In terminating decimal expansion, the prime factorization of the denominator has no other factors other than 2 and 5, In non-terminating but repeating decimal expansion, you will find that the prime factorization of the denominator has factors other than 2 and 5. (If the number of decimal digits is infinite, the number is rational only if there is a repeating pattern.) A rational number is of the form \(\dfrac{p}{q}\) where: The set of rational numbers is denoted by \(Q\) or \(\mathbb{Q}\). You can notice that the digits in the quotient keep repeating. Now suppose that   $\displaystyle{ \sqrt{2} }$   did have a fractional representation, so that We illustrate several times by What we use most often in daily life, and what our calculators produce at the touch of a button, are decimal numbers. Determine if \(\begin{align}\frac{11}{25}\end{align}\) is a terminating or a non-terminating number. Attempt the test now. Rational numbers can be written as fractions, ratios, terminating decimals, or repeating decimals. Yes, 4 is a rational number because it satisfies the condition of rational numbers. For instance, while rational numbers can be converted to decimal representation, some of them need an infinite number of digits to be represented exactly in decimal form. obtain   $\displaystyle{ (10^n - 1) \cdot x }$   as a non-repeating (terminating) decimal, because the Theorem: Rational numbers are precisely those decimal numbers whose decimal representation is either terminating or eventually repeating. Thus   $30$   is a product of three prime numbers and its square, $\900\,$ , is a product of six   primes: Numbers that have a non-repeating decimal can also be a rational number. \( \begin{align}\therefore \frac{11}{25}\end{align}\) is a terminating number. The conversion of fractions to decimals is something with which we are all familiar   –   Terminating decimals like \(0.12, 0.625, 1.325\), etc. A rational number is a number that can be written as a fraction, \(\frac{a}{b}\) where a and b are integers. long division. &=3 +0. Terminating decimal numbers can also easily be written in that form: for example 0.67 = 67/100, 3.40938 = 340938/100000, and so on. Rational numbers include natural numbers, whole numbers, and integers. and experience Cuemath’s LIVE Online Class with your child. (there may be repetition) then   $\displaystyle{ n^2 }$   is a product of   $2\, k$ If a decimal number can be expresed in the form  \(\frac{p}{q}\)  and \(q \neq 0 \), it is a rational number. We at Cuemath believe that Math is a life skill. Book a FREE trial class today! describing decimal forms of rational numbers A rational number is a number that can be written as a ratio of two integers a and b, where b is not zero. Help your child score higher with Cuemath’s proprietary FREE Diagnostic Test. repeating part of   $x$   has subtracted away. Here is a small activity for you . Before attempting to justify the claim of the above theorem   –   by showing how to move back and   {1000y = 1721.873873873...} \\  Note that in terminating decimal expansion, you will find that the prime factorization of the denominator has no other factors other than 2 and 5. Example 1: Show that is a rational number. Express \(\begin{align}\frac{1}{13}\end{align}\) in decimal form. The common feature $ 2=\frac{p^2}{q^2} \, $ , or $\displaystyle{ 2\cdot q^2 = p^2 \; }$ . It is not always possible to do this exactly.   {1000000y = 1721873.873873873...}  You can download the FREE grade-wise sample papers from below: To know more about the Maths Olympiad you can click here. $2\,$ ), either. Copyright © MathLynx 2012. To go from a fraction to its decimal representation we use long division. Hence, the number 3.14 is a rational number. A repeating decimal can be written as a fraction using algebraic methods, so any repeating decimal is a rational number. Now look at the following example questio… The terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits. As discussed earlier, the set of numbers that can be represented as fractions is denoted by   $\mathbb{Q}$ As the collection of possible non-zero remainders is limited by the denominator We have seen that some rational numbers, such as 7 16, have decimal Let us review what we have already learnt and then go further to multiplication and division of fractional numbers as well as of decimal fractions. the decimal number 1.5 is rational because it can be expressed as the fraction 3/2; the repeating decimal 0.333… is equivalent to the rational number 1/3; Traditionally, the set of all rational numbers is denoted by a bold-faced Q. Example: \( 0.25 = \dfrac{25}{100} \) is a rational number. $\displaystyle{ 100\, x-x = 72.7272... \, - \, 0.7272... \, }$ , or   $99\, x = 72\; $ . \frac{2}{9} & = 0.222222\cdots \\ & \\ \frac{8}{11} & = 0.727272\cdots If it is non-terminating and non-recurring, it is not a rational number. One of the cats seems to think she’s a dog. Irrational number cannot be expressed in the form \(\frac{p}{q}\). A number with a finite number of decimal digits is always rational. For example, 4/ 7 is a rational number, as is 0.37 because it can be written as the fraction 37/100. Let   $x$   denote our repeating decimal. \end{array}$$, In each of the above cases, on dividing one integer by another we either obtain a remainder of   $0$   While dividing a number \(a \div b \), if we get zero as the remainder, the decimal expansion of such a number is called terminating. It follows that   $\displaystyle{ x=\frac{43210}{999}\; }$ . 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