{"id":29781,"date":"2026-10-27T11:22:28","date_gmt":"2026-10-27T11:22:28","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=29781"},"modified":"2026-10-28T09:26:14","modified_gmt":"2026-10-28T09:26:14","slug":"prove-that-root-2-is-irrational","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/math-explorer-program\/prove-that-root-2-is-irrational\/","title":{"rendered":"Prove That Root 2 Is Irrational: Step-by-Step complete guide"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"29781\" class=\"elementor elementor-29781\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-caeda1c elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"caeda1c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-175e28a\" data-id=\"175e28a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-727b766\" data-id=\"727b766\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b3ed8c5 elementor-widget elementor-widget-heading\" data-id=\"b3ed8c5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">How to Prove That Root 2 Is Irrational (Step by Step)\n<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7173d71 elementor-widget elementor-widget-text-editor\" data-id=\"7173d71\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">If you\u2019ve ever wondered how to prove that \u221a2 is irrational, good news: it\u2019s one of the most famous proofs in maths, and it asks for very little. If you know what an even number is and can square a number, you\u2019ve got everything you need.<\/span><\/p><p><span style=\"font-weight: 400;\">We\u2019ll go through it slowly, look at why each step works, point out where students usually slip, and finish with a short version for exam day. Read it once, then close the page and try writing it out yourself. That\u2019s what makes it stick.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eae7c8c elementor-align-center elementor-widget elementor-widget-button\" data-id=\"eae7c8c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0c6af1e elementor-widget elementor-widget-image\" data-id=\"0c6af1e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational-1024x576.webp\" class=\"attachment-large size-large wp-image-29931\" alt=\"Prove That Root 2 Is Irrational: Step-by-Step complete guide\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational-1024x576.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational-300x169.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational-768x432.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational-1536x864.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Exploring-Why-\u221a2-Isnt-Rational.webp 1672w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-cbb94e4\" data-id=\"cbb94e4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b872164 elementor-widget elementor-widget-html\" data-id=\"b872164\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; \r\n    z-index: 1 !important;\r\n    right: 70px;\r\n    width: 100%;\r\n    max-width: 340px;\r\n    font-family: 'Poppins', sans-serif;\r\n}\r\n\r\n\/* Image Styling *\/\r\n.toc-sticky-image {\r\n    width: 100%;\r\n    height: auto;\r\n    border-radius: 22px; \r\n    margin-bottom: 15px; \r\n    box-shadow: 0 4px 12px rgba(0,0,0,0.05);\r\n    display: block;\r\n}\r\n\r\n\/* TOC Container *\/\r\n.custom-toc {\r\n    background: #fff;\r\n    border: 1px solid #dfe3ea;\r\n    border-radius: 22px;\r\n    padding: 0px 28px;\r\n    width: 100%;\r\n    max-height: 24rem; \r\n    overflow-y: auto;\r\n    box-shadow: 0 2px 8px rgba(0,0,0,0.03);\r\n}\r\n\r\n\/* Title *\/\r\n.custom-toc .toc-title {\r\n    font-size: 25px;\r\n    font-weight: 700;\r\n    color: #17233c;\r\n    position: sticky;\r\n    padding: 0.6rem 0rem;\r\n    background-color: white;\r\n    top: 0px;\r\n    margin-bottom: 12px;\r\n    display: flex;\r\n    align-items: center;\r\n    gap: 10px;\r\n    z-index: 2;\r\n}\r\n\r\n\/* List *\/\r\n.custom-toc .toc-list {\r\n    list-style: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\n\/* Links *\/\r\n.custom-toc .toc-list li {\r\n    margin-bottom: 22px;\r\n}\r\n\r\n.custom-toc .toc-list a {\r\n    text-decoration: none;\r\n    color: #6b7a99;\r\n    font-size: 14px;\r\n    font-weight: 600;\r\n    transition: 0.3s ease;\r\n    display: inline-block;\r\n}\r\n\r\n\/* Hover Color *\/\r\n.custom-toc .toc-list a:hover {\r\n    color: #8B0000;\r\n}\r\n\r\n\/* H2 Indent *\/\r\n.custom-toc .toc-list .toc-h2 {\r\n    padding-left: 18px;\r\n    font-size: 14px;\r\n}\r\n\r\n\/* Mobile Responsive Adjustments *\/\r\n@media(max-width:768px) {\r\n    .toc-wrapper {\r\n        position: relative;\r\n        top: 0;\r\n        z-index: 1;\r\n        right: 0;\r\n        max-width: 100%;\r\n        margin-bottom: 30px;\r\n    }\r\n\r\n    .custom-toc {\r\n        max-height: none; \r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7d4f431 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"7d4f431\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b19ed7d\" data-id=\"b19ed7d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a1e17af\" data-id=\"a1e17af\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dee6df6 elementor-widget elementor-widget-heading\" data-id=\"dee6df6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What does \u201cirrational\u201d actually mean?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bb8803e elementor-widget elementor-widget-text-editor\" data-id=\"bb8803e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A quick word on the terms first.<\/span><\/p><p><span style=\"font-weight: 400;\">A <\/span><b>rational number<\/b><span style=\"font-weight: 400;\"> is anything you can write as a fraction a\/b, where a and b are whole numbers and b isn\u2019t zero. 1\/2, 7\/3, 5 (which is 5\/1) and 0.25 (which is 1\/4) are all rational.<\/span><\/p><p><span style=\"font-weight: 400;\">An <\/span><b>irrational number<\/b><span style=\"font-weight: 400;\"> is one you can\u2019t write that way. As a decimal, it goes on forever without ever settling into a repeating pattern.<\/span><\/p><p><span style=\"font-weight: 400;\">\u221a2 is the positive number that gives 2 when multiplied by itself. Your calculator will show something like 1.41421356&#8230;, but that\u2019s just a rounded slice of it. What we want to show is that no fraction, no matter how huge its top and bottom numbers, equals \u221a2 exactly.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-915e794\" data-id=\"915e794\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fd6d9e0 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"fd6d9e0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-533811e\" data-id=\"533811e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4cc0d05\" data-id=\"4cc0d05\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-add6f44 elementor-widget elementor-widget-heading\" data-id=\"add6f44\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The idea: proof by contradiction\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7ae693e elementor-widget elementor-widget-text-editor\" data-id=\"7ae693e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The classic approach is called proof by contradiction. It sounds fancier than it is. Imagine someone tells you, \u201cThere\u2019s a large whole number.\u201d You say, \u201cOkay, let\u2019s suppose that\u2019s true,\u201d and then you add 1 to it. Now you\u2019ve got something bigger, so the claim couldn\u2019t have been true in the first place.<\/span><\/p><p><span style=\"font-weight: 400;\">We\u2019ll pull the same move here:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Pretend the opposite is true (that \u221a2 is rational).<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Follow the logic one step at a time.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Arrive at something impossible.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Conclude that the starting assumption was wrong, so \u221a2 must be irrational.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">This is the version most schools and universities teach, so it\u2019s the one your examiner is most likely hoping to see.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ac2e84a\" data-id=\"ac2e84a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-15a8d42 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"15a8d42\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8514a67\" data-id=\"8514a67\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-6586267\" data-id=\"6586267\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a473fd1 elementor-widget elementor-widget-heading\" data-id=\"a473fd1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The full proof<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7ec4acd elementor-widget elementor-widget-text-editor\" data-id=\"7ec4acd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Take it slowly and make sure each line makes sense before you move on.<\/span><\/p><p><b>Step 1: Make the assumption.<\/b><span style=\"font-weight: 400;\"> Suppose \u221a2 is rational. Then we can write<\/span><\/p><p><span style=\"font-weight: 400;\">\u221a2 = a\/b<\/span><\/p><p><span style=\"font-weight: 400;\">where a and b are whole numbers, b isn\u2019t zero, and the fraction is in its lowest terms, meaning a and b share no common factor other than 1. (Any fraction can be reduced like this, so assuming it costs us nothing.)<\/span><\/p><p><b>Step 2: Square both sides.<\/b><\/p><p><span style=\"font-weight: 400;\">2 = a\u00b2\/b\u00b2<\/span><\/p><p><b>Step 3: Rearrange.<\/b><span style=\"font-weight: 400;\"> Multiply both sides by b\u00b2:<\/span><\/p><p><span style=\"font-weight: 400;\">a\u00b2 = 2b\u00b2<\/span><\/p><p><b>Step 4: Spot that a\u00b2 is even.<\/b><span style=\"font-weight: 400;\"> The right side is 2 times a whole number, so a\u00b2 has to be even.<\/span><\/p><p><b>Step 5: Conclude that a is even.<\/b><span style=\"font-weight: 400;\"> This leans on a small fact: if a\u00b2 is even, then a is even. Here\u2019s why. Suppose they were odd. Then a = 2k + 1 for some whole number k, and<\/span><\/p><p><span style=\"font-weight: 400;\">a\u00b2 = (2k + 1)\u00b2 = 4k\u00b2 + 4k + 1 = 2(2k\u00b2 + 2k) + 1<\/span><\/p><p><span style=\"font-weight: 400;\">which is odd. So an odd number always has an odd square. Since our a\u00b2 is even, a can\u2019t be odd, which means it\u2019s even.<\/span><\/p><p><b>Step 6: Write as 2k.<\/b><span style=\"font-weight: 400;\"> Because a is even, we can say a = 2k for some whole number k.<\/span><\/p><p><b>Step 7: Substitute back.<\/b><span style=\"font-weight: 400;\"> Put a = 2k into a\u00b2 = 2b\u00b2:<\/span><\/p><p><span style=\"font-weight: 400;\">(2k)\u00b2 = 2b\u00b2<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\"> 4k\u00b2 = 2b\u00b2<\/span><\/p><p><span style=\"font-weight: 400;\">Divide both sides by 2 and you get:<\/span><\/p><p><span style=\"font-weight: 400;\">2k\u00b2 = b\u00b2<\/span><\/p><p><b>Step 8: Spot that b is even too.<\/b><span style=\"font-weight: 400;\"> Same reasoning as Step 5. b\u00b2 equals 2 times a whole number, so b\u00b2 is even, and therefore b is even.<\/span><\/p><p><b>Step 9: Find the contradiction.<\/b><span style=\"font-weight: 400;\"> Now both a and b are even, so both can be divided by 2. That means a\/b was <\/span><i><span style=\"font-weight: 400;\">not<\/span><\/i><span style=\"font-weight: 400;\"> in lowest terms after all, even though that\u2019s exactly what we assumed at the start. Something\u2019s gone wrong.<\/span><\/p><p><b>Step 10: Conclude.<\/b><span style=\"font-weight: 400;\"> The assumption led us to a dead end, so it has to be false. \u221a2 can\u2019t be written as a fraction, so it\u2019s irrational. \u220e<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-48e5e4b\" data-id=\"48e5e4b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-84c2e50 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"84c2e50\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-294a1b4\" data-id=\"294a1b4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-29b354d\" data-id=\"29b354d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a48989a elementor-widget elementor-widget-heading\" data-id=\"a48989a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The short version<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6ca96d0 elementor-widget elementor-widget-text-editor\" data-id=\"6ca96d0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Pressed for time, or is the question only worth a few marks? Here\u2019s the compact version:<\/span><\/p><p><i><span style=\"font-weight: 400;\">Suppose \u221a2 = a\/b in lowest terms. Squaring gives a\u00b2 = 2b\u00b2, so a\u00b2 is even, which makes an even. Write a = 2k. Then 4k\u00b2 = 2b\u00b2, so b\u00b2 = 2k\u00b2 and b is even too. But now a and b share a factor of 2, which contradicts lowest terms. So \u221a2 can\u2019t be rational.<\/span><\/i><\/p><p><span style=\"font-weight: 400;\">Those few lines are enough, as long as you don\u2019t skip the \u201clowest terms\u201d idea or the reason an even square means an even number. Those two points are where most marks are won or lost, so don\u2019t rush them.<\/span><\/p><p>Read More:<\/p><ol><li id=\"132000-in-words-spelling-place-value-chart-easy-examples\" class=\"elementor-heading-title elementor-size-default\"><strong><a href=\"https:\/\/oratrics.com\/blogs\/maths\/132000-in-words\/\">132000 in Words<\/a><\/strong><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-93f61a6\" data-id=\"93f61a6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-638632c elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"638632c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0aa7cc2\" data-id=\"0aa7cc2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-44fe1d5\" data-id=\"44fe1d5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2e4a769 elementor-widget elementor-widget-heading\" data-id=\"2e4a769\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Why this actually works\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a1bcbe1 elementor-widget elementor-widget-text-editor\" data-id=\"a1bcbe1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Understanding the \u201cwhy\u201d matters more than memorising the steps, so let\u2019s zoom out for a second.<\/span><\/p><p><span style=\"font-weight: 400;\">The whole proof rests on one clash. A fraction in lowest terms can\u2019t have an even top and an even bottom. Yet the equation a\u00b2 = 2b\u00b2 forces both to be even. Those two facts can\u2019t live together, so something has to give. The only thing we chose freely was our starting assumption, so that\u2019s what collapses.<\/span><\/p><p><span style=\"font-weight: 400;\">It also shows why the trick doesn\u2019t work for \u221a4. Try it with a\u00b2 = 4b\u00b2 and you just get a = 2b, with no contradiction anywhere. That makes sense, because \u221a4 = 2 is perfectly rational. What makes 2 special is that it\u2019s a prime appearing just once on the right-hand side.<\/span><\/p><p><span style=\"font-weight: 400;\">The result is also very old. Ancient Greek mathematicians discovered that the diagonal of a square can\u2019t be expressed as a fraction of its side, and the argument above is the modern form of that discovery. It changed the way people thought about numbers.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-deabc50\" data-id=\"deabc50\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-90ad816 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"90ad816\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-dda4f9a\" data-id=\"dda4f9a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-1fd461e\" data-id=\"1fd461e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ac1eac5 elementor-widget elementor-widget-heading\" data-id=\"ac1eac5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Common mistakes<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7e41bd0 elementor-widget elementor-widget-text-editor\" data-id=\"7e41bd0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Students tend to trip over the same few things:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Forgetting \u201clowest terms.\u201d<\/b><span style=\"font-weight: 400;\"> Without it there\u2019s no contradiction, so state it right at the start.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Skipping why an even a\u00b2 means an even a.<\/b><span style=\"font-weight: 400;\"> Examiners usually want to see the odd-number check from Step 5, even if it\u2019s only one sentence.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Treating the calculator\u2019s decimal as evidence.<\/b><span style=\"font-weight: 400;\"> A long, random-looking decimal proves nothing. Only logic settles this.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Stopping too early.<\/b><span style=\"font-weight: 400;\"> Showing a is even is only half the job. You still need b to be even to get the clash.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Writing \u201cthis is impossible\u201d without saying why.<\/b><span style=\"font-weight: 400;\"> Spell out what clashes: a and b share a factor of 2, but they were supposed to share none.<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a37d3c7\" data-id=\"a37d3c7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-721e128 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"721e128\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9476ce9\" data-id=\"9476ce9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-df1f435\" data-id=\"df1f435\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3dc9c32 elementor-widget elementor-widget-heading\" data-id=\"3dc9c32\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Other ways to prove it<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-362dab9 elementor-widget elementor-widget-text-editor\" data-id=\"362dab9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The method above is the most popular, but it\u2019s not the only one.<\/span><\/p><p><b>Prime factorisation.<\/b><span style=\"font-weight: 400;\"> Every whole number greater than 1 breaks into primes in exactly one way. In a\u00b2 = 2b\u00b2, count how many times the prime 2 appears on each side. On the left, a\u00b2 has an even number of 2s, because squaring doubles every exponent. On the right, 2b\u00b2 has an odd number. An even count can\u2019t equal an odd count, so the equation fails. The same idea works neatly for \u221a3, \u221a5, \u221a6 and others.<\/span><\/p><p><b>The geometric method.<\/b><span style=\"font-weight: 400;\"> Picture the smallest whole-number square whose area is exactly double that of another whole-number square. A clever construction shows you can always find a smaller pair, which contradicts the word \u201csmallest.\u201d This idea is known as infinite descent.<\/span><\/p><p><span style=\"font-weight: 400;\">Different roads, same destination. That\u2019s one of the nice things about maths.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0ce9409\" data-id=\"0ce9409\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5b95101 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"5b95101\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-60b0b98\" data-id=\"60b0b98\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-f7b1ee6\" data-id=\"f7b1ee6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d6b7823 elementor-widget elementor-widget-heading\" data-id=\"d6b7823\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1abbae5 elementor-widget elementor-widget-text-editor\" data-id=\"1abbae5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Learning to prove that \u221a2 is irrational is a small step that teaches a big lesson: in maths, we don\u2019t trust patterns or approximations, we trust reasoning. Once this proof feels comfortable, you\u2019ll start to see that the proofs for the square roots of other primes follow the same pattern.<\/span><\/p><p><span style=\"font-weight: 400;\">Here\u2019s my suggestion: close this page, grab a blank sheet of paper, and write out the ten steps in your own words. If you can do it without peeking, you\u2019ve really got it.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5db7918\" data-id=\"5db7918\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c54a330 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"c54a330\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-31cc473\" data-id=\"31cc473\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e117d9f\" data-id=\"e117d9f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4fc4d09 elementor-widget elementor-widget-heading\" data-id=\"4fc4d09\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Frequently Asked Questions\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-70eddef elementor-widget elementor-widget-accordion\" data-id=\"70eddef\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1181\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1181\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Is root 2 a real number?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1181\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1181\"><p><span style=\"font-weight: 400;\">Yes. Every irrational number is real, so \u221a2 has its own spot on the number line, at roughly 1.414.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1182\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1182\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Can I use the same proof for root 3?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1182\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1182\"><p><span style=\"font-weight: 400;\">Yes. Swap 2 for 3 and use the fact that if a\u00b2 is divisible by 3, then a is too. The rest of the argument follows the same pattern.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1183\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1183\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Is 1.41421356 the exact value of root 2?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1183\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1183\"><p><span style=\"font-weight: 400;\">No, it\u2019s just a rounded approximation. The true decimal never ends and never falls into a repeating pattern.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1184\" class=\"elementor-tab-title\" data-tab=\"4\" role=\"button\" aria-controls=\"elementor-tab-content-1184\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Why can we trust proof by contradiction?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1184\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"4\" role=\"region\" aria-labelledby=\"elementor-tab-title-1184\"><p><span style=\"font-weight: 400;\">In logic, a statement is either true or false. If assuming it\u2019s true leads to something impossible, it can\u2019t be true.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Is root 2 a real number?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Yes. Every irrational number is real, so \\u221a2 has its own spot on the number line, at roughly 1.414.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Can I use the same proof for root 3?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Yes. Swap 2 for 3 and use the fact that if a\\u00b2 is divisible by 3, then a is too. The rest of the argument follows the same pattern.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Is 1.41421356 the exact value of root 2?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">No, it\\u2019s just a rounded approximation. The true decimal never ends and never falls into a repeating pattern.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Why can we trust proof by contradiction?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">In logic, a statement is either true or false. If assuming it\\u2019s true leads to something impossible, it can\\u2019t be true.<\\\/span><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ecc172c\" data-id=\"ecc172c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>How to Prove That Root 2 Is Irrational (Step by Step) If you\u2019ve ever wondered how to prove that \u221a2 [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":29914,"comment_status":"open","ping_status":"open","sticky":false,"template":"rttpg_full_width","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[9],"tags":[],"class_list":["post-29781","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-explorer-program"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Prove That Root 2 Is Irrational: Step-by-Step complete guide - Oratrics<\/title>\n<meta name=\"description\" content=\"Learn how to prove that \u221a2 is irrational with this step-by-step guide. 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