+ ".3+ ll-2 Decimal Forms of Rational Numhers. o'*- f : o.a::. 0hjective To express rational numbers as decimals or fractions.

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1 ll-2 Decimal Forms of Rational Numhers 0hjective To express rational numbers as decimals or fractions. Any common fraction can be written as a decimal by dividing the numerator by the denominator. If the remainder is zero, the decimal is called a terminating, ending, or finite decimal. Example I Expressut udecimal. Solutian o.375 8)tooo The division at the left shows that can be expressed as the terminating decimal Answer If you don't reach a remainder of zero when dividing the numerator by the denominator, continue to divide until the remainders begin to repeat. Exanple 2 Express each rational number as a decimal: Solution ". 512 Chapter I I )5xoo o 48 - l8 l8 2 f : o.a::. o'*- o.6363 r l)tlooo 66 JJ " ". 3+: +-- 7)t3-ooooooo 2t t a- : l t4,o.'.3+ :

2 The decimal quotients shown in Example 2 are nonterminating' unending, or infinite. The dots indicate that the decimals continue without end. We write: f : o.s:: ft: o.oao: tj : z.ztst t4zlsi t4 They are also called repeating or periodic because the same digit or block of digits repeats unendingly. A bar is used to indicate the block of digits that repeat, as shown below. f:oa: #:o* zl: ntstt+ When you divide a positive integer n by a positive integer d, the remainder r at each step must be zero or a positive integer less than d. For example, if the divisor is 6, the remainders will be O, 1,2,3, 4, or 5, and the division will terminate or begin repeating within 5 steps after only zeros remain to be brought down. In general: For every integer r and every positive integer d, the decimal form of the rational number { either terminates or eventually repeats in a block of fewer than d d digits. To express a terminating decimal as a common fraction, express the decimal as a common fraction with a power of ten as the denominator. This fraction is then usually expressed in simplest form. EXample 3 Express each terminating decimal as a fraction in simplest form. a Solution a. 0.38: #: * 0.42s : ffi: # The following examples show how to express a repeating decimal as a common fraction. Example 4 Express as a fraction in simplest form. Solution Let N: rhe number Let n : the number of digits in the block of repeating digits. Multiply N by l0'. Since has 2 digits in the repeating block, n:2. Therefore, multiply both sides of the equation N : O.5m by r00n: 100(0.542) (Solution conlinues on the next page.) Rational and lrrational Numbers 513

3 Since : Then Subtract N from 100N. Solve for N.,0.542 can also be written as I 00(0.547) : t0o, ) : l00n = N : N : 53.7 ^,_53.7 _537 _l7g '" o.sn:# Answer EXample 5 Express as a fraction in simplest form. SOlUtiOn First, express as a common fraction. LetN:thenumber. 1000N : N: O.37s 999N : 375 Since there are 3 digits in the repeating block, multiply N by 103, or Then subtract "/v: Since : 3nl2t, _0.375 : Answer All terminating decimals and all repeating decimals represent rational numbers that can be written in the form f, where n is an integer and d is a positive ind' teger. It is often convenient to use an approximation of a lengthy decimal. For example, you may approxi.ut. as , or fr To round a decimal: 1. If the first digit dropped is greater than or equal to 5, add I to the last digit retained. 2. ff the first digit dropped is less than 5, don't change the last digit retained. Example 6 shows decimals being rounded to various decimal places. The symbol: means "is approximately equal to." 514 Chapter I I

4 Exanple 6 a. o.aft :. - O :0.416 :0.42 c :0.5 d J e l (to the nearest thousandth) (to the nearest thousandth) (to the nearest unit) (to the nearest unit) Oral Enercises Round each number to the nearest tenth. r , Round the numbers in Exercises 1-5 to the nearest hundredth. Tell whether the decimal form terminates or repeats. B.+ 14.,k n.+, s Written Exercises Express each rational number as a terminating or repeating decimal. A t.".? + -.lt 5.8 e ,. ".2 6.# # _ t.".-t 4. a. -15 /' l l o tt. 3;o 15.-# _ Express each rational number as a fraction in simplest form. 8.+ n.2+,6. + J -5 _ o o O.U r : t 25. o.857t4' Rational and lrrational Numbers 515

5 Find the number halfway between the given numbers. Sample ] and 0.1s6 Solution ;: #:0.7s 0.7s +!<o.tso- 0.7s) : jto.oool : : Answer B 2e.! and 0.2se 30. * and and O.77 and and f anA Express both numbers as fractions. Then find their product. 35. J and 0.7s s and f ana $ 3s. + and , 0.7 and O.2i and C 41. a. c. 42. a. c Express ;, ;. and as repeating decimals. Express +, +, ana fr as repeating decimals. What is the relationship between the numbers in (a) and (b)? -l.6 Express 7 and I as repeating decimals. What is the relationship between the blocks of digits that repeat in (a)? Express ;, ;. and f as decimals. 43. Since #: O.t, #: "Q.r) for I < n ( 100. a. Confirm the fact above by expressing ana #,# $ Express f as S to show that 0.9 : l. c. Use the method of Example 4 to show that 0.9: 1. as decimals. Mixed Review Exercises Find the prime factorization of each number Solve. 7.(y+3Xy-4):0 lo. k3-16ft:0 516 Chapter 1l 8. (c + 5)2 :9 11. l-r + 2l: t0 9. Y2 : k+4<t6