{"id":156558,"date":"2022-03-26T01:53:23","date_gmt":"2022-03-25T20:23:23","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/irrational-numbers-3\/"},"modified":"2024-12-13T13:43:08","modified_gmt":"2024-12-13T08:13:08","slug":"irrational-numbers-3","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/","title":{"rendered":"Irrational Numbers"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Irrational_Numbers\" title=\"Irrational Numbers\">Irrational Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Irrational_Numbers_Definition\" title=\"Irrational Numbers Definition\">Irrational Numbers Definition<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#How_do_you_know_a_number_is_irrational\" title=\"How do you know a number is irrational?\">How do you know a number is irrational?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Is_Pi_an_irrational_number\" title=\"Is Pi an irrational number?\">Is Pi an irrational number?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Symbol\" title=\"Symbol\">Symbol<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Properties\" title=\"Properties\">Properties<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#List_of_Irrational_Numbers\" title=\"List of Irrational Numbers\">List of Irrational Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Are_Irrational_Numbers_Real_Numbers\" title=\"Are Irrational Numbers Real Numbers?\">Are Irrational Numbers Real Numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Sum_and_Product_of_Two_Irrational_Numbers\" title=\"Sum and Product of Two Irrational Numbers\">Sum and Product of Two Irrational Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Irrational_Number_Proof\" title=\"Irrational Number Proof\">Irrational Number Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#How_to_Find_an_Irrational_Number\" title=\"How to Find an Irrational Number?\">How to Find an Irrational Number?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#Problems_and_Solutions\" title=\"Problems and Solutions\">Problems and Solutions<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Irrational_Numbers\"><\/span>Irrational Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Irrational Numbers- Introduction: <b>Irrational numbers<\/b> are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio, such as p\/q, where p and q are integers, q\u22600. It is a contradiction of <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/rational-numbers\/\" target=\"_blank\" rel=\"noopener\">rational numbers<\/a>.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156557 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg\" alt=\"Irrational Numbers\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg?v=1648239802 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers-300x212.jpg?v=1648239802 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<p>Irrational numbers are expressed usually in the form of R\\Q, where the backward slash symbol denotes \u2018set minus\u2019. it can also be expressed as R \u2013 Q, which states the difference between a set of real numbers and a set of rational numbers.<\/p>\n<p>The calculations based on these numbers are a bit complicated. For example, \u221a5, \u221a11, \u221a21, etc., are irrational. If such numbers are used in arithmetic operations, then first we need to evaluate the values under root. These values could be sometimes recurring also. Now let us find out its definition, lists of irrational numbers, how to find them, etc., in this article.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Irrational_Numbers_Definition\"><\/span>Irrational Numbers Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An<b> irrational number<\/b> is a real number that cannot be expressed as a ratio of integers, for example, \u221a 2 is an irrational number. Again, the decimal expansion of an <b>irrational number is <\/b><b><i>neither terminating nor recurring<\/i><\/b>.<\/p>\n<p><b>Irrational Meaning:<\/b> The meaning of irrational is not having a ratio or no ratio can be written for that number. That means the number which cannot be expressed other than by means of roots. In other words, we can say that irrational numbers cannot be represented as the ratio of two integers.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"How_do_you_know_a_number_is_irrational\"><\/span>How do you know a number is irrational?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The real numbers which cannot be expressed in the form of p\/q, where p and q are integers and q \u2260 0 are known as irrational numbers. For example  \u221a 2 and \u221a 3 etc. are irrational. Whereas any number which can be represented in the form of p\/q, such that, p and q are integers and q \u2260 0 is known as a rational number.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Is_Pi_an_irrational_number\"><\/span>Is Pi an irrational number?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Pi (\u03c0) is an irrational number because it is non-terminating. The approximate value of pi is 22\/7. Also, the value of \u03c0 is 3.14159 26535 89793 23846 264\u2026<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Symbol\"><\/span>Symbol<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Generally, the symbol used to represent the irrational symbol is \u201cP\u201d.  Since the irrational numbers are defined negatively, the set of real numbers (R) that are not the rational number (Q), is called an irrational number. The symbol P is often used because of the association with the real and rational number. (i.e) because of the alphabetic sequence P, Q, R. But mostly, it is represented using the set difference of the real minus rationals, in a way R- Q or R\\Q.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Properties\"><\/span>Properties<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Since irrational numbers are the subsets of the real numbers, irrational numbers will obey all the properties of the real number system. The following are the properties of irrational numbers:<\/p>\n<ul>\n<li>The addition of an irrational number and a rational number gives an irrational number.  For example, let us assume that x is an irrational number, y is a rational number and the addition of both the numbers x +y gives a rational number z.<\/li>\n<li>Multiplication of any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational, then x =z\/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.<\/li>\n<li>The least common multiple (LCM) of any two irrational numbers may or may not exist.<\/li>\n<li>The addition or the multiplication of two irrational numbers may be rational; for example, \u221a2. \u221a2 = 2. Here, \u221a2 is an irrational number. If it is multiplied twice, then the final product obtained is a rational number. (i.e) 2.<\/li>\n<li>The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.<\/li>\n<\/ul>\n<h2>List of Irrational Numbers<\/h2>\n<p>The famous irrational numbers consist of Pi, Euler\u2019s number, Golden ratio. Many square roots and cube roots numbers are also irrational, but not all of them. For example, \u221a3 is an irrational number but \u221a4 is a rational number. Because 4 is a perfect square, such as 4 = 2 x 2 and \u221a4 = 2, which is a rational number.  It should be noted that there are infinite irrational numbers between any two real numbers. For example, say 1 and 2, there are infinitely many irrational numbers between 1 and 2. Now, let us have a look at the values of famous irrational numbers.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table-bordered\">\n<tbody>\n<tr>\n<td>Pi, \u03c0<\/td>\n<td>3.14159265358979\u2026<\/td>\n<\/tr>\n<tr>\n<td>Euler\u2019s Number, e<\/td>\n<td>2.71828182845904\u2026<\/td>\n<\/tr>\n<tr>\n<td>Golden ratio, \u03c6<\/td>\n<td>1.61803398874989\u2026.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"Are_Irrational_Numbers_Real_Numbers\"><\/span>Are Irrational Numbers Real Numbers?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers. It means that irrational numbers cannot be expressed as the ratio of two numbers. The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Sum_and_Product_of_Two_Irrational_Numbers\"><\/span>Sum and Product of Two Irrational Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Now, let us discuss the sum and the product of the irrational numbers.<\/p>\n<p><b>Product of Two Irrational Numbers<\/b><\/p>\n<p><b>Statement: <\/b>The product of two irrational numbers is sometimes rational or irrational<\/p>\n<p>For example, \u221a2 is an irrational number, but when \u221a2  is multiplied by \u221a2, we get the result 2, which is a rational number.<\/p>\n<p>(i.e.,) \u221a2 x \u221a2 = 2<\/p>\n<p>We know that \u03c0 is also an irrational number, but if \u03c0 is multiplied by \u03c0, the result is \u03c0<sup>2<\/sup>, which is also an irrational number.<\/p>\n<p>(i.e..) \u03c0 x \u03c0 = \u03c0<sup>2<\/sup><\/p>\n<p>It should be noted that while multiplying the two irrational numbers, it may result in an irrational number or a rational number.<\/p>\n<p><b>Sum of Two Irrational Numbers<\/b><\/p>\n<p><b>Statement: <\/b>The sum of two irrational numbers is sometimes rational or irrational.<\/p>\n<p>Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.<\/p>\n<p>For example, if we add two irrational numbers, say 3\u221a2+ 4\u221a3, a sum is an irrational number.<\/p>\n<p>But, let us consider another example, (3+4\u221a2) + (-4\u221a2 ), the sum is 3, which is a rational number.<\/p>\n<p>So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational number.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Irrational_Number_Proof\"><\/span>Irrational Number Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The following theorem is used to prove the above statement<\/p>\n<p><strong>Theorem<\/strong>: Given <em>p<\/em> is a prime number and <span class=\"latex-for-amp\">a<sup>2<\/sup><\/span> is divisible by <em>p, <\/em>(where <em>a<\/em> is any positive integer), then it can be concluded that p also divides <em>a<\/em>.<\/p>\n<p><strong>Proof:<\/strong> Using the Fundamental Theorem of Arithmetic, the positive integer can be expressed in the form of the product of its primes as:<\/p>\n<p>a = p<sub>1<\/sub> \u00d7 p<sub>2 <\/sub>\u00d7 p<sub>3\u2026\u2026\u2026..  <\/sub>\u00d7 p<sub>n \u2026..(1)<\/sub><\/p>\n<p>Where, p<sub>1,<\/sub> p<sub>2<\/sub><sub>, <\/sub>p<sub>3<\/sub>,<sub> \u2026\u2026, <\/sub>p<sub>n<\/sub> represent all the prime factors of <em>a<\/em>.<\/p>\n<p>Squaring both the sides of equation (1),<\/p>\n<p>a<sup>2<\/sup> = ( p<sub>1<\/sub> \u00d7 p<sub>2 <\/sub>\u00d7 p<sub>3\u2026\u2026\u2026..  <\/sub>\u00d7 p<sub>n) (<\/sub> p<sub>1<\/sub> \u00d7 p<sub>2<\/sub> \u00d7 p<sub>3<\/sub>\u2026\u2026\u2026..  \u00d7 p<sub>n<\/sub>)<\/p>\n<p>\u21d2a<sup>2<\/sup> = (p<sub>1<\/sub>)<sup>2<\/sup> \u00d7 (p<sub>2<\/sub>)<sup>2<\/sup><sub> <\/sub>\u00d7 (<sup>p<\/sup><sub><sup>3<\/sup><\/sub><sup> <\/sup>)<sup>2<\/sup><sub><sup>\u2026\u2026\u2026..<\/sup><\/sub>\u00d7 (p<sub>n<\/sub>)<sup>2<\/sup><\/p>\n<p>The only prime factors of <span class=\"latex-for-amp\">a<sup>2<\/sup><\/span> are p<sub>1<\/sub>, p<sub>2,<\/sub> p<sub>3\u2026\u2026\u2026..,<\/sub> p<sub>n<\/sub>. If <em>p<\/em> is a prime number and a factor of <span class=\"latex-for-amp\">a<sup>2<\/sup><\/span>, then p is one of  p<sub>1<\/sub>, p<sub>2 ,<\/sub> p<sub>3\u2026\u2026\u2026..,<\/sub> p<sub>n<\/sub>. So, <em>p<\/em> will also be a factor of <em>a<\/em>.<\/p>\n<p>Hence, if <span class=\"latex-for-amp\">a<sup>2<\/sup><\/span>  is divisible by <em>p<\/em>, then <em>p<\/em> also divides <em>a<\/em>.<\/p>\n<p>Now, using this theorem, we can prove that <strong>\u221a<\/strong> 2 is irrational.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_Find_an_Irrational_Number\"><\/span>How to Find an Irrational Number?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us find the irrational numbers between 2 and 3.<br \/>\nWe know, square root of 4 is 2; \u221a4 =2<br \/>\nand the square root of 9 is 3; \u221a9 = 3<br \/>\nTherefore, the number of irrational numbers between 2 and 3 are \u221a5, \u221a6, \u221a7, and \u221a8, as these are not perfect squares and cannot be simplified further. Similarly, you can also find the irrational numbers, between any other two perfect square numbers.<\/p>\n<p><strong>Another case:<\/strong><\/p>\n<p>Let us assume a case of <strong>\u221a<\/strong>2. Now, how can we find if <strong>\u221a<\/strong>2 is an irrational number?<\/p>\n<p>Suppose, <strong>\u221a<\/strong>2 is a rational number. Then, by the definition of rational numbers, it can be written that,<\/p>\n<p><strong>\u221a<\/strong> 2 =p\/q    \u2026\u2026.(1)<\/p>\n<p>Where <em>p<\/em> and <em>q<\/em> are co-prime integers and <em>q<\/em> \u2260 0 (Co-prime numbers are those numbers whose common factor is 1).<\/p>\n<p>Squaring both the sides of equation (1), we have<\/p>\n<p>2 = p<sup>2<\/sup>\/q<sup>2<\/sup><\/p>\n<p>\u21d2 p<sup>2<\/sup> = 2 q <sup>2<\/sup>    \u2026\u2026\u2026. (2)<\/p>\n<p>From the theorem stated above, if 2 is a prime factor of <span class=\"latex-for-amp\">p<sup>2<\/sup><\/span>, then 2 is also a prime factor of <em>p<\/em>.<\/p>\n<p>So, <em>p <\/em>= 2 \u00d7 <em>c<\/em>, where <em>c<\/em> is an integer.<\/p>\n<p>Substituting this value of <em>p<\/em> in equation (3), we have<\/p>\n<p><span class=\"latex-for-amp\">(2c)<sup>2<\/sup><\/span> = 2 q <sup>2<\/sup><\/p>\n<p>\u21d2 <span class=\"latex-for-amp\">q<sup>2<\/sup><\/span> = <span class=\"latex-for-amp\">2c <sup>2<\/sup> <\/span><\/p>\n<p>This implies that 2 is a prime factor of <span class=\"latex-for-amp\">q<sup>2<\/sup><\/span> also. Again from the theorem, it can be said that 2 is also a prime factor of <em>q<\/em>.<\/p>\n<p>According to the initial assumption, <em>p<\/em> and <em>q<\/em> are co-primes but the result obtained above contradicts this assumption as <em>p<\/em> and <em>q<\/em> have 2 as a common prime factor other than 1. This contradiction arose due to the incorrect assumption that <strong>\u221a<\/strong>2  is rational.<\/p>\n<p><strong>So, root 2 is irrational.<\/strong><\/p>\n<p>Similarly, we can justify the statement discussed in the beginning that if <em>p<\/em> is a prime numbers  then <strong>\u221a<\/strong> p  is an irrational number. Similarly, it can be proved that for any prime number <em>p<\/em>,<strong>\u221a<\/strong> p is irrational<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Problems_and_Solutions\"><\/span>Problems and Solutions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Question 1<\/strong>: <strong>Which of the following are Rational Numbers or Irrational Numbers?<\/strong><\/p>\n<p>2, -.45678\u2026, 6.5, <strong>\u221a<\/strong> 3, <strong>\u221a<\/strong> 2<\/p>\n<p><strong>Solution<\/strong>: Rational Numbers \u2013 2, 6.5 as these have terminating decimals.<\/p>\n<p>Irrational Numbers \u2013 -.45678\u2026, <strong>\u221a<\/strong> 3, <strong>\u221a<\/strong> 2 as these have a non-terminating non-repeating decimal expansion.<\/p>\n<p><b>Question 2:<\/b><strong> Check if below numbers are rational or irrational. <\/strong><\/p>\n<p>2, 5\/11, -5.12, 0.31<\/p>\n<p><b>Solution:<\/b> Since the decimal expansion of a rational number either terminates or repeats. So, 2, 5\/11, -5.12, 0.31 are all rational numbers.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Irrational Numbers Irrational Numbers- Introduction: Irrational numbers are the real numbers that cannot be represented as a simple fraction. It [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Irrational Numbers- Introduction","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.","custom_permalink":"maths\/irrational-numbers\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Irrational Numbers - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Irrational Numbers - Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"og:description\" content=\"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-25T20:23:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-12-13T08:13:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"vipin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Irrational Numbers - Infinity Learn by Sri Chaitanya","description":"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/","og_locale":"en_US","og_type":"article","og_title":"Irrational Numbers - Infinity Learn by Sri Chaitanya","og_description":"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.","og_url":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2022-03-25T20:23:23+00:00","article_modified_time":"2024-12-13T08:13:08+00:00","og_image":[{"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"vipin","Est. reading time":"8 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#primaryimage","inLanguage":"en-US","url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg?v=1648239802","contentUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg?v=1648239802","width":606,"height":428},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/","name":"Irrational Numbers - Infinity Learn by Sri Chaitanya","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"primaryImageOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#primaryimage"},"datePublished":"2022-03-25T20:23:23+00:00","dateModified":"2024-12-13T08:13:08+00:00","description":"Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Irrational Numbers"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2"},"headline":"Irrational Numbers","datePublished":"2022-03-25T20:23:23+00:00","dateModified":"2024-12-13T08:13:08+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#webpage"},"wordCount":1485,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/irrational-numbers\/#primaryimage"},"thumbnailUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/irrational-numbers.jpg","articleSection":["Maths"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2","name":"vipin","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","caption":"vipin"},"sameAs":["http:\/\/surge.infinitylearn.com"],"url":"https:\/\/infinitylearn.com\/surge\/author\/vipin\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/156558"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=156558"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/156558\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=156558"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=156558"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=156558"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=156558"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}