdot b+r}{b}}={\frac {a}{b}}.}"></span> Наприклад, для дробу <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 {\frac {8}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89734036c466a92239649275d0ba29c9d8b52ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {8}{3}}}"></span> матимемо <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 k={\frac {\overbrace {8} ^{a}-\overbrace {2} ^{r}}{\underbrace {3} _{b}}}+{\frac {\overbrace {2} ^{r}}{\underbrace {3} _{b}}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>8</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mover> <mo>−</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>2</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mover> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mn>3</mn> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munder> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>2</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mover> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mn>3</mn> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munder> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {\overbrace {8} ^{a}-\overbrace {2} ^{r}}{\underbrace {3} _{b}}}+{\frac {\overbrace {2} ^{r}}{\underbrace {3} _{b}}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3594cd1b36b05e28763ed82b3dacf6c2d10c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:23.863ex; height:11.676ex;" alt="{\displaystyle k={\frac {\overbrace {8} ^{a}-\overbrace {2} ^{r}}{\underbrace {3} _{b}}}+{\frac {\overbrace {2} ^{r}}{\underbrace {3} _{b}}};}"></span> здійснюючи обчислення, отримуємо <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 {\frac {8}{3}}={\frac {8-2}{3}}+{\frac {2}{3}}=2+{\frac {2}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>−</mo> <mn>2</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8}{3}}={\frac {8-2}{3}}+{\frac {2}{3}}=2+{\frac {2}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aeb50379bf3f97c79a01a039d0df51056e70be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.684ex; height:5.176ex;" alt="{\displaystyle {\frac {8}{3}}={\frac {8-2}{3}}+{\frac {2}{3}}=2+{\frac {2}{3}}.}"></span> Зворотне перетворення: <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 {\frac {\overbrace {2} ^{k}\cdot \overbrace {3} ^{b}+\overbrace {2} ^{r}}{\underbrace {3} _{b}}}={\frac {8}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>2</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mover> <mo>⋅</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>3</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mover> <mo>+</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mn>2</mn> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mover> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mn>3</mn> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munder> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\overbrace {2} ^{k}\cdot \overbrace {3} ^{b}+\overbrace {2} ^{r}}{\underbrace {3} _{b}}}={\frac {8}{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ca0d64dafa4934c380ffefa93a52e7257d291a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:22.653ex; height:12.009ex;" alt="{\displaystyle {\frac {\overbrace {2} ^{k}\cdot \overbrace {3} ^{b}+\overbrace {2} ^{r}}{\underbrace {3} _{b}}}={\frac {8}{3}}.}"></span> </p><p>Отже, щоб виділити цілу частину (неповну частку) з неправильного дробу <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 {\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"></span>, необхідно поділити чисельник <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, зменшений на остачу <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, тобто <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 a-r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e83c59fa09bb23316efd41e70922c9ed5090fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.766ex; height:2.343ex;" alt="{\displaystyle a-r,}"></span> на знаменник <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>; одержана неповна частка <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 k={\frac {a-r}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>r</mi> </mrow> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {a-r}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ecb53def77fa6eea1a2971cffd4dcf8347d479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.265ex; height:5.176ex;" alt="{\displaystyle k={\frac {a-r}{b}}}"></span> буде цілою частиною, яка у сумі із лишковим дробовим доданком <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 {\frac {r}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64eaa183e493402fe964f53692d6f96a45550bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.885ex; height:4.843ex;" alt="{\displaystyle {\frac {r}{b}}}"></span> дорівнюють даному дробові <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 {\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ca6e3e42a0093600ae2abfd5587ae52136547a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}.}"></span> </p><p>Навпаки, якщо у чисельнику знаменник <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> помножити на неповну частку <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> та до отриманого добутку додати остачу <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, а знаменник залишити тим самим <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 b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"></span> то дістанемо неправильний дріб <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 {\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbb66e57f89debc3cde3213de12228971148a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}"></span>. </p><p>Важливо зауважити, що дріб із виділеною цілою частиною записують без знаку додавання; зокрема, дріб <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 3+{\frac {7}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+{\frac {7}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea5dfbfda8b3b9f7aedae5b19bbbd128bbeeffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.164ex; height:5.176ex;" alt="{\displaystyle 3+{\frac {7}{10}}}"></span> записується просто <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 3{\frac {7}{10}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{\frac {7}{10}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb31b8c813a6e9f08253defe24006826f6958391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.97ex; height:5.176ex;" alt="{\displaystyle 3{\frac {7}{10}}.}"></span> Тобто ціла частина завжди записується ліворуч, а лишковий дробовий доданок - праворуч від цілої частини. </p> <div class="mw-heading mw-heading3"><h3 id="Додавання_дробів_та_віднімання_дробів"><span id=".D0.94.D0.BE.D0.B4.D0.B0.D0.B2.D0.B0.D0.BD.D0.BD.D1.8F_.D0.B4.D1.80.D0.BE.D0.B1.D1.96.D0.B2_.D1.82.D0.B0_.D0.B2.D1.96.D0.B4.D0.BD.D1.96.D0.BC.D0.B0.D0.BD.D0.BD.D1.8F_.D0.B4.D1.80.D0.BE.D0.B1.D1.96.D0.B2"></span>Додавання дробів та віднімання дробів</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=4" title="Редагувати розділ: Додавання дробів та віднімання дробів"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Щоб додати (або відняти) два дроби з різними знаменниками, необхідно звести ці дроби до спільного знаменника, а потім застосувати правило додавання (віднімання) дробів із рівними знаменниками. </p><p>Наприклад, розгляньмо суму дробів <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 {\frac {4}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ca7a90f2e65f751186364ba382c61bb14b2f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{9}}}"></span> та <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 {\frac {4}{5}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{5}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b6c2786d0bda04efb2140deadf39a612a2f96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.645ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{5}}.}"></span> Знайдімо <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 {\frac {4}{9}}+{\frac {4}{5}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{9}}+{\frac {4}{5}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c223cfd392fdf15bd6825c477543abc48356b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.484ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{9}}+{\frac {4}{5}}.}"></span> Спільний знаменник цих дробів дорівнює <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 45.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>45.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90ebc331106205ea6ad126a3ed5f3b5dd0b718a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 45.}"></span> Одержуємо: <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 {\frac {4\cdot 5}{9\cdot 5}}+{\frac {4\cdot 9}{5\cdot 9}}={\frac {20+36}{45}}={\frac {56}{45}}=1{\frac {11}{45}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>9</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>9</mn> </mrow> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>9</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>20</mn> <mo>+</mo> <mn>36</mn> </mrow> <mn>45</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>56</mn> <mn>45</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>45</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4\cdot 5}{9\cdot 5}}+{\frac {4\cdot 9}{5\cdot 9}}={\frac {20+36}{45}}={\frac {56}{45}}=1{\frac {11}{45}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bab283a4e0d66e296365d80569267868262a4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.274ex; height:5.343ex;" alt="{\displaystyle {\frac {4\cdot 5}{9\cdot 5}}+{\frac {4\cdot 9}{5\cdot 9}}={\frac {20+36}{45}}={\frac {56}{45}}=1{\frac {11}{45}}.}"></span> </p><p>Розгляньмо різницю дробів <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 {\frac {3}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e9422e650e87d5ed42ed3c97142390aa4992da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.999ex; height:5.343ex;" alt="{\displaystyle {\frac {3}{7}}}"></span> та <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 {\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca269377f18d1b032279be1559cb3e7c3623e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.645ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}.}"></span> Знайдімо <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 {\frac {3}{7}}-{\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{7}}-{\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7513201fae1da7ec369a0cf356b0c645e222508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.484ex; height:5.343ex;" alt="{\displaystyle {\frac {3}{7}}-{\frac {1}{2}}.}"></span> Спільний знаменник цих дробів дорівнює <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 14.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>14.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 14.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a90241cc85148423e0015e5c288c4adc1aabb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 14.}"></span> Одержуємо: <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 {\frac {6-7}{14}}=-{\frac {1}{14}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mo>−</mo> <mn>7</mn> </mrow> <mn>14</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>14</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6-7}{14}}=-{\frac {1}{14}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0b1639a2c7bb42bde87c713def96290c83e15f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.716ex; height:5.176ex;" alt="{\displaystyle {\frac {6-7}{14}}=-{\frac {1}{14}}.}"></span> </p><p>Для дробів справджуються наступні властивості додавання: </p> <ul><li>переставна властивість: <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 {\frac {a}{b}}+{\frac {c}{d}}={\frac {c}{d}}+{\frac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {c}{d}}+{\frac {a}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ede7c4d0cc24242055e32ef2e5cff8acf99645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.015ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {c}{d}}+{\frac {a}{b}}}"></span>;</li> <li>сполучна властивість: <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 ({\frac {a}{b}}+{\frac {c}{d}})+{\frac {p}{q}}={\frac {a}{b}}+({\frac {c}{d}}+{\frac {p}{q}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\frac {a}{b}}+{\frac {c}{d}})+{\frac {p}{q}}={\frac {a}{b}}+({\frac {c}{d}}+{\frac {p}{q}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abab47e98059a96b43818c985d0a405ede07add" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.972ex; height:5.343ex;" alt="{\displaystyle ({\frac {a}{b}}+{\frac {c}{d}})+{\frac {p}{q}}={\frac {a}{b}}+({\frac {c}{d}}+{\frac {p}{q}}).}"></span></li></ul> <p>Розгляньмо декілька прикладів: </p> <ul><li><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 4{\frac {5}{12}}+2{\frac {3}{4}}=4{\frac {5}{12}}+2{\frac {3\cdot 3}{4\cdot 3}}=4{\frac {5}{12}}+2{\frac {9}{12}}=6{\frac {14}{12}}=6{\frac {7}{6}}=7{\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>12</mn> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>12</mn> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4{\frac {5}{12}}+2{\frac {3}{4}}=4{\frac {5}{12}}+2{\frac {3\cdot 3}{4\cdot 3}}=4{\frac {5}{12}}+2{\frac {9}{12}}=6{\frac {14}{12}}=6{\frac {7}{6}}=7{\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816fbc8ffb42888270e435ccfcf641f9c3d90aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.117ex; height:5.343ex;" alt="{\displaystyle 4{\frac {5}{12}}+2{\frac {3}{4}}=4{\frac {5}{12}}+2{\frac {3\cdot 3}{4\cdot 3}}=4{\frac {5}{12}}+2{\frac {9}{12}}=6{\frac {14}{12}}=6{\frac {7}{6}}=7{\frac {1}{6}}}"></span>;</li> <li><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 4{\frac {1}{6}}-2{\frac {4}{9}}=5{\frac {1\cdot 3}{6\cdot 3}}-2{\frac {4\cdot 2}{9\cdot 2}}=5{\frac {3}{18}}-2{\frac {8}{18}}=4{\frac {21}{18}}-2{\frac {8}{18}}=2{\frac {13}{18}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>6</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>9</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>18</mn> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>18</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>18</mn> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>18</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>13</mn> <mn>18</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4{\frac {1}{6}}-2{\frac {4}{9}}=5{\frac {1\cdot 3}{6\cdot 3}}-2{\frac {4\cdot 2}{9\cdot 2}}=5{\frac {3}{18}}-2{\frac {8}{18}}=4{\frac {21}{18}}-2{\frac {8}{18}}=2{\frac {13}{18}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e3c7f817471276dbf29de508612848a0ad066c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.7ex; height:5.176ex;" alt="{\displaystyle 4{\frac {1}{6}}-2{\frac {4}{9}}=5{\frac {1\cdot 3}{6\cdot 3}}-2{\frac {4\cdot 2}{9\cdot 2}}=5{\frac {3}{18}}-2{\frac {8}{18}}=4{\frac {21}{18}}-2{\frac {8}{18}}=2{\frac {13}{18}}}"></span>;</li> <li><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 {\frac {3a}{8b}}-{\frac {a}{5b}}={\frac {15a-8a}{40b}}={\frac {7a}{40b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>a</mi> </mrow> <mrow> <mn>8</mn> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>5</mn> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>15</mn> <mi>a</mi> <mo>−</mo> <mn>8</mn> <mi>a</mi> </mrow> <mrow> <mn>40</mn> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mi>a</mi> </mrow> <mrow> <mn>40</mn> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3a}{8b}}-{\frac {a}{5b}}={\frac {15a-8a}{40b}}={\frac {7a}{40b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903358580235f91f3184915b16211a1998332889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.044ex; height:5.343ex;" alt="{\displaystyle {\frac {3a}{8b}}-{\frac {a}{5b}}={\frac {15a-8a}{40b}}={\frac {7a}{40b}}}"></span>;</li> <li><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 {\frac {11x}{14y}}+{\frac {z}{21y}}={\frac {3\cdot 11x}{3\cdot 14y}}+{\frac {2\cdot z}{2\cdot 21y}}={\frac {33x+2z}{42y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mi>x</mi> </mrow> <mrow> <mn>14</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>21</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>11</mn> <mi>x</mi> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>14</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mi>z</mi> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>21</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>33</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> </mrow> <mrow> <mn>42</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11x}{14y}}+{\frac {z}{21y}}={\frac {3\cdot 11x}{3\cdot 14y}}+{\frac {2\cdot z}{2\cdot 21y}}={\frac {33x+2z}{42y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1921244cca340f5e75d25a4e258f34dbe59dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.404ex; height:5.676ex;" alt="{\displaystyle {\frac {11x}{14y}}+{\frac {z}{21y}}={\frac {3\cdot 11x}{3\cdot 14y}}+{\frac {2\cdot z}{2\cdot 21y}}={\frac {33x+2z}{42y}}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Множення_дробів"><span id=".D0.9C.D0.BD.D0.BE.D0.B6.D0.B5.D0.BD.D0.BD.D1.8F_.D0.B4.D1.80.D0.BE.D0.B1.D1.96.D0.B2"></span>Множення дробів</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=5" title="Редагувати розділ: Множення дробів"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Щоб помножити дріб на ціле число, необхідно його чисельник помножити на це число, а знаменник залишити без зміни. Наприклад, <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 0\cdot {\frac {15}{20}}={\frac {15\cdot 0}{20}}={\frac {0}{20}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>20</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>15</mn> <mo>⋅</mo> <mn>0</mn> </mrow> <mn>20</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <mn>20</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot {\frac {15}{20}}={\frac {15\cdot 0}{20}}={\frac {0}{20}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f94ccf692c408e956294f111b8f6720059a3b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.271ex; height:5.176ex;" alt="{\displaystyle 0\cdot {\frac {15}{20}}={\frac {15\cdot 0}{20}}={\frac {0}{20}}=0.}"></span> Зрозуміло, що число <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> не містить частин (не може бути розбитим на <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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span> частин діленням на число <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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span>). Використовуючи літери, можна записати правило множення дробів на числа: <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 {\frac {a}{b}}\cdot m={\frac {a\cdot m}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>m</mi> </mrow> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\cdot m={\frac {a\cdot m}{b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13946d486b13f4f6d6aedb88afde7e2845773ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.316ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}\cdot m={\frac {a\cdot m}{b}}.}"></span> Для <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 m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span> будемо мати <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 0\cdot {\frac {a}{b}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot {\frac {a}{b}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7489fc2d10704d772f59998468a8cd30c757018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.815ex; height:4.843ex;" alt="{\displaystyle 0\cdot {\frac {a}{b}}=0.}"></span> Так само <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 {\frac {a}{b}}\cdot 0=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\cdot 0=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2d8732494dac66bcd0c9667cc858570a879e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.815ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}\cdot 0=0.}"></span> Розгляньмо декілька прикладів: </p> <ul><li><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 {\frac {5}{6}}\cdot -5=-{\frac {25}{6}}=-4{\frac {1}{6}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{6}}\cdot -5=-{\frac {25}{6}}=-4{\frac {1}{6}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a80a7932043db7d32a1f19f02f9b3d3aec9c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.43ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{6}}\cdot -5=-{\frac {25}{6}}=-4{\frac {1}{6}};}"></span></li> <li><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 (-{\frac {12}{7}})\cdot (-7)={\frac {84}{7}}=12;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>7</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>84</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mn>12</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-{\frac {12}{7}})\cdot (-7)={\frac {84}{7}}=12;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491a46938e84fdcb1a096b7d8e1f1a9151795113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.567ex; height:5.343ex;" alt="{\displaystyle (-{\frac {12}{7}})\cdot (-7)={\frac {84}{7}}=12;}"></span></li> <li><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 (-3)\cdot (4+2):7={\frac {(-3)\cdot 4+(-3)\cdot 2}{7}}=-{\frac {18}{7}}=-2{\frac {4}{7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mn>7</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>18</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-3)\cdot (4+2):7={\frac {(-3)\cdot 4+(-3)\cdot 2}{7}}=-{\frac {18}{7}}=-2{\frac {4}{7}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2e69fc5e771df693592a6e516cefbf7866bcfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:55.333ex; height:5.843ex;" alt="{\displaystyle (-3)\cdot (4+2):7={\frac {(-3)\cdot 4+(-3)\cdot 2}{7}}=-{\frac {18}{7}}=-2{\frac {4}{7}}.}"></span></li></ul> <p>При множенні двох дробів перемножуються їхні чисельники та знаменники. Іншими словами, добуток двох дробів представляє собою дріб, чисельник якого дорівнює добутку чисельників, а знаменник - добутку знаменників. Зокрема, <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 {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a\cdot c}{b\cdot d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a\cdot c}{b\cdot d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08cc857f174f59d3c2e49e13fe429f2652e5ccf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.294ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a\cdot c}{b\cdot d}}.}"></span> При цьому варто зауважити, що множення дробів підпорядковується комутативному, асоціативному та дистрибутивному законам: </p> <ul><li>переставний закон: <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 {\frac {x}{y}}\cdot {\frac {w}{z}}={\frac {w}{z}}\cdot {\frac {x}{y}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}\cdot {\frac {w}{z}}={\frac {w}{z}}\cdot {\frac {x}{y}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54aecf029736145b39c2fed56894c96d8daf50e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.436ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{y}}\cdot {\frac {w}{z}}={\frac {w}{z}}\cdot {\frac {x}{y}};}"></span> наприклад, <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 {\frac {5}{6}}\cdot {\frac {2}{3}}={\frac {5\cdot 2}{6\cdot 3}}={\frac {2\cdot 5}{3\cdot 6}}={\frac {10}{18}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>6</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>18</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{6}}\cdot {\frac {2}{3}}={\frac {5\cdot 2}{6\cdot 3}}={\frac {2\cdot 5}{3\cdot 6}}={\frac {10}{18}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6ebe9f1ec29b5e9d31dbdc0d83a7afdc9c861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.46ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{6}}\cdot {\frac {2}{3}}={\frac {5\cdot 2}{6\cdot 3}}={\frac {2\cdot 5}{3\cdot 6}}={\frac {10}{18}};}"></span></li> <li>сполучний закон: <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 {\frac {a}{b}}({\frac {c}{d}}\cdot {\frac {w}{z}})=({\frac {a}{b}}\cdot {\frac {c}{d}})\cdot {\frac {w}{z}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}({\frac {c}{d}}\cdot {\frac {w}{z}})=({\frac {a}{b}}\cdot {\frac {c}{d}})\cdot {\frac {w}{z}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9211ff69079fd8d234553e57d7e76592e4e521a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.638ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}({\frac {c}{d}}\cdot {\frac {w}{z}})=({\frac {a}{b}}\cdot {\frac {c}{d}})\cdot {\frac {w}{z}};}"></span> наприклад, <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 {\frac {2}{5}}\cdot ({\frac {1}{2}}\cdot {\frac {4}{7}})=({\frac {2}{5}}\cdot {\frac {1}{2}})\cdot {\frac {4}{7}}={\frac {2\cdot 1\cdot 4}{5\cdot 2\cdot 7}}={\frac {8}{70}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>4</mn> </mrow> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>7</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>70</mn> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{5}}\cdot ({\frac {1}{2}}\cdot {\frac {4}{7}})=({\frac {2}{5}}\cdot {\frac {1}{2}})\cdot {\frac {4}{7}}={\frac {2\cdot 1\cdot 4}{5\cdot 2\cdot 7}}={\frac {8}{70}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b91bb2f2f84572f3bbc040e43768c988b8b6ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.112ex; height:5.343ex;" alt="{\displaystyle {\frac {2}{5}}\cdot ({\frac {1}{2}}\cdot {\frac {4}{7}})=({\frac {2}{5}}\cdot {\frac {1}{2}})\cdot {\frac {4}{7}}={\frac {2\cdot 1\cdot 4}{5\cdot 2\cdot 7}}={\frac {8}{70}};}"></span></li> <li>розподільний закон: <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 {\frac {a}{b}}\cdot ({\frac {c}{d}}+{\frac {w}{z}})={\frac {a\cdot c}{b\cdot d}}+{\frac {a\cdot w}{b\cdot z}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>w</mi> </mrow> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}\cdot ({\frac {c}{d}}+{\frac {w}{z}})={\frac {a\cdot c}{b\cdot d}}+{\frac {a\cdot w}{b\cdot z}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807c8d78ebe56930215388eb6c355bd024c9ffe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.694ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}\cdot ({\frac {c}{d}}+{\frac {w}{z}})={\frac {a\cdot c}{b\cdot d}}+{\frac {a\cdot w}{b\cdot z}};}"></span> наприклад, <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 {\frac {2}{3}}\cdot ({\frac {1}{2}}+{\frac {3}{4}})=({\frac {2}{3}}\cdot {\frac {1}{2}})\cdot {\frac {3}{4}}={\frac {2\cdot 1\cdot 3}{3\cdot 2\cdot 4}}={\frac {6}{24}}={\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>24</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\cdot ({\frac {1}{2}}+{\frac {3}{4}})=({\frac {2}{3}}\cdot {\frac {1}{2}})\cdot {\frac {3}{4}}={\frac {2\cdot 1\cdot 3}{3\cdot 2\cdot 4}}={\frac {6}{24}}={\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9bf24fcc818dee2aac5925cccee10e4334b169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.37ex; height:5.343ex;" alt="{\displaystyle {\frac {2}{3}}\cdot ({\frac {1}{2}}+{\frac {3}{4}})=({\frac {2}{3}}\cdot {\frac {1}{2}})\cdot {\frac {3}{4}}={\frac {2\cdot 1\cdot 3}{3\cdot 2\cdot 4}}={\frac {6}{24}}={\frac {1}{4}}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ділення_дробів"><span id=".D0.94.D1.96.D0.BB.D0.B5.D0.BD.D0.BD.D1.8F_.D0.B4.D1.80.D0.BE.D0.B1.D1.96.D0.B2"></span>Ділення дробів</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=6" title="Редагувати розділ: Ділення дробів"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Щоб розділити один дріб на другий, необхідно діене помножити на число, обернене до дільника. Засосовуючи літери, це можна записати наступним чином: </p><p><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 {\frac {a}{b}}:{\frac {c}{d}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}:{\frac {c}{d}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09f98b4d32d019170ff7ebafe0d4d23a221cbe70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.597ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{b}}:{\frac {c}{d}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}"></span> </p><p>Необхідно відзначити, що <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 1:{\frac {a}{b}}={\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1:{\frac {a}{b}}={\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97e34aeab12bd5c05c6d1bc38dd16466ba2545b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.33ex; height:5.509ex;" alt="{\displaystyle 1:{\frac {a}{b}}={\frac {b}{a}}}"></span> та <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 0:{\frac {a}{b}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0:{\frac {a}{b}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4176da8390b89c6051d181b827edc62856d29dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.073ex; height:4.843ex;" alt="{\displaystyle 0:{\frac {a}{b}}=0.}"></span> На нуль ділити не можна. </p><p>Розгляньмо декілька прикладів: </p> <ul><li><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 {\frac {4}{21}}:{\frac {12}{7}}={\frac {\not {4}^{1}}{21}}\cdot {\frac {7}{\not {12}^{3}}}={\frac {1\cdot \not {7}^{1}}{\not {21}^{3}\cdot 3}}={\frac {1\cdot 1}{3\cdot 3}}={\frac {1}{9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>21</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mn>21</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mrow> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> <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> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> </mrow> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{21}}:{\frac {12}{7}}={\frac {\not {4}^{1}}{21}}\cdot {\frac {7}{\not {12}^{3}}}={\frac {1\cdot \not {7}^{1}}{\not {21}^{3}\cdot 3}}={\frac {1\cdot 1}{3\cdot 3}}={\frac {1}{9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc593d8f1dfedf9d97bba56353786b4a5c3388df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.431ex; height:6.843ex;" alt="{\displaystyle {\frac {4}{21}}:{\frac {12}{7}}={\frac {\not {4}^{1}}{21}}\cdot {\frac {7}{\not {12}^{3}}}={\frac {1\cdot \not {7}^{1}}{\not {21}^{3}\cdot 3}}={\frac {1\cdot 1}{3\cdot 3}}={\frac {1}{9}}}"></span>;</li> <li><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 {\frac {3}{7}}:1{\frac {1}{5}}={\frac {3}{7}}:{\frac {6}{5}}={\frac {1}{7}}\cdot {\frac {5}{2}}={\frac {5}{14}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> <mo>:</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>14</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{7}}:1{\frac {1}{5}}={\frac {3}{7}}:{\frac {6}{5}}={\frac {1}{7}}\cdot {\frac {5}{2}}={\frac {5}{14}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b80ccbb5bfe8135859ccae3ce8e79c8360e78b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.164ex; height:5.343ex;" alt="{\displaystyle {\frac {3}{7}}:1{\frac {1}{5}}={\frac {3}{7}}:{\frac {6}{5}}={\frac {1}{7}}\cdot {\frac {5}{2}}={\frac {5}{14}}}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Раціональні_вирази"><span id=".D0.A0.D0.B0.D1.86.D1.96.D0.BE.D0.BD.D0.B0.D0.BB.D1.8C.D0.BD.D1.96_.D0.B2.D0.B8.D1.80.D0.B0.D0.B7.D0.B8"></span>Раціональні вирази</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=7" title="Редагувати розділ: Раціональні вирази"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Вирази, які містять додавання, віднімання, множення, ділення та піднесення до степеня із натуральним показником чисел та змінних, називаються раціональними. У випадку, коли раціональний вираз не містить ділення на вираз із змінною, його називають цілим, у протилежному випадку - дробовим. Серед виразів </p><p><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 2x+y^{2},\quad \quad {\frac {7b(a+3)}{12}},\quad \quad {\frac {a}{13}},\quad \quad x^{3}-2x,\quad \quad {\frac {2h-g}{z-14z}},\quad \quad x^{2}+axy+{\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mi>b</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>13</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mi>g</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mn>14</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+y^{2},\quad \quad {\frac {7b(a+3)}{12}},\quad \quad {\frac {a}{13}},\quad \quad x^{3}-2x,\quad \quad {\frac {2h-g}{z-14z}},\quad \quad x^{2}+axy+{\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09383d3f5f311a6b2ba3bc24d4e1f829d5c56fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:78.982ex; height:5.843ex;" alt="{\displaystyle 2x+y^{2},\quad \quad {\frac {7b(a+3)}{12}},\quad \quad {\frac {a}{13}},\quad \quad x^{3}-2x,\quad \quad {\frac {2h-g}{z-14z}},\quad \quad x^{2}+axy+{\frac {1}{x}}}"></span> </p><p>вирази <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 2x+y^{2},\,\,{\frac {7b(a+3)}{12}},\,\,{\frac {a}{13}},\,\,x^{3}-2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mi>b</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>13</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+y^{2},\,\,{\frac {7b(a+3)}{12}},\,\,{\frac {a}{13}},\,\,x^{3}-2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0205341b72177758fa845644d904b1e4fbfebba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.887ex; height:5.676ex;" alt="{\displaystyle 2x+y^{2},\,\,{\frac {7b(a+3)}{12}},\,\,{\frac {a}{13}},\,\,x^{3}-2x}"></span> є цілими раціональними, а вирази <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 {\frac {2h-g}{z-14z}},\,\,x^{2}+axy+{\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mi>g</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mn>14</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2h-g}{z-14z}},\,\,x^{2}+axy+{\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fe5abcc84d11dcd25179b5c5ca4a7badd00c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.931ex; height:5.509ex;" alt="{\displaystyle {\frac {2h-g}{z-14z}},\,\,x^{2}+axy+{\frac {1}{x}}}"></span> - дробово-раціональні. </p><p>Тотожно рівними є вирази, якщо за усіх припустими значень змінних значення цих виразів є рівними. Наприклад, тотожно рівними є вирази <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 2a^{2}-18}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>18</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{2}-18}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbca955441de5dae1d024a7f06da76a8b554da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.612ex; height:2.843ex;" alt="{\displaystyle 2a^{2}-18}"></span> та <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 (2a+6)(x-3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2a+6)(x-3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0385fc483c5e5fc4d213e1aa33e6a25ad8edb233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.993ex; height:2.843ex;" alt="{\displaystyle (2a+6)(x-3).}"></span> </p><p>Раціональним дробом називається вираз <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 {\frac {P_{1}}{P_{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P_{1}}{P_{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa7e818a31d5d2b6d00d5d42b1e15f76f3486e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.029ex; height:5.509ex;" alt="{\displaystyle {\frac {P_{1}}{P_{2}}},}"></span> де <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 P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> та <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 P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> - поліноми. Якщо чисельник й знаменник раціонального дробу помножити чи поділити на поліном <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 P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> (відмінний від нуль-полінома, <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 P_{3}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1c1b62dc014ee9dbc464b3a711ca88e62fd7b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.807ex; height:2.676ex;" alt="{\displaystyle P_{3}\neq 0}"></span>), то отримаємо дріб, який є тотожно рівним даному: <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 {\frac {P_{1}}{P_{2}}}={\frac {P_{1}\cdot P_{3}}{P_{2}\cdot P_{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P_{1}}{P_{2}}}={\frac {P_{1}\cdot P_{3}}{P_{2}\cdot P_{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd9d9096f3f1d74aceebe148833e647ab3a97d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.736ex; height:5.843ex;" alt="{\displaystyle {\frac {P_{1}}{P_{2}}}={\frac {P_{1}\cdot P_{3}}{P_{2}\cdot P_{3}}},}"></span> де <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 P_{3}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a0141ff30e85a57c2b73ce0fe0c304b1343b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.454ex; height:2.676ex;" alt="{\displaystyle P_{3}\neq 0.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Дії_над_раціональними_виразами"><span id=".D0.94.D1.96.D1.97_.D0.BD.D0.B0.D0.B4_.D1.80.D0.B0.D1.86.D1.96.D0.BE.D0.BD.D0.B0.D0.BB.D1.8C.D0.BD.D0.B8.D0.BC.D0.B8_.D0.B2.D0.B8.D1.80.D0.B0.D0.B7.D0.B0.D0.BC.D0.B8"></span>Дії над раціональними виразами</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=8" title="Редагувати розділ: Дії над раціональними виразами"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>На основі властивості дробу скорочуватися (ділення чисельника та знаменника на спільний множник) можно скорочувати й раціональні вирази. Щоб скоротити дріб, необхідно спершу чисельник й знаменник розкласти на множники. Зокрема, розгляньмо дріб <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 {\frac {12-3a^{2}}{15a+30}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>15</mn> <mi>a</mi> <mo>+</mo> <mn>30</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12-3a^{2}}{15a+30}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6764340c44419eafa1b676387037c0d7545a7178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.203ex; height:5.843ex;" alt="{\displaystyle {\frac {12-3a^{2}}{15a+30}}.}"></span> Застосовуючи основну властивість дробу, одержимо: <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 {\frac {12-3a^{2}}{15a+30}}={\frac {3(4-a^{2})}{15(a+2)}}={\frac {3(2-a)(2+a)}{15(2+a)}}={\frac {2-a}{5}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>12</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>15</mn> <mi>a</mi> <mo>+</mo> <mn>30</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>15</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>15</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>a</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12-3a^{2}}{15a+30}}={\frac {3(4-a^{2})}{15(a+2)}}={\frac {3(2-a)(2+a)}{15(2+a)}}={\frac {2-a}{5}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f48463b314fb7c4eb9eb7d5d745845ec3e20c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.853ex; height:6.676ex;" alt="{\displaystyle {\frac {12-3a^{2}}{15a+30}}={\frac {3(4-a^{2})}{15(a+2)}}={\frac {3(2-a)(2+a)}{15(2+a)}}={\frac {2-a}{5}}.}"></span> </p><p>Для зведення дробів до спільного знаменника потрібно: </p> <ul><li>розкласти на множники знаменники дробів; <ul><li>віднайти спільний знаменник за внесення до нього усіх різних множників у найвищих степенях, у яких вони входять до розкладів знаменників; <ul><li>віднайти додатковий множник для кожного дробу (при цьому спільник знаменник необхідно розділити на знаменник дробу); <ul><li>помножини чисельник та знаменник кожного дробу на знайдений додатковий множник.</li></ul></li></ul></li></ul></li></ul> <p>Для того, щоб додати або відняти дроби з різними знаменниками, потрібно для початку звести їх до спільного знаменника. Зокрема, розгляньмо різницю <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 {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997d25501d317bdb587b4c1b34d6c1d378eb84f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.027ex; height:6.343ex;" alt="{\displaystyle {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}.}"></span> Одержуємо: <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 {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x(x-y)}{(x-y)(x+y)}}={\frac {x^{2}-2y-x^{2}+xy}{(x-y)(x+y)}}={\frac {-2y+xy}{x^{2}-y^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x(x-y)}{(x-y)(x+y)}}={\frac {x^{2}-2y-x^{2}+xy}{(x-y)(x+y)}}={\frac {-2y+xy}{x^{2}-y^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcc8b5fbae3e541369f241fed460760b4933903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:116.797ex; height:6.509ex;" alt="{\displaystyle {\frac {x^{2}-2y}{x^{2}-y^{2}}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x}{x+y}}={\frac {x^{2}-2y}{(x-y)(x+y)}}-{\frac {x(x-y)}{(x-y)(x+y)}}={\frac {x^{2}-2y-x^{2}+xy}{(x-y)(x+y)}}={\frac {-2y+xy}{x^{2}-y^{2}}}.}"></span> </p><p>Добутком двох раціональних дробів є дріб, чисельник якого дорівнює добутку чисельників, а знаменник - добутку знаменників. Зокрема, розгляньмо добуток двох дробів <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 {\frac {2x^{2}}{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>y</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x^{2}}{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0823ede5a2ef4f27590028380f8434868b1b3fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.383ex; height:6.176ex;" alt="{\displaystyle {\frac {2x^{2}}{y}}}"></span> та <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 {\frac {y}{8xz}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y}{8xz}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780d930f2eb7f602a202df6945820317bf9f9034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.063ex; height:4.843ex;" alt="{\displaystyle {\frac {y}{8xz}}.}"></span> Одержимо: <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 {\frac {2x^{2}}{y}}\cdot {\frac {y}{8xz}}={\frac {2x^{2}\cdot y}{y\cdot 8xz}}={\frac {x}{4z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>y</mi> </mrow> <mrow> <mi>y</mi> <mo>⋅</mo> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>4</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x^{2}}{y}}\cdot {\frac {y}{8xz}}={\frac {2x^{2}\cdot y}{y\cdot 8xz}}={\frac {x}{4z}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d1e0e7357dd9bfb63636dc2506896b65d7fc47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.659ex; height:6.176ex;" alt="{\displaystyle {\frac {2x^{2}}{y}}\cdot {\frac {y}{8xz}}={\frac {2x^{2}\cdot y}{y\cdot 8xz}}={\frac {x}{4z}}.}"></span> </p><p>Тепер розділимо <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 {\frac {2x^{2}}{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>y</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x^{2}}{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0823ede5a2ef4f27590028380f8434868b1b3fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.383ex; height:6.176ex;" alt="{\displaystyle {\frac {2x^{2}}{y}}}"></span> на <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 {\frac {y}{8xz}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y}{8xz}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780d930f2eb7f602a202df6945820317bf9f9034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.063ex; height:4.843ex;" alt="{\displaystyle {\frac {y}{8xz}}.}"></span> Одержимо: <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 {\frac {2x^{2}}{y}}:{\frac {y}{8xz}}={\frac {2x^{2}}{y}}\cdot {\frac {8xz}{y}}={\frac {16x^{3}z}{y^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>x</mi> <mi>z</mi> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>16</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x^{2}}{y}}:{\frac {y}{8xz}}={\frac {2x^{2}}{y}}\cdot {\frac {8xz}{y}}={\frac {16x^{3}z}{y^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9838c21fdacfdc647e0599ff13700387961e0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.691ex; height:6.343ex;" alt="{\displaystyle {\frac {2x^{2}}{y}}:{\frac {y}{8xz}}={\frac {2x^{2}}{y}}\cdot {\frac {8xz}{y}}={\frac {16x^{3}z}{y^{2}}}.}"></span> </p><p>Щоб піднести до степеня із цілим показником раціональний дріб, необхідно піднести чисельник й знаменник до цього степеня. Зокрема, <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 ({\frac {P_{1}}{P_{2}}})^{m}={\frac {P_{1}^{m}}{P_{2}^{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\frac {P_{1}}{P_{2}}})^{m}={\frac {P_{1}^{m}}{P_{2}^{m}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056339e759e751c24083f17c63604f5ac47aec7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.945ex; height:6.509ex;" alt="{\displaystyle ({\frac {P_{1}}{P_{2}}})^{m}={\frac {P_{1}^{m}}{P_{2}^{m}}}.}"></span> Наприклад, <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 ({\frac {2z+7h}{y^{4}}})^{2}={\frac {(2z+7h)^{2}}{(y^{4})^{2}}}={\frac {4z^{2}+49h^{2}}{y^{8}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mn>7</mn> <mi>h</mi> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mn>7</mn> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>49</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\frac {2z+7h}{y^{4}}})^{2}={\frac {(2z+7h)^{2}}{(y^{4})^{2}}}={\frac {4z^{2}+49h^{2}}{y^{8}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1213ae199dd7bf6becdd3c45cf259aa7ea01d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.13ex; height:6.676ex;" alt="{\displaystyle ({\frac {2z+7h}{y^{4}}})^{2}={\frac {(2z+7h)^{2}}{(y^{4})^{2}}}={\frac {4z^{2}+49h^{2}}{y^{8}}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Цілі_раціональні_рівняння"><span id=".D0.A6.D1.96.D0.BB.D1.96_.D1.80.D0.B0.D1.86.D1.96.D0.BE.D0.BD.D0.B0.D0.BB.D1.8C.D0.BD.D1.96_.D1.80.D1.96.D0.B2.D0.BD.D1.8F.D0.BD.D0.BD.D1.8F"></span>Цілі раціональні рівняння</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=9" title="Редагувати розділ: Цілі раціональні рівняння"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Лінійні рівняння із однією змінною - рівняння, яке має вигляд <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 ax=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbc5298ece09330178fcf486ab4c6418ca9818c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.302ex; height:2.509ex;" alt="{\displaystyle ax=b,}"></span> де <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> - змінна, а <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> та <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> є відомими числами. </p><p>Властивості лінійних рівнянь: </p> <ul><li>якщо <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 a\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2068d811f09c9ca1437234192c052786eae4e41f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.138ex; height:2.676ex;" alt="{\displaystyle a\neq 0,}"></span> то <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={\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8b8b99e6c9833827215e807e0c3939e320e155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.494ex; height:5.343ex;" alt="{\displaystyle x={\frac {b}{a}}}"></span> є єдиним коренем рівняння <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 ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b82cb3720d07ecbcab36145af8fee0b7a3519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.656ex; height:2.176ex;" alt="{\displaystyle ax=b}"></span>;</li> <li>якщо <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 a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"></span> та <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 b\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69238caa691e75f448aae578c1d7d93e8c9840b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.905ex; height:2.676ex;" alt="{\displaystyle b\neq 0,}"></span> то <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 0\cdot x=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot x=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d40a3d54006cac00d8d7b588bfc19904496ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.267ex; height:2.176ex;" alt="{\displaystyle 0\cdot x=b}"></span> рівняння коренів немає</li> <li>якщо <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 a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"></span> та <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 b=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f202a65bf9433acd9a3c75335d0784bcf14c2a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.905ex; height:2.509ex;" alt="{\displaystyle b=0,}"></span> тоді <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 0\cdot x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb80a874f8485bf303e1abf696f7a99e03c05179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; 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