{"id":67235,"date":"2022-02-12T12:40:59","date_gmt":"2022-02-12T07:10:59","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=67235"},"modified":"2025-02-28T16:36:16","modified_gmt":"2025-02-28T11:06:16","slug":"study-material-maths-properties-of-rational-numbers","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-material\/maths\/properties-of-rational-numbers\/","title":{"rendered":"Properties of Rational 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\/study-material\/maths\/properties-of-rational-numbers\/#Closure_Property\" title=\"Closure Property\">Closure Property<\/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\/study-material\/maths\/properties-of-rational-numbers\/#Commutative_Property\" title=\"Commutative Property\">Commutative Property<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/maths\/properties-of-rational-numbers\/#Associative_Property\" title=\"Associative Property\">Associative Property<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/maths\/properties-of-rational-numbers\/#Distributive_Property\" title=\"Distributive Property\">Distributive Property<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/maths\/properties-of-rational-numbers\/#Identity_and_Inverse_Properties_of_Rational_Numbers\" title=\"Identity and Inverse Properties of Rational Numbers\">Identity and Inverse Properties of Rational Numbers<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Closure_Property\"><\/span><span style=\"font-size: 14pt;\">Closure Property<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>For two rational numbers, x, and y, the addition, subtraction, and multiplication results always yield a rational number. The Closure Property isn\u2019t applicable for the division as division by zero isn\u2019t defined. In other words, we can say that closure property is applicable for division too other than zero.<\/p>\n<ul style=\"list-style-type: square;\">\n<li>4\/7 + 2\/3 =26\/21<\/li>\n<li>4\/3 \u2013 2\/4 = 6\/12<\/li>\n<li>3\/5. 2\/3 = 6\/15<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Commutative_Property\"><\/span><span style=\"font-size: 14pt;\">Commutative Property<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Considering two rational numbers x y, the addition and multiplication are always commutative. Subtraction doesn\u2019t obey the commutative property. You can get a clear idea of this property by looking at the solved examples.<\/p>\n<ul style=\"list-style-type: square;\">\n<li>Commutative Law of Addition: x+y = y+x <strong>Ex:<\/strong> 1\/3+2\/3 = 3\/3<\/li>\n<li>Commutative Law of Multiplication: x.y = y.x <strong>Ex:<\/strong> 1\/2.2\/3 =2\/3.1\/2 =2\/6<\/li>\n<li>Subtraction x-y\u2260y-x <strong>Ex: <\/strong>4\/3-1\/3 = 3\/3 whereas 1\/3-4\/3=-3\/3<\/li>\n<li>Division isn\u2019t commutative x\/y \u2260y\/x <strong>Ex: <\/strong>3\/9\u00f71\/2=6\/9 whereas 1\/2 \u00f73\/9 =9\/6<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Associative_Property\"><\/span><span style=\"font-size: 14pt;\">Associative Property<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z<\/p>\n<p>whereas for Multiplication x(yz)=(xy)z<\/p>\n<p><strong>Ex:<\/strong> 1\/3 + (1\/4 + 3\/3) = (1\/3+ 1\/4) + 3\/3<\/p>\n<p>\u21d219\/12 =19\/12<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Distributive_Property\"><\/span><span style=\"font-size: 14pt;\">Distributive Property<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.<\/p>\n<p><strong>Ex:<\/strong> 1\/3.(1\/4+2\/5) =(1\/3.1\/4)+(1\/3.2\/5)<\/p>\n<p>1\/3. (17\/20)= 1\/12+2\/10<\/p>\n<p>17\/60 =17\/60<\/p>\n<p>Thus, L.H.S = R.H.S<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Identity_and_Inverse_Properties_of_Rational_Numbers\"><\/span>Identity and Inverse Properties of Rational Numbers<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Identity Property:<\/strong> We know 0 is called Additive Identity, and 1 is called Multiplicative Identity of Rational Numbers.<\/p>\n<p><strong>Ex:<\/strong> 1\/4+0 = 1\/4(Additive Identity)<\/p>\n<p>5\/3.1 = 5\/3(Multiplicative Identity)<\/p>\n<p><strong>Inverse Property:<\/strong> For a Rational Number, the x\/y additive inverse is -x\/y, and the multiplicative inverse is y\/x.<\/p>\n<p><strong>Ex:<\/strong> Additive Inverse of 2\/3 is -2\/3<\/p>\n<p>Multiplicative Inverse of 4\/5 is 5\/4<\/p>\n<p>You need to be aware of a few other properties of Rational Numbers, and they are explained below.<\/p>\n<p><strong>Property 1:<\/strong><\/p>\n<p>If a\/b is a rational number and m is a non-zero integer then a\/b =(a*m)\/(b*m).<\/p>\n<p>In other words, we can say that the rational number remains unaltered if we multiply both the numerator and denominator with the same integer.<\/p>\n<p><strong>Ex:<\/strong> 2\/3 = 2*2\/3*2 = 4\/6, 2*3\/3*3 = 6\/9, 2*4\/3*4 = 8\/12\u2026.<\/p>\n<p><strong>Property 2:<\/strong><\/p>\n<p>If a\/b is a rational number and m is a common divisor then a\/b = (a\u00f7m)\/(b\u00f7m)<\/p>\n<p>The rational number remains unchanged when dividing the numerator and denominator of a rational number with a common divisor.<\/p>\n<p><strong>Ex:<\/strong> 36\/42 =36\u00f76\/42\u00f76 = 6\/7<\/p>\n<p><strong>Property 3:<\/strong><\/p>\n<p>Consider a\/b, c\/d to be two rational numbers.<\/p>\n<p>Then a\/b = c\/d \u21d2 a*d = b*c<\/p>\n<p><strong>Ex:<\/strong> 2\/4 =4\/8 \u21d2 2.8=4.4<\/p>\n<p><strong>Property 4:<\/strong><\/p>\n<p>For every Rational Number n, any of the following conditions hold.<\/p>\n<p>(i) n&gt;0, (ii) n=0, (iii) n&lt;0<\/p>\n<p><strong>Ex:<\/strong> 3\/4 is greater than 0.<\/p>\n<p>0\/5 is equal to 0.<\/p>\n<p>-3\/4 is less than 0.<\/p>\n<p><strong>Property 5:<\/strong><\/p>\n<p>For any two rational numbers a, b, any one condition is true<\/p>\n<p>(i) a&gt;b, (ii) a=b, (iii) a&lt;b<\/p>\n<p><strong>Ex: <\/strong>2\/3 and 2\/5 are two rational numbers, and 2\/3 is greater than 2\/5<\/p>\n<p>If 4\/8 and 8\/16 are two rational numbers, then 4\/8 = 8\/16<\/p>\n<p>If -4\/7 and 3\/4 are two rational numbers, then -4\/7 is less than 3\/4<\/p>\n<p><strong>Property 6:<\/strong><\/p>\n<p>In the case of three rational numbers a &gt; b, b &gt; c, then a&gt;c<\/p>\n<p>If 4\/5, 16\/30, and -8\/15 are three rational numbers, then 4\/5 &gt;16\/30, and 16\/30 is greater than -8\/15, then 4\/5 is also greater than -8\/15.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Closure Property For two rational numbers, x, and y, the addition, subtraction, and multiplication results always yield a rational number. 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