240 CHAPTER 5 Number Theory and the Real Number System Converting a repeating decimal number to a quotient of integers is more difficult than converting a terminating decimal number to a quotient of integers. To do so, we must “create” another repeating decimal number with the same repeating digits so that when one repeating decimal number is subtracted from the other repeating decimal number, the difference will be a whole number. To create a number with the same repeating digits, multiply the original repeating decimal number by an appropriate power of 10. Examples 8 through 10 demonstrate this procedure. Example 7 Converting Decimal Numbers to Fractions Convert the following terminating decimal numbers to a quotient of integers. Reduce the quotient to lowest terms. a) 0.7 b) 0.35 c) 0.016 d) 3.41 Solution When converting a terminating decimal number to a quotient of integers, we observe the last digit to the right of the decimal point. The position of this digit will indicate the denominator of the quotient of integers. a) = 0.7 7 10 because the 7 is in the tenths position. The fraction 7 10 is in its lowest term. b) = 0.35 35 100 because the right-most digit, 5, is in the hundredths position. We finish by reducing the fraction to lowest terms: = ÷ ÷ = 35 100 35 5 100 5 7 20 c) = 0.016 16 1000 because the right-most digit, 6, is in the thousandths position. We finish by reducing the fraction to lowest terms: = ÷ ÷ = 16 1000 16 8 1000 8 2 125 d) We begin by writing 3.41 as the mixed number 3 41 100 because the right-most digit, 1, is in the hundredths position. To write the mixed number as a ratio of two integers, we follow the procedure shown in Example 2: = ⋅ + = 3 41 100 100 3 41 100 341 100 The fraction 341 100 cannot be reduced further. 7 Now try Exercise 41 Example 8 Converting a Repeating Decimal Number to a Fraction Convert 0.3 to a quotient of integers. Solution = = 0.3 0.33 0.333, and so on. Let the original repeating decimal number be n; thus, = n 0.3. Because one digit repeats, we multiply both sides of the equation by 10, which gives = n 10 3.3. Then we subtract. = − = = n n n 10 3.3 0.3 9 3.0 Note that − = n n n 10 9 and − = 3.3 0.3 3.0. Next, we solve for n by dividing both sides of the equation by 9. n n 9 9 3.0 9 3 9 1 3 = = = Therefore, = 0.3 . 1 3 Evaluate ÷1 3 on a calculator now and see what value you get. 7 Now try Exercise 49
RkJQdWJsaXNoZXIy NjM5ODQ=