27 R.3 Real Number Operations and Properties Division Involving 0 For any nonzero real number a, a 0 is undefined, 0 a =0, and 0 0 is indeterminate. (c) -12 - 4 = -12 + 1-42 = -16 (d) 5 6 - a3 8b = 5 6 + 3 8 To subtract a - b, add the opposite of b to a. = 20 24 + 9 24 The least common denominator is 24. 5 6 # 4 4 = 20 24 ; 3 8 # 3 3 = 9 24 = 29 24 Add numerators. Keep the same denominator. S Now Try Exercises 27, 39, and 45. EXAMPLE 2 Adding and Subtracting Real Numbers Find each sum or difference. (a) -6 + 1-32 (b) -2.3 + 5.6 (c) -12 - 4 (d) 5 6 - a3 8b SOLUTION Add the absolute values. The numbers have the same sign, both negative, so the sum will be negative. (a) -6 + 1-32 = -1 -6 + -3 2 = -16 + 32 = -9 (b) -2.3 + 5.6 = 5.6 - 2.3 = 5.6 - 2.3 = 3.3 The numbers have different signs. Subtract the lesser absolute value from the greater. Change to addition. The opposite of 4 is -4. What happens if we try to divide a nonzero number by 0? For example, to find 4 0 , we look for some number n that when multiplied by 0 will give 4. But n # 0 will always give a product of 0, never 4. Therefore, division by 0 is undefined. On the other hand, 0 4 = 0 because 0 # 4 = 0. Thus, dividing 0 by any nonzero number will always yield 0. Trying to divide 0 by 0 is problematic for another reason. If 0 0 = n, then n # 0 = 0. This statement is true for all numbers n, which means that 0 0 is not unique. We say that 0 0 is indeterminate—that is, there is no unique answer. This discussion can be summarized as follows.
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