{"id":4023,"date":"2023-02-01T06:54:06","date_gmt":"2023-02-01T06:54:06","guid":{"rendered":"https:\/\/theexampillar.com\/ncert\/?p=4023"},"modified":"2023-02-01T06:54:06","modified_gmt":"2023-02-01T06:54:06","slug":"ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-3","status":"publish","type":"post","link":"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-3\/","title":{"rendered":"NCERT Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.3"},"content":{"rendered":"<h2 style=\"text-align: center;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">NCERT Solutions Class 9 Maths\u00a0<\/span><br \/>\n<span style=\"color: #000000; font-family: georgia, palatino, serif;\">Chapter &#8211; 1 (Number Systems )\u00a0<\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">The NCERT Solutions in English Language for Class 9 Mathematics\u00a0<strong>Chapter &#8211; 1 Number Systems\u00a0<\/strong>Exercise 1.3 has been provided here to help the students in solving the questions from this exercise.\u00a0<\/span><\/p>\n<p><span style=\"color: #000000;\"><strong>Chapter 1: Number Systems<\/strong><\/span><\/p>\n<ul>\n<li><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-1\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 1.1<\/a><\/li>\n<li><span style=\"color: #000000;\"><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-2\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 1.2<\/a><\/span><\/li>\n<li><span style=\"color: #000000;\"><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-4\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 1.4<\/a><\/span><\/li>\n<li><span style=\"color: #000000;\"><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-5\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 1.5<\/a><\/span><\/li>\n<li><span style=\"color: #000000;\"><a href=\"https:\/\/theexampillar.com\/ncert\/ncert-solutions-class-9-maths-chapter-1-number-systems-ex-1-6\" target=\"_blank\" rel=\"noopener\">NCERT Solution Class 9 Maths Ex &#8211; 1.6<\/a><\/span><\/li>\n<\/ul>\n<h2 style=\"text-align: center;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Exercise &#8211; 1.3\u00a0<\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>1. Write the following in decimal form and say what kind of decimal expansion each has :<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4188\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"579\" height=\"117\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1.png 579w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1-300x61.png 300w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1-480x97.png 480w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><\/strong><br \/>\n<strong>Answer &#8211;<\/strong><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(i) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{36}{100}}\" alt=\"\\mathbf{\\frac{36}{100}}\" align=\"absmiddle\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4194\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1i.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"112\" height=\"172\" \/><br \/>\n<\/strong>= 0.36 (Terminating)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(ii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{1}{11}}\" alt=\"\\mathbf{\\frac{1}{11}}\" align=\"absmiddle\" \/> <\/strong><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4192\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1ii.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"96\" height=\"229\" \/><br \/>\n<\/strong>= 0.0909&#8230; = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{09}\" alt=\"0.\\overline{09}\" align=\"absmiddle\" \/> (Non Terminating and repeating)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(iii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{4\\frac{1}{8}}\" alt=\"\\mathbf{4\\frac{1}{8}}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{33}{8}}\" alt=\"\\mathbf{\\frac{33}{8}}\" align=\"absmiddle\" \/><\/strong><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4193\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iii.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"109\" height=\"217\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iii.png 158w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iii-151x300.png 151w\" sizes=\"auto, (max-width: 109px) 100vw, 109px\" \/><br \/>\n= 4.125 (Terminating)<\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(iv) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{3}{13}}\" alt=\"\\mathbf{\\frac{3}{13}}\" align=\"absmiddle\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4191\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iv.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"150\" height=\"304\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iv.png 204w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1iv-148x300.png 148w\" sizes=\"auto, (max-width: 150px) 100vw, 150px\" \/><br \/>\n<\/strong> = 0.230769&#8230; = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{230769}\" alt=\"0.\\overline{230769}\" align=\"absmiddle\" \/> (Non Terminating and repeating)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(v)<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{2}{11}}\" alt=\"\\mathbf{\\frac{2}{11}}\" align=\"absmiddle\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4189\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1v.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"105\" height=\"194\" \/><\/strong><br \/>\n= 0.1818&#8230; = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{18}\" alt=\"0.\\overline{18}\" align=\"absmiddle\" \/> (Non Terminating and repeating)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>(vi) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{329}{400}}\" alt=\"\\mathbf{\\frac{329}{400}}\" align=\"absmiddle\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4190\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Q1vi.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"155\" height=\"195\" \/><br \/>\n<\/strong>= 0.8225 (Terminating)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><strong>2. You know that <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{1}{7}}\" alt=\"\\mathbf{\\frac{1}{7}}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.\\overline{142857}}\" alt=\"\\mathbf{0.\\overline{142857}}\" align=\"absmiddle\" \/>\u00a0Can you predict what the decimal expansions of <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{2}{7},&amp;space;\\frac{3}{7},\\frac{4}{7},\\frac{5}{7},\\frac{6}{7}}\" alt=\"\\mathbf{\\frac{2}{7}, \\frac{3}{7},\\frac{4}{7},\\frac{5}{7},\\frac{6}{7}}\" align=\"absmiddle\" \/> <\/strong><strong>are, without actually doing the long division? If so, how?<br \/>\n<\/strong><strong>[Hint: Study the remainders while finding the value of <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{1}{7}}\" alt=\"\\mathbf{\\frac{1}{7}}\" align=\"absmiddle\" \/> carefully.]<br \/>\n<\/strong><strong>Answer &#8211;<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{7}&amp;space;=&amp;space;0.\\overline{142857}\" alt=\"\\frac{1}{7} = 0.\\overline{142857}\" align=\"absmiddle\" \/><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{7}&amp;space;=&amp;space;2\\times&amp;space;\\frac{1}{7}&amp;space;=\" alt=\"\\frac{2}{7} = 2\\times \\frac{1}{7} =\" align=\"absmiddle\" \/> 2 \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{142857}\" alt=\"0.\\overline{142857}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{285714}\" alt=\"0.\\overline{285714}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{3}{7}&amp;space;=&amp;space;3\\times&amp;space;\\frac{1}{7}&amp;space;=\" alt=\"\\frac{3}{7} = 3\\times \\frac{1}{7} =\" align=\"absmiddle\" \/> 3 \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{142857}\" alt=\"0.\\overline{142857}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{428571}\" alt=\"0.\\overline{428571}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{7}&amp;space;=&amp;space;4\\times&amp;space;\\frac{1}{7}&amp;space;=\" alt=\"\\frac{4}{7} = 4\\times \\frac{1}{7} =\" align=\"absmiddle\" \/> 4 \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{142857}\" alt=\"0.\\overline{142857}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{571428}\" alt=\"0.\\overline{571428}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{5}{7}&amp;space;=&amp;space;5\\times&amp;space;\\frac{1}{7}&amp;space;=\" alt=\"\\frac{5}{7} = 5\\times \\frac{1}{7} =\" align=\"absmiddle\" \/> 5 \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{142857}\" alt=\"0.\\overline{142857}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{714285}\" alt=\"0.\\overline{714285}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: georgia, palatino, serif; color: #000000;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{6}{7}&amp;space;=&amp;space;6\\times&amp;space;\\frac{1}{7}&amp;space;=\" alt=\"\\frac{6}{7} = 6\\times \\frac{1}{7} =\" align=\"absmiddle\" \/> 6 \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{142857}\" alt=\"0.\\overline{142857}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{857142}\" alt=\"0.\\overline{857142}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>3. Express the following in the form <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{p}{q}}\" alt=\"\\mathbf{\\frac{p}{q}}\" align=\"absmiddle\" \/>, where p and q are integers and q \u2260 0.<\/strong><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\"><strong>(i) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.\\overline{6}}\" alt=\"\\mathbf{0.\\overline{6}}\" align=\"absmiddle\" \/><br \/>\n(ii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.4\\overline{7}}\" alt=\"\\mathbf{0.4\\overline{7}}\" align=\"absmiddle\" \/><br \/>\n(iii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.\\overline{001}}\" alt=\"\\mathbf{0.\\overline{001}}\" align=\"absmiddle\" \/><br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\"><strong>Answer &#8211;<\/strong><br \/>\n<strong>(i) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.\\overline{6}}\" alt=\"\\mathbf{0.\\overline{6}}\" align=\"absmiddle\" \/> = <\/strong>0.666..<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Assume that \u00a0<\/span><em style=\"font-family: georgia, palatino, serif;\">x<\/em><span style=\"font-family: georgia, palatino, serif;\"> = 0.666\u2026\u00a0 \u00a0 \u00a0 \u00a0 &#8212;&#8212;&#8212;&#8212;- (i)\u00a0<\/span><br \/>\n<span style=\"font-family: georgia, palatino, serif;\">Multiplying both sides of the equation (i) by 10<\/span><span style=\"font-family: georgia, palatino, serif;\">,<\/span><span style=\"font-family: georgia, palatino, serif;\"><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Then,<br \/>\n10<em>x<\/em> = 6.666\u2026\u00a0 &#8212;&#8212;&#8212;&#8212;- (ii)<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Subtracting the equation (i) from (ii)<\/span><span style=\"font-family: georgia, palatino, serif;\">,<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">\u00a0 10x = 6.666\u2026<br \/>\n<span style=\"text-decoration: underline;\">&#8211;\u00a0 \u00a0 x = 0.666\u2026\u00a0<\/span><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">9<em>x<\/em> = 6<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Therefore, x = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{3}\" alt=\"\\frac{2}{3}\" align=\"absmiddle\" \/><br \/>\nSo, the decimal number becomes<br \/>\n<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{6}\" alt=\"0.\\overline{6}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{3}\" alt=\"\\frac{2}{3}\" align=\"absmiddle\" \/> and it is in the required <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{p}{q}\" alt=\"\\frac{p}{q}\" align=\"absmiddle\" \/> form.<br \/>\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(ii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.4\\overline{7}}\" alt=\"\\mathbf{0.4\\overline{7}}\" align=\"absmiddle\" \/> = 0.47777&#8230;..<br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">Assume that\u00a0<em>x<\/em> = 0.4777\u2026 &#8212;&#8212;&#8212;&#8212;- (i)<br \/>\nMultiplying both sides of the equation (i) by 10,<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Then,<br \/>\n10<em>x<\/em> = 4.777\u2026 \u00a0 &#8212;&#8212;&#8212;&#8212;- (ii)<br \/>\nSubtracting the equation (i) from (ii),<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">\u00a010<em>x<\/em> = 4.777\u2026<br \/>\n<span style=\"text-decoration: underline;\"><em>&#8211;\u00a0 \u00a0x<\/em> = 0.4777\u2026<\/span><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">9<em>x<\/em> = 4.3<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Therefore, <em>x<\/em>\u00a0= <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{4.3}{9}\" alt=\"\\frac{4.3}{9}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{4.3}{9}\" alt=\"\\frac{4.3}{9}\" align=\"absmiddle\" \/> \u00d7 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{10}{10}\" alt=\"\\frac{10}{10}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{43}{10}\" alt=\"\\frac{43}{10}\" align=\"absmiddle\" \/><br \/>\nSo, the decimal number becomes<br \/>\n<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.4\\overline{7}\" alt=\"0.4\\overline{7}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{43}{10}\" alt=\"\\frac{43}{10}\" align=\"absmiddle\" \/>\u00a0and it is in the required <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{p}{q}\" alt=\"\\frac{p}{q}\" align=\"absmiddle\" \/> form.<br \/>\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(iii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{0.\\overline{001}}\" alt=\"\\mathbf{0.\\overline{001}}\" align=\"absmiddle\" \/> = 0.001001&#8230;<br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">Assume that \u00a0<em>x<\/em> = 0.001001\u2026 &#8212;&#8212;&#8212;&#8212;- (i)<br \/>\nMultiplying both sides of the equation (i) by 1000,<br \/>\nThen,<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Then, 1000<em>x<\/em> = 1.001001\u2026 &#8212;&#8212;&#8212;&#8212;- (ii)<br \/>\nSubtracting the equation (i) from (ii)<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">1000<em>x<\/em> = 1.001001..<br \/>\n<span style=\"text-decoration: underline;\"><em>&#8211;\u00a0 \u00a0 \u00a0 \u00a0x<\/em> = 0.001001<br \/>\n<\/span><\/span><span style=\"font-family: georgia, palatino, serif;\">999<em>x<\/em> = 1<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Therefore, <em>x<\/em> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{999}\" alt=\"\\frac{1}{999}\" align=\"absmiddle\" \/><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">So, the decimal number becomes<br \/>\n<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{001}\" alt=\"0.\\overline{001}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{999}\" alt=\"\\frac{1}{999}\" align=\"absmiddle\" \/>\u00a0and it is in the required <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{p}{q}\" alt=\"\\frac{p}{q}\" align=\"absmiddle\" \/> form.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>4. Express 0.99999\u2026. in the form <span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{p}{q}}\" alt=\"\\mathbf{\\frac{p}{q}}\" align=\"absmiddle\" \/><\/span><\/span>. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.<br \/>\n<\/strong><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>Answer &#8211;<\/strong> \u00a0<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Assume that\u00a0<em>x<\/em> = 0.9999 \u2026..\u00a0 &#8212;&#8212;&#8212;&#8212;&#8211; (i)<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Multiplying both sides by 10,<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">10<em>x<\/em> = 9.9999\u2026. &#8212;&#8212;&#8212;&#8212;&#8211; (ii)<br \/>\n<span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\">Subtracting the equation (i) from (ii)<\/span><\/span><br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">10<em>x<\/em> = 9.9999..<br \/>\n<\/span><span style=\"text-decoration: underline;\"><span style=\"color: #000000; font-family: georgia, palatino, serif; text-decoration: underline;\">\u2013 <em>x<\/em> = 0.9999..<br \/>\n<\/span><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">9<em>x<\/em>\u00a0= 9<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\">Therefore, <\/span><\/span>x = 1<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">The difference between 1 and 0.999999 is 0.000001 which is negligible.<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the answer can be justified.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{1}{17}}\" alt=\"\\mathbf{\\frac{1}{17}}\" align=\"absmiddle\" \/> ? Perform the division to check your answer.<br \/>\n<\/strong><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>Answer &#8211;<\/strong> <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{17}\" alt=\"\\frac{1}{17}\" align=\"absmiddle\" \/><br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Dividing 1 by 17:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4195\" src=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Ans5.png\" alt=\"NCERT Class 9 Solutions Maths\" width=\"314\" height=\"616\" srcset=\"https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Ans5.png 314w, https:\/\/theexampillar.com\/ncert\/wp-content\/uploads\/2023\/01\/NCERT-Solutions-Class-9-Maths-Ex-1.3-Ans5-153x300.png 153w\" sizes=\"auto, (max-width: 314px) 100vw, 314px\" \/><br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{17}\" alt=\"\\frac{1}{17}\" align=\"absmiddle\" \/> = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?0.\\overline{0588235294117647}\" alt=\"0.\\overline{0588235294117647}\" align=\"absmiddle\" \/><br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">There are 16 digits in the repeating block of the decimal expansion of <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{17}\" alt=\"\\frac{1}{17}\" align=\"absmiddle\" \/>.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>6. Look at several examples of rational numbers in the form <span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{p}{q}}\" alt=\"\\mathbf{\\frac{p}{q}}\" align=\"absmiddle\" \/><\/span><\/span> (q \u2260 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy? <\/strong><br \/>\n<\/span><strong><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Answer &#8211;\u00a0 <\/span><\/strong><span style=\"color: #000000; font-family: georgia, palatino, serif;\">We observe that when q is\u00a02, 4, 5, 8, 10\u2026 Then the decimal expansion is terminating.<br \/>\nFor example:<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{1}{2}\" alt=\"\\frac{1}{2}\" align=\"absmiddle\" \/> = 0. 5, denominator q = 2<sup>1<\/sup><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{7}{8}\" alt=\"\\frac{7}{8}\" align=\"absmiddle\" \/> = 0. 875, denominator q =2<sup>3<\/sup><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{4}{5}\" alt=\"\\frac{4}{5}\" align=\"absmiddle\" \/> = 0. 8, denominator q = 5<sup>1<br \/>\n<\/sup><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">We can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>7. Write three numbers whose decimal expansions are non-terminating non-recurring.<br \/>\n<\/strong><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>Answer &#8211;\u00a0<\/strong><\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">\u221a2 = 1.41421&#8230;.<br \/>\n\u221a3 = 1.73205&#8230;.<br \/>\n<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">\u221a7 = 2.645751&#8230;.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><strong>8. Find three different irrational numbers between the rational numbers <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{5}{7}}\" alt=\"\\mathbf{\\frac{5}{7}}\" align=\"absmiddle\" \/> and <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\frac{9}{11}}\" alt=\"\\mathbf{\\frac{9}{11}}\" align=\"absmiddle\" \/>.<\/strong><br \/>\n<\/span><strong><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Answer &#8211;<\/span><\/strong><span style=\"color: #000000; font-family: georgia, palatino, serif;\">\u00a0<\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">Converting <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{5}{7}\" alt=\"\\frac{5}{7}\" align=\"absmiddle\" \/> and <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{9}{11}\" alt=\"\\frac{9}{11}\" align=\"absmiddle\" \/> into the decimal form gives<br \/>\n<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{5}{7}\" alt=\"\\frac{5}{7}\" align=\"absmiddle\" \/> = 0.714285&#8230;..<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000; font-family: georgia, palatino, serif;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{9}{11}\" alt=\"\\frac{9}{11}\" align=\"absmiddle\" \/> = 0.818181&#8230;.. <\/span><\/p>\n<p style=\"text-align: justify;\">Therefore, 3 irrational numbers that are contained between <span style=\"color: #000000; font-family: georgia, palatino, serif;\">0.714285&#8230;..\u00a0 <\/span>and <span style=\"color: #000000; font-family: georgia, palatino, serif;\">0.818181&#8230;.. <\/span>are:<br \/>\n<span style=\"color: #000000; font-family: georgia, palatino, serif;\">0.73073007300073000073\u2026, <\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">0.75075007300075000075\u2026 and <\/span><span style=\"color: #000000; font-family: georgia, palatino, serif;\">0.76076007600076000076\u2026<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>9. \u00a0Classify the following numbers as rational or irrational according to their type:<br \/>\n<\/strong><\/span><strong><span style=\"font-family: georgia, palatino, serif;\">(i) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\sqrt{23}}\" alt=\"\\mathbf{\\sqrt{23}}\" align=\"absmiddle\" \/><br \/>\n(ii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\sqrt{225}}\" alt=\"\\mathbf{\\sqrt{225}}\" align=\"absmiddle\" \/><br \/>\n(iii) 0.3796<br \/>\n(iv) 7.478478<br \/>\n(v) 1.101001000100001\u2026<br \/>\n<\/span><\/strong><span style=\"font-family: georgia, palatino, serif;\"><strong>Answer &#8211;<\/strong><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\"><strong>(i) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\sqrt{23}}\" alt=\"\\mathbf{\\sqrt{23}}\" align=\"absmiddle\" \/> <\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">= 4.79583152331\u2026<br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Since the number is non-terminating and non-recurring therefore, it is an irrational number.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(ii) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\mathbf{\\sqrt{225}}\" alt=\"\\mathbf{\\sqrt{225}}\" align=\"absmiddle\" \/> = <\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">15 = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{15}{1}\" alt=\"\\frac{15}{1}\" align=\"absmiddle\" \/><br \/>\n<\/span><span style=\"font-family: georgia, palatino, serif;\">Since the number can be represented in <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{p}{q}\" alt=\"\\frac{p}{q}\" align=\"absmiddle\" \/> form, it is a rational number.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(iii) 0.3796<br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">Since the number,0.3796, is terminating, it is a rational number.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(iv) 7.478478<br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">The number,7.478478, is non-terminating but recurring, it is a rational number.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"font-family: georgia, palatino, serif;\"><strong>(v) 1.101001000100001\u2026<br \/>\n<\/strong><\/span><span style=\"font-family: georgia, palatino, serif;\">Since the number,1.101001000100001\u2026, is non-terminating non-repeating (non-recurring), it is an irrational number.<\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000; font-family: georgia, palatino, serif;\">\u00a0<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions Class 9 Maths\u00a0 Chapter &#8211; 1 (Number Systems )\u00a0 The NCERT Solutions in English Language for Class 9 Mathematics\u00a0Chapter &#8211; 1 Number Systems\u00a0Exercise 1.3 has been provided here to help the students in solving the questions from this exercise.\u00a0 Chapter 1: Number Systems NCERT Solution Class 9 Maths Ex &#8211; 1.1 NCERT Solution [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[698],"tags":[699,702,186,700,701,5],"class_list":["post-4023","post","type-post","status-publish","format-standard","hentry","category-class-9-maths","tag-class-9th-chapter-1-number-systems-in-english","tag-ncert-solution-class-9","tag-ncert-solutions","tag-ncert-solutions-class-9-maths-chapter-1-in-english","tag-ncert-solutions-class-9-maths-in-english","tag-ncert-solutions-in-english"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.9 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>NCERT Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.3 | TheExamPillar NCERT<\/title>\n<meta name=\"description\" content=\"NCERT Solutions Class 9 Maths\u00a0Chapter - 1 (Number Systems )\u00a0The NCERT Solutions in English 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