The square root of 23 is an irrational number because it cannot be expressed as a ratio of two integers. In other words, there are no two integers aaa and bbb such that 23=ab\\sqrt{23} = \\frac{a}{b}23\u200b=ba\u200b, where b\u22600b \\ne 0b\ue020=0 and the fraction is in simplest form.<\/p>\n\n\n\n
A number is considered irrational if its decimal expansion neither terminates nor repeats. For 23\\sqrt{23}23\u200b, the decimal form is approximately 4.79583152…, and it continues infinitely without forming any repeating pattern. This non-repeating, non-terminating nature confirms that the number is irrational.<\/p>\n\n\n\n
The rational root theorem, often used in number theory, indicates that if a square root of a natural number results in a rational number, then the original number must be a perfect square. Examples of perfect squares include 1, 4, 9, 16, 25, and so on. These numbers have whole numbers as their square roots: 16=4\\sqrt{16} = 416\u200b=4, 25=5\\sqrt{25} = 525\u200b=5, etc. However, 23 is not a perfect square. It lies between 16 and 25, whose square roots are 4 and 5 respectively. Therefore, 23\\sqrt{23}23\u200b must be between 4 and 5, but not equal to either.<\/p>\n\n\n\n
Furthermore, a proof by contradiction can demonstrate the irrationality of square roots of non-perfect square integers. Suppose 23\\sqrt{23}23\u200b were rational, and equal to ab\\frac{a}{b}ba\u200b in simplest terms. Squaring both sides would yield 23=a2b223 = \\frac{a^2}{b^2}23=b2a2\u200b, or a2=23b2a^2 = 23b^2a2=23b2. This implies that a2a^2a2 is divisible by 23, so aaa must also be divisible by 23. Letting a=23ka = 23ka=23k, and substituting back gives (23k)2=23b2(23k)^2 = 23b^2(23k)2=23b2, which leads to 529k2=23b2529k^2 = 23b^2529k2=23b2, or b2=23k2b^2 = 23k^2b2=23k2. This implies that bbb is also divisible by 23, contradicting the assumption that ab\\frac{a}{b}ba\u200b is in lowest terms. Hence, 23\\sqrt{23}23\u200b must be irrational.<\/p>\n\n\n\n Why is the square root of 23 an irrational number The Correct Answer and Explanation is: The square root of 23 is an irrational number because it cannot be expressed as a ratio of two integers. In other words, there are no two integers aaa and bbb such that 23=ab\\sqrt{23} = \\frac{a}{b}23\u200b=ba\u200b, where b\u22600b \\ne […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","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":"","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-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":"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":""},"mobile":{"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":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-236193","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=236193"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236193\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236193"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236193"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-654.jpeg\")
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