228 CHAPTER 5 Number Theory and the Real Number System We will develop the rules for multiplication of integers using number patterns. The four possible cases are 1. × positive integer positive integer, 2. × positive integer negative integer, 3. × negative integer positive integer, and 4. × negative integer negative integer. CASE 1: × POSITIVE INTEGER POSITIVE INTEGER The product of two positive integers can be defined as repeated addition of a positive integer. Thus, ⋅ 3 2 means + + 2 2 2. This sum will always be positive. Thus, a positive integer times a positive integer is a positive integer . CASE 2: × POSITIVE INTEGER NEGATIVE INTEGER Consider the following patterns: = = = 3(3) 9 3(2) 6 3(1) 3 Note that each time the second factor is reduced by 1, the product is reduced by 3. Continuing the process gives = 3(0) 0 What comes next? 3( 1) 3 3( 2) 6 − = − − = − The pattern indicates that a positive integer times a negative integer is a negative integer . We can confirm this result by using the number line. The expression − 3( 2) means − + − + − ( 2) ( 2) ( 2). Adding − + − + − ( 2) ( 2) ( 2) on the number line, we obtain a sum of 6. − 22 22 22 26 25 24 23 22 21 0 1 CASE 3: × NEGATIVE INTEGER POSITIVE INTEGER A procedure similar to that used in case 2 will indicate that a negative integer times a positive integer is a negative integer . Solution We obtain the vertical difference by subtracting the lower elevation from the higher elevation. − − = + = 20,237 ( 282) 20,237 282 20,519 The vertical difference is 20,519 ft. 7 Now try Exercise 61 Multiplication Property of Zero ⋅ = ⋅ = a a 0 0 0 RECREATIONAL MATH Four 4’s The game of Four 4’s is a challenging way to learn about the operations on integers. In this game, you must use exactly four 4’s, and no other digits, along with one or more of the operations of addition, subtraction, multiplication, and division * to write expressions. You may also use as many grouping symbols (that is, parentheses and brackets) as you wish. The object of the game is to first write an expression that is equal to 0. Then write a second expression that is equal to 1, then write a third expression that is equal to 2, and so on up to 9. For example, one way to represent the number 2 is as follows: = = ⋅ + 2. 4 4 4 4 16 8 There are several other acceptable ways as well. For more information see MathIsFun.com/Puzzles/Four-Fours. Solutions for the whole numbers 0–9 can be found in the back of this book. See also Exercise 82 for an expansion of the rules needed to represent larger whole numbers. *This game will be expanded to include other operations such as exponents and square roots later in the book. Multiplication of Integers The multiplication property of zero is important in our discussion of multiplication of integers. It indicates that the product of 0 and any number is 0.
RkJQdWJsaXNoZXIy NjM5ODQ=