A Rational Number is represented as a fraction. For example, x \/ y is a rational number. Here, the upper term of the fraction is called a numerator that is \u2018x\u2019, and the lower term of the fraction is called a denominator that is \u2018y\u2019. Both terms numerator (x) and denominator (y) must be integers. Integer numbers are both positive and negative numbers like -4, -3, -2, -1, 1, 2, 3, 4…etc\u2026 The most important thing in the rational numbers is the denominator of the rational number cannot be equal to zero.<\/p>\n
Rational Numbers are 2\/3, 4\/5, 6\/7, 8\/9, 10\/15, 20\/4, 125\/100, \u2026.etc\u2026 Check out the complete concept of Rational Numbers in Terminating and Non-Terminating Decimals below.<\/p>\n
By simplifying the rational numbers, we will get the result in the form of decimal fractions. We have two types of decimal fractions. They are<\/p>\n
When we converted the rational numbers into a decimal fraction, we will get either finite numbers of digits or infinite numbers of digits after the decimal point. If we get the decimal fraction with the finite number of digits, then it is called Terminating Numbers. If we get the decimal fraction with the infinite number of digits after the decimal point, then it is called Non \u2013 Terminating Numbers.<\/p>\n
Example for Terminating Numbers are 1.25, 0.68, 1.234, 2.456, 3. 4567, 5.687, 6.24, 8. 46, \u2026.etc\u2026.The below examples are in the form of rational numbers and we need to convert that numbers into the form of decimal numbers.
\n(1) x \/ y = 100 \/ 25 = 0.4
\n(2) x \/ y = 644 \/ 8 = 8.5
\n(3) x \/ y = 5 \/ 4 = 1.25<\/p>\n
Example for Non \u2013 Terminating Numbers are 1.23333, 2.566666, 5.8678888, 3.467777, 4.6899999,\u2026..etc\u2026 The below-mentioned x \/ y fraction indicates the rational numbers and by simplifying it, we will get the decimal numbers.
\n(1) x \/ y = 256 \/ 6 =42.66666\u2026
\n(2) x \/ y = 10 \/ 3 = 3.33333\u2026
\n(3) x \/ y = 20 \/ 9 = 2.222222\u2026<\/p>\n
Note: If a rational number (\u2260 integer) can be expressed in the form p\/(2^n \u00d7 5^m) where p \u2208 Z, n \u2208 W, and m \u2208 W then the rational number will become a terminating decimal. If not, the rational number becomes the Non \u2013 Terminating Numbers.<\/p>\n
1. Find out the conversion of rational numbers to terminating decimal fractions?<\/p>\n
(i) 1\/4 is a rational fraction of form p\/q. When this rational fraction is converted to decimal it becomes 0.25, which is a terminating decimal fraction.<\/p>\n
(ii) 1\/8 is a rational fraction of form p\/q. When this rational fraction is converted to decimal fraction it becomes 0.125, which is also an example of a terminating decimal fraction.<\/p>\n
(iii) 4\/40 is a rational fraction of form p\/q. When this rational fraction is converted to decimal fraction it becomes 0.1, which is an example of a terminating decimal fraction.<\/p>\n
2. Find out the conversion of rational numbers to nonterminating decimal fractions.<\/p>\n
(i) 1\/11 is a rational fraction of form p\/q. When we convert this rational fraction into a decimal, it becomes 0.090909\u2026 which is a non-terminating decimal.<\/p>\n
(ii) 1\/13 is a rational fraction of form p\/q. When we convert this rational fraction into a decimal, it becomes 0.0769230769230\u2026 which is a non-terminating decimal.<\/p>\n
(iii) 2\/3 is a rational fraction of form p\/q. When this is converted to a decimal number it becomes 0.66666667\u2026 which is a non-terminating decimal fraction.<\/p>\n
You may see different types of numbers such as real numbers, whole numbers, rational numbers, etc. Now, let us check out the irrational numbers. Irrational numbers are also real numbers that are represented as a simple fraction. There is no repeating or no terminate pattern available in Irrational Numbers. the numbers which do not consist of exact square roots of integers treats as Irrational Numbers. Also, the Irrational Number is pi and that is equal to the value of 3.14.<\/p>\n
Add the two rational numbers. For example x \/ y = 1 \/ 2 and p \/ q = 2 \/ 6
\nTo add the two rational numbers, we need to find out the LCM of the denominators.
\nThat is, LCM of 2 and 6 is 6.
\nx \/ y + p \/ q = 1 \/ 2 + 2 \/ 6 = (3 + 2) \/ 6 = 5 \/ 6.<\/p>\n
Multiply the two rational numbers such as 2 \/ 3 and 5 \/ 6.
\nx \/ y = 2 \/ 3 and p \/ q = 5 \/ 6
\nx \/ y X p \/ q = 2 \/ 3 x 5 \/ 6
\n(xX p) \/ (y X q) = (2 x 5) \/ 3 x 6)
\npx \/ yq = 10 \/ 18 = 5 \/ 9.<\/p>\n
Subtract the two rational numbers. Here, 5 \/ 8 and 12 \/ 5 are rational numbers.
\nx \/ y = 5 \/ 8 and p \/ q = 12 \/ 5
\nx \/ y \u2013 p \/ q = 5 \/ 8 \u2013 12 \/ 5
\nFor subtraction, we need to find out the LCM of denominator values.
\nLCM of 8 and 5 is 40.
\n5 \/ 8 \u2013 12 \/ 5 = [(5 x 5) \u2013 (12 x 8)] \/ 40 = (25 \u2013 96) \/ 40 = -71 \/ 40.<\/p>\n
To divide the two rational numbers, we need to cross multiply the terms. For example, x \/ y and p \/ q are two rational numbers.
\n(x \/ y) \u00f7( p \/ q) = (2 \/ 5) \u00f7 ( 3 \/ 7)
\nCross Multiply the first fraction numerator with second fraction denominator and vice versa.
\nxq \/ py = (2 x 7) \/ (5 x 3 )
\nxq \/ py= 14 \/ 15.<\/p>\n","protected":false},"excerpt":{"rendered":"
A Rational Number is represented as a fraction. For example, x \/ y is a rational number. Here, the upper term of the fraction is called a numerator that is \u2018x\u2019, and the lower term of the fraction is called a denominator that is \u2018y\u2019. Both terms numerator (x) and denominator (y) must be integers. … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":54902,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/03\/Rational-Numbers-in-Terminating-and-Non-Terminating-Decimals.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/54898"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=54898"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/54898\/revisions"}],"predecessor-version":[{"id":92479,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/54898\/revisions\/92479"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/54902"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=54898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=54898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=54898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}