1.3

1
1.3 Properties of Real Numbers

2
1.3 Properties of Real Numbers

• Real Numbers

3
1.3 Properties of Real Numbers

• 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 (?)

7
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)

8
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)

9
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• Integers

10
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• Integers (-6)

11
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)

12
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)

13
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• Whole s

14
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• Whole s (0)

15
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)

16
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)

17
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)
• Natural s

18
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)
• Natural s (7)

19
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)
• (N) Natural s (7)

20
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?)
• (Z) Integers (-6)
• (W) Whole s (0)
• (N) Natural s (1)

21
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?) Irrational
• (Z) Integers (-6)
• (W) Whole s (0)
• (N) Natural s (1)

22
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?) Irrational v 5
• (Z) Integers (-6)
• (W) Whole s (0)
• (N) Natural s (1)

23
1.3 Properties of Real Numbers

• Real Numbers (R)
• (Q) Rational (?) (I) Irrational v 5
• (Z) Integers (-6)
• (W) Whole s (0)
• (N) Natural s (1)

24
Example 1

25
Example 1

• 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) v 16

29
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4

30
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4 - N

31
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4 - N, W

32
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4 - N, W, Z

33
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4 - N, W, Z, Q

34
Example 1

• Name the sets of numbers to which each apply.
• (a) v 16 4 - N, W, Z, Q, R

35
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185

36
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z

37
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q

38
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R

39
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20

40
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R

41
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -?

42
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -? - Q

43
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -? - Q, R

44
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -? - Q, R
• __
• (e) 0.45

45
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -? - Q, R
• __
• (e) 0.45 - Q

46
Example 1

• Name the sets of numbers to which each apply.
• v 16 4 - N, W, Z, Q, R
• -185 - Z, Q, R
• v 20 - I, R
• (d) -? - Q, R
• __
• (e) 0.45 - Q, R

47
Properties of Real Numbers
Property Addition Multiplication
Commutative a b b a ab ba
Associative (ab)c a(bc) (ab)c a(bc)
Identity a0 a 0a a1 a 1a
Inverse a(-a) 0 -aa a1 1 1a a a
Distributive a(bc)abac and (bc)abaca a(bc)abac and (bc)abaca
48
Example 2
49
Example 2

• 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) (34)x

53
Example 2

• Name the property used in each equation.
• (a) (5 7) 8 8 (5 7)
• Commutative Addition
• (b) 3(4x) (34)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