5.3 The Rational Numbers 239 Example 5 Terminating Decimal Numbers Show that the following rational numbers can be expressed as terminating decimal numbers. a) 4 5 b) − 13 20 c) 135 20 Now try Exercise 33 Solution To express the rational number in decimal form, divide the numerator by the denominator. If you use a calculator or long division, you will see that each fraction results in a terminating decimal number. a) = 4 5 0.8 b) − = − 13 20 0.65 c) = 135 20 6.75 7 Example 6 Repeating Decimal Numbers Show that the following rational numbers can be expressed as repeating decimal numbers. a) 2 3 b) 14 99 c) 1 5 36 Now try Exercise 39 Solution If you use a calculator or long division, you will see that each fraction results in a repeating decimal number. a) ÷ = … 2 3 0.6666 or 0.6 b) ÷ = … 14 99 0.141414 or 0.14 c) = + = + …= … 1 1 1 0.138888 1.138888 or 1.138 5 36 5 36 7 Note that in each part of Example 6, the quotient when expressed as a decimal number has no final digit and continues indefinitely. Each number is a repeating decimal number. When a fraction is converted to a decimal number, the maximum number of digits that can repeat is − n 1, where n is the denominator of the fraction. For example, when 2 7 is converted to a decimal number, the maximum number of digits that can repeat is − 7 1, or 6. Converting Decimal Numbers to Fractions We can convert a terminating or repeating decimal number into a quotient of integers. The explanation of the procedure will refer to the positional values to the right of the decimal point, as illustrated here: 0 . 2 8 5 7 1 4 Units position Tenths position Hundredths position Thousandths position Ten-thousandths position Hundred-thousandths position Millionths position Example 7 demonstrates how to convert from a decimal number to a fraction.
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