226 CHAPTER 5 Number Theory and the Real Number System Addition of Integers Addition of integers can be represented geometrically using a number line. To do so, begin at 0 on the number line. Represent the first addend (the first number to be added) by an arrow starting at 0. Draw the arrow to the right if the addend is positive. If the addend is negative, draw the arrow to the left. From the tip of the first arrow, draw a second arrow to represent the second addend. Draw the second arrow to the right or left, as just explained. The sum of the two integers is found at the tip of the second arrow. Example 3 Adding Integers Evaluate the following using the number line. a) + − 3 ( 5) b) − + − 1 ( 4) c) 6 4 − + d) + − 3 ( 3) Solution a) 23 22 25 21 0 1 2 3 3 4 5 24 25 Thus, + − = − 3 ( 5) 2. b) 24 21 22 21 24 23 25 0 1 2 3 4 5 Thus, − + − = − 1 ( 4) 5. c) 23 22 26 21 0 1 2 4 3 4 24 25 26 Thus, 6 4 2. − + = − d) 23 3 22 21 24 23 25 0 1 2 3 4 5 6 Thus, + − = 3 ( 3) 0. 7 Now try Exercise 11 In Example 3(d), the number −3 is said to be the additive inverse of 3 and 3 is said to be the additive inverse of −3 because their sum is 0. In general, the additive inverse of the number n is −n, since + − = n n ( ) 0. Inverses are discussed more formally in Chapter 9. Notice in the paragraph above we used the letter n to make a general statement about additive inverses. In mathematics, and throughout this textbook, we will often use letters called variables to represent numbers. Variables are usually shown in italics. Subtraction of Integers Any subtraction problem can be rewritten as an addition problem. To do so, we use the following rule of subtraction.
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