5.2 The Integers 229 CASE 4: × NEGATIVE INTEGER NEGATIVE INTEGER We have illustrated that a positive integer times a negative integer is a negative integer. We make use of this fact in the following pattern: 4( 4) 16 3( 4) 12 2( 4) 8 1( 4) 4 − = − − = − − = − − = − In this pattern, each time the first factor is decreased by 1, the product is increased by 4. Continuing this process gives 0( 4) 0 ( 1)( 4) 4 ( 2)( 4) 8 − = − − = − − = This pattern illustrates that a negative integer times a negative integer is a positive integer . The examples were restricted to integers. The rules for multiplication, however, can be used for any numbers. We summarize them as follows. Rules for Multiplication 1. The product of two numbers with like signs (positive × positive or negative × negative) is a positive number. 2. The product of two numbers with unlike signs (positive × negative or negative × positive) is a negative number. Example 7 Multiplying Integers Evaluate a) ⋅ 8 7 b) ⋅ − 8 ( 7) c) − ⋅ ( 8) 7 d) − ⋅ − ( 8) ( 7) Solution a) ⋅ = 8 7 56 b) ⋅ − = − 8 ( 7) 56 Now try Exercise 27 c) − ⋅ = − ( 8) 7 56 d) − ⋅ − = ( 8) ( 7) 56 7 Division of Integers You may already realize that a relationship exists between multiplication and division. 6 2 3 means that 3 2 6 20 10 2 means that 2 10 20 ÷ = ⋅ = = ⋅ = Timely Tip Why can’t we divide by 0? Suppose we are trying to determine the quotient 5 0 . If such an expression were defined, then we could write = x 5 0 . If true, this would mean that = 5 ⋅ x 0 , or = 5 0, which is false. Thus, there is no number, x, that makes the equation = x 5 0 true. Therefore, we say a quotient of any nonzero number divided by zero is undefined. Division For any a b , , and c where ≠ = b a b c 0, means that ⋅ = c b a. These examples demonstrate that division is the reverse process of multiplication. In the definition of division, note that we cannot divide by 0 (see the Timely Tip in the margin). The quotient of a nonzero number divided by 0 is said to be undefined . For example, 5 0 is undefined.
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