{"id":2375,"date":"2022-12-29T10:00:55","date_gmt":"2022-12-29T04:30:55","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=2375"},"modified":"2022-12-30T09:43:45","modified_gmt":"2022-12-30T04:13:45","slug":"conversion-of-decimal-numbers-into-rational-numbers","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/conversion-of-decimal-numbers-into-rational-numbers\/","title":{"rendered":"How To Convert Decimal Number Into Rational Number"},"content":{"rendered":"
Case I:<\/strong> When the decimal number is of terminating nature. Case II:<\/strong> When decimal representation is of non-terminating repeating nature. Algorithm:<\/strong><\/p>\n Algorithm:<\/strong><\/p>\n Example 1: \u00a0 \u00a0<\/strong>Express each of the following numbers in the form p\/q. Example 2: \u00a0 \u00a0<\/strong>Express each of the following decimals in the form p\/q. Example 3: \u00a0 \u00a0<\/strong>Convert the following decimal numbers in the form p\/q. Example 4: \u00a0 \u00a0<\/strong>Express the following decimals in the form Example 5: \u00a0<\/strong> \u00a0Express each of the following mixed recurring decimals in the form p\/q Example 6: \u00a0 \u00a0<\/strong>Represent 3.765 on the number line. Example 7: \u00a0 \u00a0<\/strong>Visualize \\(4.\\overline{26}\\)\u00a0on the number line, upto 4 decimal places. Example 8: \u00a0 \u00a0<\/strong>Express the decimal \\(0.003\\overline{52}\\)\u00a0in the form\u00a0p\/q Example 9: \u00a0 \u00a0<\/strong>Give an example of two irrational numbers, the product of which is Example 10: \u00a0 \u00a0<\/strong> Insert a rational and an irrational number between 2 and 3. Example 11: \u00a0 \u00a0<\/strong> Find two irrational numbers between 2 and 2.5. Example 12: \u00a0 \u00a0<\/strong>Find two irrational numbers lying between\u00a0\u221a2 and\u00a0\u221a3. Example 13: \u00a0 \u00a0<\/strong>Find two irrational numbers between 0.12 and 0.13. Example 14: \u00a0 \u00a0<\/strong>Find two rational numbers between 0.232332333233332…. and 0.252552555255552…… Example 15: \u00a0 \u00a0<\/strong>Find a rational number and also an irrational number between the numbers a and b given below: Example 16: \u00a0 \u00a0<\/strong>Find one irrational number between the number a and b given below :
\nAlgorithm:<\/strong><\/p>\n\n
\nIn a non terminating repeating decimal, there are two types of decimal representations<\/p>\n\n
\nFor Example:<\/strong> \\(0.\\overline{6},\\,\\,0.\\overline{16},\\,\\,0.\\overline{123}\\) are pure recurring decimals.<\/li>\n
\nFor Example:<\/strong> \\(2.1\\overline{6},\\,\\,0.3\\overline{5},\\,\\,0.7\\overline{85}\\) are mixed recurring decimals.<\/li>\n<\/ol>\nConversion Of A Pure Recurring Decimal To The Form<\/strong> p\/q<\/strong><\/h2>\n
\n
Conversion Of A Mixed Recurring Decimal To The Form p\/q<\/strong><\/h2>\n
\n
Conversion Of Decimal Numbers Into Rational Numbers Example Problems With Solutions<\/strong><\/h2>\n
\n(i) 0.15 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(ii) 0.675\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (iii) \u201325.6875
\nSolution: \u00a0 \u00a0<\/strong>
\n<\/p>\n
\n\\(\\text{(i) 0}\\text{.}\\overline{\\text{6}}\\text{ (ii) 0}\\text{.}\\overline{\\text{35}}\\text{ (iii) 0}\\text{.}\\overline{\\text{585}}\\)
\nSolution: \u00a0 \u00a0<\/strong>
\n
\n
\nThe above example suggests us the following rule to convert a pure recurring decimal into a rational number in the form p\/q.<\/p>\n
\n\\(\\text{(i) }5.\\bar{2}\\text{ (ii) }23.\\overline{43}\\)
\nSolution: \u00a0 \u00a0<\/strong>
\n<\/p>\n
\n\\(\\text{(i) }0.3\\overline{2}\\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (ii) }0.12\\overline{3}\\)
\nSolution: \u00a0 \u00a0<\/strong>
\n
\n<\/p>\n
\n\\(\\text{(i) }4.3\\overline{2}\\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (ii) }15.7\\overline{12}\\)
\nSolution: \u00a0 \u00a0<\/strong>
\n
\n<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>This number lies between 3 and 4. The distance 3 and 4 is divided into 10 equal parts. Then the first mark to the right of 3 will represent 3.1 and second 3.2 and so on. Now, 3.765 lies between 3.7 and 3.8. We divide the distance between 3.7 and 3.8 into 10 equal parts 3.76 will be on the right of 3.7 at the sixth\u00a0<\/sup>mark, and 3.77 will be on the right of 3.7 at the 7th<\/sup> mark and 3.765 will lie between 3.76 and 3.77 and soon.
\n<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>We have, \u00a0 \\(4.\\overline{26}\\) = 4.2626
\nThis number lies between 4 and 5. The distance between 4 and 5 is divided into 10 equal parts. Then the first mark to the right of 4 will represent 4.1 and second 4.2 and soon. Now, 4.2626 lies between 4.2 and 4.3. We divide the distance between 4.2 and 4.3 into 10 equal parts 4.2626 lies between 4.26 and 4.27. Again we divide the distance between 4.26 and 4.27 into 10 equal parts. The number 4.2626 lies between 4.262 and 4.263. The distance between 4.262 and 4.263 is again divided into 10 equal parts. Sixth mark from right to the 4.262 is 4.2626.
\n<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>Let x = \\(0.003\\overline{52}\\)
\nClearly, there is three digit on the right side of the decimal point which is without bar. So, we multiply both sides of x by 103<\/sup> = 1000 so that only the repeating decimal is left on the right side of the decimal point.
\n<\/p>\n
\n(i) a rational number
\n(ii) an irrational number
\nSolution: \u00a0 \u00a0<\/strong>(i) The product of \u221a27 and\u00a0\u221a3 is \u221a81= 9, which is a rational number.
\n(ii) The product of \u221a2 and \u221a3 is \u221a6, which is an irrational number.<\/p>\n
\nSolution: \u00a0 <\/strong>\u00a0If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then \\(\\sqrt{ab}\\) is an irrational number lying between a and b. Also, if a,b are rational numbers, then \\(\\frac { a+b }{ 2 }\\) is a rational number between them.
\n\u2234 A rational number between 2 and 3 is
\n\\(\\frac { 2+3 }{ 2 }\\) = 2.5
\nAn irrational number between 2 and 3 is
\n\\(\\sqrt{2\\times 3}=\\sqrt{6}\\)<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>If a and b are two distinct positive rational numbers such that ab is not a perfect square of a rational number, then \\(\\sqrt{ab}\\) is an irrational number lying between a and b.
\n\u2234 Irrational number between 2 and 2.5 is
\n\\( \\sqrt{2\\times 2.5}=\\sqrt{5} \\)
\nSimilarly, irrational number between 2 and \\(\\sqrt{5}\\) is\u00a0\\( \\sqrt{2\\times \\sqrt{5}} \\)
\nSo, required numbers are \\(\\sqrt{5}\\) and \\( \\sqrt{2\\times \\sqrt{5}} \\).<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>We know that, if a and b are two distinct positive irrational numbers, then is an irrational number lying between a and b.
\n\u2234 Irrational number between \u221a2 and\u00a0\u221a3 is \\( \\sqrt{\\sqrt{2}\\times \\sqrt{3}}=\\sqrt{\\sqrt{6}} \\)\u00a0=\u00a061\/4<\/sup>
\nIrrational number between \u221a2 and 61\/4<\/sup>\u00a0is \\( \\sqrt{\\sqrt{2}\\times {{6}^{1\/4}}} \\) = 21\/4<\/sup> \u00d7 61\/8<\/sup>.
\nHence required irrational number are 61\/4<\/sup> and 21\/4<\/sup> \u00d7 61\/8<\/sup><\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>Let a = 0.12 and b = 0.13. Clearly, a and b are rational numbers such that a < b.
\nWe observe that the number a and b have a 1 in the first place of decimal. But in the second place of decimal a has a 2 and b has 3. So, we consider the numbers
\nc = 0.1201001000100001 ……
\nand, d = 0.12101001000100001…….
\nClearly, c and d are irrational numbers such that a < c < d < b.<\/p>\n
\nSolution: \u00a0 \u00a0<\/strong>Let a = 0.232332333233332…. \u00a0and b = 0.252552555255552…..
\nThe numbers c = 0.25 and d = 0.2525
\nClearly, c and d both are rational numbers such that a < c < d < b.<\/p>\n
\na = 0.101001000100001…., \u00a0b = 0.1001000100001…
\nSolution: \u00a0 <\/strong>Since the decimal representations of a and b are non-terminating and non-repeating. So,
\na and b are irrational numbers.
\nWe observed that in the first two places of decimal a and b have the same digits. But in the third place of decimal a has a 1 whereas b has zero.
\n\u2234 a > b
\nConstruction of a rational number between a and b : As mentioned above, first two digits after the decimal point of a and b are the same. But in the third place a has a 1 and b has a zero. So, if we consider the number c given by
\nc = 0.101
\nThen, c is a rational number as it has a terminating decimal representation.
\nSince b has a zero in the third place of decimal and c has a 1.
\n\u2234 b < c
\nWe also observe that c < a, because c has zeros in all the places after the third place of decimal whereas the decimal representation of a has a 1 in the sixth place.
\nThus, c is a rational number such that
\nb < c < a.
\nHence , c is the required rational number between a and b.
\nConstruction of an irrational number between a and b : Consider the number d given by
\nd = 0.1002000100001……
\nClearly, d is an irrational number as its decimal representation is non-terminating and non-repeating.
\nWe observe that in the first three places of their decimal representation b and d have the same digits but in the fourth place d and a 2 whereas b has only a 1.
\n\u2234 d > b
\nAlso, comparing a and d, we obtain a > d
\nThus, d is an irrational number such that
\nb < d < a.<\/p>\n
\na = 0.1111….. = \\(0.\\bar{1}\\) and b = 0.1101
\nSolution: \u00a0 \u00a0<\/strong>Clearly, a and b are rational numbers, since a has a repeating decimal and b has a terminating decimal. We observe that in the third place of decimal a has a 1, while b has a zero.
\n\u2234 a > b
\nConsider the number c given by
\nc = 0.111101001000100001…..
\nClearly, c is an irrational number as it has non-repeating and non-terminating decimal representation.
\nWe observe that in the first two places of their decimal representations b and c have the same digits. But in the third place b has a zero whereas c has a 1.
\n\u2234 b < c
\nAlso, c and a have the same digits in the first four places of their decimal representations but in the fifth place c has a zero and a has a 1.
\n\u2234 c < a
\nHence, b < c < a
\nThus, c is the required irrational number between a and b.<\/p>\n