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Chapter 1 11 Glencoe Algebra 2

1-2 Study Guide and InterventionProperties of Real Numbers

Real Numbers All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and integers.

R real numbers {all rationals and irrationals}

Q rational numbers {all numbers that can be represented in the form m − n , where m and n are integers and

n is not equal to 0}

I irrational numbers {all nonterminating, nonrepeating decimals}

Z integers {…, -3, -2, -1, 0, 1, 2, 3, …}

W whole numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, …}

N natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, …}

Name the sets of numbers to which each number belongs.

a. - 11 −

3 rationals (Q), reals (R)

b. √ ��

25

√

��

25 = 5 naturals (N), wholes (W), integers (Z), rationals (Q), reals (R)

Exercises

Name the sets of numbers to which each number belongs.

1. 6 −

7 2. -

√

��

81 3. 0 4. 192.0005

5. 73 6. 34 1 −

2 7. √�

36 −

9 8. 26.1

9. π 10. 15 −

3 11. - 4.

−−

17

12. √

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25 −

2 13. -1 14. √�

42

15. -11.2 16. - 8 −

13 17.

√

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5 −

2

18. 33. −

3 19. 894,000 20. -0.02

Example

Q, R Z, Q, R W, Z, Q, R Q, R

N, W, Z, Q, R Q, R Q, R Q, R

I, R N, W, Z, Q, R Q, R

Q, R Z, Q, R I, R

Q, R Q, R I, R

Q, R N, W, Z, Q, R Q, R

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Chapter 1 12 Glencoe Algebra 2

Study Guide and Intervention (continued)

Properties of Real Numbers

1-2

Properties of Real Numbers

Simplify 9x + 3y + 12y - 0.9x.

9x + 3y + 12y - 0.9x = 9x + (- 0.9x) + 3y + 12y Commutative Property (+)

= (9 + (- 0.9))x + (3 + 12)y Distributive Property

= 8.1x + 15y Simplify.

ExercisesSimplify each expression.

1. 8(3a - b) + 4(2b - a) 2. 40r + 18t - 5t + 11r 3. 1 − 5 (4j + 2k -6j + 3k)

4. 10(6g + 3h) + 4(5g - h) 5. 12( a − 3 - b −

4 ) 6. 8(2.4r - 3.1t) - 6(1.5r + 2.4t)

7. 4(20 - 4p) - 3 − 4 (4 - 16p) 8. 5.5j + 8.9k - 4.7k -10.9j 9. 1.2(7x - 5y) - (10y - 4.3x)

10. 9(7d - 4f ) - 0.6(d + 5f ) 11. 2.5(12m - 8.5p) 12. 3 − 4 p - 1 −

5 r - 3 −

5 r - 1 −

2 p

13. 4(10g + 80h) - 20(10h - 5g) 14. 2(15d + 45c) + 5 − 6 (12d + 18c)

15. (7y - 2.1x)3 + 2(3.5x - 6y) 16. 2 − 3 (18m - 6p + 12m + 3p)

17. 14( j - 2k) - 3j(4 - 7k) 18. 50(3a - b) - 20(b - 2a)

Example

Real Number Properties

For any real numbers a, b, and c

Property Addition Multiplication

Commutative a + b = b + a a � b = b · a

Associative (a + b) + c = a + (b + c) (a � b) � c = a � (b � c)

Identity a + 0 = a = 0 + a a � 1 = a = 1 � a

Inverse a + (-a) = 0 = (-a) + a a � 1 − a = 1 = 1 − a · a, a ≠ 0.

Closure a + b is a real number. a � b is a real number.

Distributive a(b + c) = ab + ac and (b + c)a = ba + ca

20a k - 2 − 5 j

80g + 26h 4a - 3b

77 - 4p 12.7x - 16y

62.4d - 39f 30m - 21.25p

140g + 120h

0.7x + 9y

2j - 7k

51r + 13t

10.2r - 39.2t

4.2k - 5.4j

1 − 4 p - 4 −

5 r

40d + 105c

20m - 2p

190a - 70b

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