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1.3 – Properties of Real Numbers. Real Numbers 1.3 – Properties of Real Numbers.

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Presentation on theme: "1.3 – Properties of Real Numbers. Real Numbers 1.3 – Properties of Real Numbers."— Presentation transcript:

1 1.3 – Properties of Real Numbers

2 Real Numbers 1.3 – Properties of Real Numbers

3 Real Numbers (R)

4 1.3 – Properties of Real Numbers Real Numbers (R)

5 1.3 – Properties of Real Numbers Real Numbers (R) Rational

6 1.3 – Properties of Real Numbers Real Numbers (R) Rational (⅓)

7 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

8 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

9 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers

10 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers (-6)

11 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

12 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

13 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s

14 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s (0)

15 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

16 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

17 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s

18 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s (7)

19 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (7)

20 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

21 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

22 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

23 1.3 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (I) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

24 Example 1

25 Name the sets of numbers to which each apply.

26 Example 1 Name the sets of numbers to which each apply.

27 Example 1 Name the sets of numbers to which each apply.

28 Example 1 Name the sets of numbers to which each apply. (a) √ 16

29 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4

30 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N

31 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W

32 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z

33 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q

34 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q, R

35 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185

36 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z

37 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q

38 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R

39 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20

40 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R

41 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞

42 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q

43 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R

44 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45

45 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) Q

46 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) Q, R

47 Properties of Real Numbers PropertyAdditionMultiplication Commutativea + b = b + aa·b = b·a Associative (a+b)+c = a+(b+c) (a · b) · c = a · (b · c) Identitya+0 = a = 0+aa·1 = a = 1·a Inversea+(-a) =0= -a+aa·1 =1= 1·a a a Distributivea(b+c)=ab+ac and (b+c)a=ba+ca

48 Example 2

49 Name the property used in each equation.

50 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7)

51 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition

52 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x

53 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x Associative Multiplication

54 Example 3 What is the additive and multiplicative inverse for -1¾?

55 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾

56 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + = 0

57 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0

58 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾

59 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾ · = 1

60 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: (-1¾)(- 4 / 7 ) = 1

61 1.4 – The Distributive Property

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

63 Example 4

64 Simplify 2(5m+n)+3(2m–4n).

65 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

66 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

67 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)

68 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

69 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

70 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)

71 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

72 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

73 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)

74 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

75 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

76 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n)

77 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m

78 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m +

79 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n

80 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n +

81 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m

82 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m –

83 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

84 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

85 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

86 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n

87 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m

88 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m – 10n


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