Copyright © Cengage Learning. All rights reserved. Real Numbers and Their Basic Properties 1

Copyright © Cengage Learning. All rights reserved. Section 1.7 Properties of Real Numbers

3 Objectives Apply the closure properties by evaluating an expression for given values for variables. Apply the commutative and associative properties. Apply the distributive property of multiplication over addition to rewrite an expression

4 Objectives Recognize the identity elements and find the additive and multiplicative inverse of a nonzero real number. Identify the property that justifies a given statement

5 Apply the closure properties by evaluating an expression for given values for variables 1.

6 Closure Property The closure properties guarantee that the sum, difference, product, or quotient (except for division by 0) of any two real numbers is also a real number. Closure Properties If a and b are real numbers, then a + b is a real number. a – b is a real number. ab is a real number. is a real number (b  0).

7 Example 1 Let x = 8 and y = –4. Find the real-number answers to show that each expression represents a real number. a. x + y b. x – y c. xy d. Solution: We substitute 8 for x and –4 for y in each expression and simplify. a. x + y = 8 + (–4) = 4

8 Example 1 – Solution b. x – y = 8 – (–4) = = 12 c. xy = 8(–4) = –32 d. = = –2 cont’d

9 Apply the commutative and associative properties 2.

10 Commutative Property of Real Numbers The commutative properties (from the word commute, which means to go back and forth) guarantee that addition or multiplication of two real numbers can be done in either order. Commutative Properties If a and b are real numbers, then a + b = b + a commutative property of addition ab = ba commutative property of multiplication Comment Since 5 – 3  3 – 5 and 5  3  3  5, the commutative property cannot be applied to a subtraction or a division.

11 Example 2 Let x = –3 and y = 7. Show that a. x + y = y + x b. xy = yx Solution: a. We can show that the sum x + y is the same as the sum y + x by substituting –3 for x and 7 for y in each expression and simplifying. x + y = –3 + 7 = 4 and y + x = 7 + (–3) = 4 b. We can show that the product xy is the same as the product yx by substituting –3 for x and 7 for y in each expression and simplifying. xy = –3(7) = –21 and yx = 7(–3) = –21

12 Associative Property of Real Numbers The associative properties guarantee that three real numbers can be regrouped in an addition or multiplication. Associative Properties If a, b, and c are real numbers, then (a + b) + c = a + (b + c) associative property of addition (ab)c = a(bc) associative property of multiplication Because of the associative property of addition, we can group (or associate) the numbers in a sum in any way that we wish.

13 Apply the commutative and associative properties For example, (3 + 4) + 5 = (4 + 5) = = 12 = 12 The answer is 12 regardless of how we group the three numbers. The associative property of multiplication permits us to group (or associate) the numbers in a product in any way that we wish.

14 Apply the commutative and associative properties For example, (3  4)  7 = 12  7 3  (4  7) = 3  28 = 84 = 84 The answer is 84 regardless of how we group the three numbers. Comment Since (2 – 5) – 3  2 – (5 – 3) and (2  5)  3  2  (5  3), the associative property cannot be applied to subtraction or division.

15 Apply the distributive property of multiplication over addition to rewrite an expression 3.

16 Distributive Property of Real Numbers The distributive property shows how to multiply the sum of two numbers by a third number. Because of this property, we can often add first and then multiply, or multiply first and then add. For example, 2(3 + 7) can be calculated in two different ways. We will add and then multiply, or we can multiply each number within the parentheses by 2 and then add. 2(3 + 7) = 2(10) 2(3 + 7) = 2   7 = 20 = = 20 Either way, the result is 20.

17 Apply the distributive property of multiplication over addition to rewrite an expression In general, we have the following property. Distributive Property of Multiplication Over Addition If a, b, and c are real numbers, then a(b + c) = ab + ac Because multiplication is commutative, the distributive property also can be written in the form (b + c)a = ba + ca

18 Apply the distributive property of multiplication over addition to rewrite an expression We can interpret the distributive property geometrically. Since the area of the largest rectangle in Figure 1-25 is the product of its width a and its length b + c, its area is a(b + c). The areas of the two smaller rectangles are ab and ac. Since the area of the largest rectangle is equal to the sum of the areas of the smaller rectangles, we have a(b + c) = ab + ac. Figure 1-25

19 Apply the distributive property of multiplication over addition to rewrite an expression The previous discussion shows that multiplication distributes over addition. Multiplication also distributes over subtraction. For example, 2(3 – 7) can be calculated in two different ways. We will subtract and then multiply, or we can multiply each number within the parentheses by 2 and then subtract. 2(3 – 7) = 2(–4) 2(3 – 7) = 2  3 – 2  7 = –8 = 6 – 14 = –8 Either way, the result is –8. In general, we have a(b – c) = ab – ac

20 Example 3 Evaluate each expression in two different ways: a. 3(5 + 9) b. 4(6 – 11) c. –2(–7 + 3) Solution: a. 3(5 + 9) = 3(14) 3(5 + 9) = 3   9 = 42 = = 42 b. 4(6 – 11) = 4(–5) 4(6 – 11) = 4  6 – 4  11 = –20 = 24 – 44 = –20

21 Example 3 – Solution c. –2(–7 + 3) = –2(–4) –2(–7 + 3) = –2(–7) + (–2)(3) = 8 = 14 + (–6) = 8 cont’d

22 Apply the distributive property of multiplication over addition to rewrite an expression The distributive property can be extended to three or more terms. For example, if a, b, c, and d are real numbers, then a(b + c + d) = ab + ac + ad

23 Recognize the identity elements and find the additive and multiplicative inverse of a nonzero real number 4.

24 Additive Identity (0) and Multicative Identity (1) The numbers 0 and 1 play special roles in mathematics. The number 0 is the only number that can be added to another number (say, a) and give an answer that is the same number a: 0 + a = a + 0 = a The number 1 is the only number that can be multiplied by another number (say, a) and give an answer that is the same number a: 1  a = a  1 = a

25 Additive Inverse (-a) and Multicative Inverse (1/a) Given a, its additive inverse is –a, because a + (-a) = 0 3 and -3 are additive inverses of each other Given a, its multicative inverse is (1/a), because a ∙ (1/a) = 1 3 and (1/3) are multicative inverses of each other. (Also called the reciprocal of each other)

26 Your Turn Find the additive and multiplicative inverses of. Solution: The additive inverse of is because. The multiplicative inverse of is because.

27 Identify the property that justifies a given statement 5.

28 Example 6 The property in the right column justifies the statement in the left column. a is a real number closure property of addition b. is a real number closure property of division c = commutative property of addition d. –3 + (2 + 7) = (–3 + 2) + 7 associative property of addition e. (5)(–4) = (–4)(5) commutative property of multiplication

29 Example 6 f. (ab)c = a(bc) associative property of multiplication g. 3(a + 2) = 3a + 3  2 distributive property h = 3 additive identity property i. 3(1) = 3 multiplicative identity property j. 2 + (–2) = 0 additive inverse property k. multiplicative inverse property cont’d

30 Identify the property that justifies a given statement The properties of the real numbers are summarized as follows. Properties of Real Numbers