26 CHAPTER R Review of Basic Concepts Operations on Real Numbers Recall that the answer to an addition problem is a sum. The answer to a subtraction problem is a difference. The answer to a multiplication problem is a product, and the answer to a division problem is a quotient. Other phrases that indicate these operations are given in the table. Words or Phrases That Indicate Operations on Real Numbers Operation Word or Phrase Addition Sum, added to, more than, increased by, plus Subtraction Difference of, subtracted from, less than, decreased by, minus Multiplication Product of, times, twice, triple, of, percent of Division Quotient of, divided by, ratio of To perform operations on real numbers, we use the following sign rules. Sign Rules for Operations on Real Numbers Operation Examples To add two numbers with the same sign, add their absolute values. The sum has the same sign as the given numbers. 2 + 7 = 9 -2 + 1-72 = -9 To add two numbers with different signs, find the absolute values of the numbers, and subtract the lesser absolute value from the greater. The sum has the same sign as the number with the greater absolute value. -12 + 5 = -7 4 + 1-102 = -6 -3 + 8 = 5 To subtract a number b from another number a, change to addition and replace b by its additive inverse (opposite), -b. a −b =a + 1 −b2 Then use the sign rules for addition. 5 - 7 = 5 + 1-72 = -2 -8 - 4 = -8 + 1-42 = -12 -17 - 1-42 = -17 + 4 = -13 To multiply or divide two numbers with the same sign, multiply or divide their absolute values. The answer will be positive. 5172 = 35, -51-72 = 35 35 5 = 7, -35 -5 = 7 To multiply or divide two numbers with different signs, multiply or divide their absolute values. The answer will be negative. -8192 = -72, 61-72 = -42 -45 9 = -5, 36 -9 = -4 Consider the quotient 15 , 3 and the product 15 # 1 3. 15 , 3 = 15 3 = 5 and 15 # 1 3 = 15 3 = 5 Thus division can be written in terms of multiplication using the multiplicative inverse (reciprocal). For all real numbers a and b1b≠02, a ÷b = a b =a # 1 b . To divide a by b, multiply a (the dividend) by 1 b, the reciprocal of b (the divisor). For this reason, the sign rules for division are the same as those for multiplication. Consider the quotient a b. To find a b, we look for a number n that when multiplied by b will give a. For example, 10 5 = 2 because 2 # 5 = 10.
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