Properties of Real Numbers

1
Properties of Real Numbers

• The properties of real numbers
• allow us to manipulate expressions
• and equations and find the values
• of a variable.

2
Number Classification

• Natural numbers are the counting numbers.
• Whole numbers are natural numbers and zero.
• Integers are whole numbers and their opposites.
• Rational numbers can be written as a fraction.
• Irrational numbers cannot be written as a
  fraction.
• All of these numbers are real numbers.

3
Number Classifications

• Subsets of the Real Numbers

Q - Rational
Z - Integers
I - Irrational
W - Whole
N - Natural
4
Classify each number

• -1

• real, rational, integer
• real, rational, integer, whole, natural
• real, irrational
• real, rational
• real, rational, integer, whole
• real, rational

6
0
-2.222
5
Properties of Real Numbers

• Commutative Property
• Think commuting to work.
• Deals with ORDER. It doesnt matter what order
  you ADD or MULTIPLY.

• ab ba
• 4 6 6 4

6
Properties of Real Numbers

• Associative Property
• Thinkthe people you associate with, your group.
• Deals with grouping when you Add or Multiply.
• Order does not change.

7
Properties of Real Numbers

• Associative Property

• (a b) c a ( b c)
• (nm)p n(mp)

8
Properties of Real Numbers

• Additive Identity Property
• s 0 s
• Multiplicative Identity Property
• 1(b) b

9
Properties of Real Numbers

• Distributive Property
• a(b c) ab ac
• (r s)9 9r 9s

10
Name the Property

• 5 5 0
• 5(2x 7) 10x 35
• 8 7 7 8
• 24(2) 2(24)
• (7 8) 2 2 (7 8)

• Additive Identity
• Distributive
• Commutative
• Commutative
• Commutative

11
Name the Property

• Associative
• Multiplicative
• Identity
• Distributive
• Distributive

• 7 (8 2) (7 8) 2
• 1 v -4 v -4
• (6 - 3a)b 6b - 3ab
• 4(a b) 4a 4b

12
Properties of Real Numbers

• Reflexive Property

• a b a b

The same expression is written on both sides
of the equal sign.
13
Properties of Real Numbers
Symmetric Property

• If a b then b a
• If 4 5 9 then 9 4 5

14
Properties of Real Numbers

• Transitive Property

• If a b and b c then a c
• If 3(3) 9 and 9 4 5, then 3(3) 4 5

15
Properties of Real Numbers

• Substitution Property

• If a b, then a can be replaced by b.
• a(3 2) a(5)

16
Name the property

• 5(4 6) 20 30
• 5(4 6) 5(10)
• 5(4 6) 5(4 6)
• If 5(4 6) 5(10) then 5(10) 5(4 6)
• 5(4 6) 5(6 4)
• If 5(10) 5(4 6) and 5(4 6) 20 30 then
  5(10) 20 30

• Distributive
• Substitution
• Reflexive
• Symmetric
• Commutative
• Transitive