Download Properties of Real numbers

1.8
PROPERTIES OF REAL NUMBERS
Introduction
 Properties of Real Numbers allow you to
write equivalent expressions and to simplify
expressions.
 The following list of properties applies to
addition and multiplication.
 What about Subtraction and Division?
 Remember our rules for addition and
multiplication:
 We can think of all subtraction problems as addition
problems (add the opposite) and all division
problems can be turned into multiplication problems
(multiply by the reciprocal.)
The Commutative Property
 Think: Change the order of the numbers;
move the numbers around
 The Commutative Property of Addition
 a+b=b+a
 Example: 6 + 2 = 2 + 6
 Is that true?
 The Commutative Property of Multiplication
 ab=ba
 Example: 4  5 = 5  4
 Is that true?
The Associative Property
 Think: Move the parentheses to associate (or
combine) different numbers
 Associative Property of Addition
 (a + b) + c = a + (b + c)
 Example: (1 + 2) + 3 = 1 + (2 + 3)
 Is that true?
 Associative Property of Multiplication
 (a  b) c = a (b  c)
 Example: (4  2) 3 = 4(2  3)
 Is that true?
Identity Properties
 Identity Property of Addition
 Think: What can I add to a number without changing its
identity?
 Add 0 and I’ll get what I started with
 a+0 =a
 Example: 14 + 0 = 14
 Identity Property of Multiplication
 Think: What can I multiply a number by without changing
its identity?
 Multiply by 1 and I’ll get what I started with
 a1=a
 Example: 26  1 = 26
Inverse Properties
 Inverse Property of Addition
 Think: adding opposites = 0
 a + (-a) = 0
 Example: 6 + -6 = 0
 Inverse Property of Multiplication
 Think: multiplying reciprocals = 1
 a(
)=1
 Example: 7 (
)=1
The Distributive Property
 Think: Distribute your outside number to
each of your inside numbers
 Multiply both of your inside #’s by your outside
#’s, then add or subtract.
 a( b + c) = ab + ac
 a (b – c) = ab – ac
 Example: 10(20 – 2) = 10(20) – 10(2)
10(18) = 200 – 20
180 = 180
Multiplication Properties
 Multiplication Property of Zero
 Think: Multiply anything by 0 and you’ll get 0
 n0=0
 Example: 245.5  0 = 0
 Multiplication Property of –1
 Think: Multiply by -1 means you switch the sign
 -1  n = –n
 -1  68 = – 68
Identifying These Properties
 Think: What’s happening with the numbers?
What operation is involved?
1. 6 + 2 = 2 + 6


We’re switching the order
Commutative Property of Addition
2. 5 + 0 = 5


We’re adding zero. We get what we started with
Identity Property of Addition
 Think: What’s happening with the numbers?
What operation is involved?
3. -3 + (5 + 6) = (-3 + 5) + 6


We’re not changing the order, but we’re moving
around the parentheses. We’re adding.
Associative Property of Addition
4. 3  1 = 3


We’re multiplying by 1, and we get what we
started with.
Identity Property of Multiplication
 Think: What’s happening with the numbers?
What operation is involved?
5. - 8 + 8 = 0


We’re adding opposites and we get 0.
Inverse Property of Addition
6. 10 (


)=1
We’re multiplying reciprocals and we get 1.
Inverse Property of Multiplication
 Think: What’s happening with the numbers?
What operation is involved?
7. 6(2 – a) = 12 – 6a


We’re multiplying both of our inside numbers by
our outside number.
The Distributive Property
8. -1  5 = -5


We’re multiplying 5 by -1, and we get the
opposite of 5.
Multiplication Property of -1
 Think: What’s happening with the numbers?
What operation is involved?
9. 3(a  4) = (3  a)  4


We aren’t changing the order, but we’re moving
around the parentheses. We’re multiplying.
The Associative Property of Multiplication
10. 8  0 = 0


We’re multiplying by 0 and we get 0.
Multiplication Property of Zero
Before you Leave…
 On your notes page, give me an example of
the Identity Property of Addition.
 Explain the difference between the
Commutative Property of Multiplication and
the Associative Property of Multiplication.