Section 1'5 Properties of Real Numbers

1
Section 1.5Properties of Real Numbers

• Commutative
• Associative
• Distributive
• Identity
• Inverse

2
Commutative Property of Addition

• Changing the order of the numbers in a sum will
  not change the result.
• a b b a
• 7 2 2 7

3
Commutative Property of Multiplication

• Changing the order of the numbers in a product
  will not change the result.
• a b b a
• 3 4 4 3

4
Associative Property of Addition

• Changing the grouping of the numbers in a sum
  will not change the result.
• a (b c) (a b) c
• 3 (4 5) (3 4) 5
• 3 9 7 5
• 12 12

5
Associative Property of Multiplication

• Changing the grouping of the numbers in a product
  will not change the result.
• a(bc) (ab)c
• 2(3 4) (2 3)4
• 2(12) (6)4
• 24 24

6
Distributive Property

• Multiplication distributes over addition.
• a(b c) ab ac
• 3(2 5) 32 35
• 3(7) 6 15
• 21 21

7
Compute the area of the rectangle two ways.

• First compute the result directly
• 3 5 15

5
3
3
2
8
Compute the area of the rectangle two ways.

• Now compute each region.
• 3 3 3 2
• 9 6
• 15

9
6
3
3
2
9
Compute the area of the rectangle two ways.

• 3(3 2) 33 32
• 3(5) 9 6
• 15 15

10
Additive Identity Property

• There exists a unique number 0 such that zero
  preserves identities under addition.
• a 0 a and 0 a a
• In other words adding zero to a number does not
  change its value.

11
Multiplicative Identity Property

• There exists a unique number 1 such that the
  number 1 preserves identities under
  multiplication.
• a 1 a and 1 a a
• In other words multiplying a number by 1 does not
  change the value of the number.

12
Additive Inverse Property

• For each real number a there exists a unique real
  number a such that their sum is zero.
• a (-a) 0
• In other words opposites add to zero.

13
Multiplicative Inverse Property

• For each real number a, except 0, there exists a
  unique real number
• such that their product is 1.

14
Getting Ready for Class

• What is the commutative property of addition?
• Do you know from your experience with numbers
  that the commutative property is true?
• Write the commutative property of multiplication
  in symbols and words.
• How do you rewrite expressions using the
  distributive property?

15
Homework

• Problem Set 1.5
• Page 45
• 1-89 odd

16
1 State the property or properties that justify
the following.

• 3 2 2 3
• Commutative property

17
4 State the property or properties that justify
the following.

• 10(0.1) 1
• Multiplicative inverse property

18
6 State the property or properties that justify
the following.

• 3(x 10) 3x 30
• Distributive property

19
8 State the property or properties that justify
the following.

• 3 (4 5) (3 4) 5
• Associative property

20
10 State the property or properties that justify
the following.

• (5 2) 9 (2 5) 9
• Commutative property

21
16 State the property or properties that justify
the following.

• a(5b) 5(ab)
• Commutative property
• Associative property

22
22 Each of the following problems has a mistake
in it. Correct the right-hand side.

• 5(4 x) 4 5x
• 20 5x

23
24 Each of the following problems has a mistake
in it. Correct the right-hand side.

• 2(y 1) 2y 1
• 2y 2

24
32 Use the associative property to rewrite each
of the following expressions and then simplify
the result.

• 5 (6 x)
• (5 6) x
• 11 x

25
62 Apply the distributive property to each of
the following expressions. Simplify when
possible.
26
76 Apply the distributive property to each of
the following expressions. Simplify when
possible.

• 6(a 2) 8
• 6a 12 8
• 6a 20