238 CHAPTER 5 Number Theory and the Real Number System Quotient Dividend Remainder Divisor Quotient Remainder Divisor Now try Exercise 27 Up to this point, we have only worked with positive mixed numbers and positive improper fractions. When converting a negative mixed number to an improper fraction, or a negative improper fraction to a mixed number, it is best to ignore the negative sign temporarily. Perform the calculation as described earlier and then reattach the negative sign. Example 4 Negative Mixed Numbers and Improper Fractions a) Convert −3 5 6 to an improper fraction. b) Convert − 8 5 to a mixed number. Solution a) First ignore the negative sign and examine 3 5 6 . We learned in Example 2(a) that = 3 5 6 23 6 . To convert −3 5 6 we reattach the negative sign. Thus, − = − 3 5 6 23 6 b) We learned in Example 3(a) that = 8 5 1 3 5 . Therefore, − = − 8 5 1 3 5 . 7 Now try Exercise 29 Terminating or Repeating Decimal Numbers Note the following important property of the rational numbers. Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. Examples of terminating decimal numbers are 0.5, 0.75, and 4.65. Examples of repeating decimal numbers are … … 0.333 , 0.2323 , and … 8.13456456 . One way to indicate that a number or group of numbers repeats is to place a bar above the number or group of numbers that repeats. Thus, … 0.333 may be written … 0.3, 0.2323 may be written 0.23, and … 8.13456456 may be written 8.13456. b) Divide the numerator, 225, by the denominator, 8. ) 28 8 225 16 65 64 1 Therefore, = 225 8 28 1 8 The mixed number is 28 . 1 8 7
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