Download 1.3 Notes: Properties of Numbers

Notes: Properties of Numbers
1.3: I can apply algebraic properties to simplify algebraic expressions. (C.1.c)
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Equivalent Expressions: ____________________________________________________

The properties below allow you to write an equivalent expression for a given expression.
Properties of Equality
Symbols
For any number
a, a = a.
Property
Reflexive
Property
Words
Any quantity is equal
to itself.
Symmetric
Property
If one quantity equals
a second quantity,
then the second
quantity equals the
first.
For any numbers
a and b, if a = b,
then b = a.
Transitive
Property
If one quantity equals
a second quantity and
the second quantity
equals a third
quantity, then the first
quantity equals the
third quantity.
For any numbers
a, b, and c, if a =
b and b = c, then
a = c.
A quantity may be
substituted for its
equal in any
expression.
If a = b, then a
may be replaced
by b in any
expression.
Substitution
Property
Examples
Key Words
 Name the Property of Equality illustrated in each example below.
1. The distance from Sarah’s house to the school is equal to the distance from the school to Sarah’s house.
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2. If the area of a triangle is calculated by 2 𝑏ℎ. If you have a triangle with a base of 5 ft. and a height of 10
1
ft. you can calculate the area by solving (5)(10).
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3. Having 2 apples plus 8 apples is the same as having 10 apples, and having 10 apples is the same as having
2 apples plus 8 apples.
4. If two quarters is equal to five dimes, and five dimes is equal to ten nickels, then two quarters is equal to
ten nickels.

There are special properties associated with both addition and multiplication.
Property
Words
Commutative The order in which
Property
you add or multiply
numbers does not
change the sum or
product.
Symbols
For any numbers
a and b,
a+b = b+a
and
a•b = b•a
Associative
Property
The way you group
three or more
numbers when
adding or multiplying
does not change
their sum or product.
Property
Additive
Identity
Or
Identity
Property of
Addition
Multiplicative
Identity
Or
Identity
Property of
Multiplication
Words
For any number a,
the sum of a and 0 is
a.
Property
Additive
Inverse
Or
Inverse
Property of
Addition
Words
A number and its
opposite are additive
inverses of each
other.
Multiplicative
Inverse
Or
Inverse
Property of
Multiplication
For every number ,
𝑏
where a ≠ 0 and
b ≠ 0, there is exactly
𝑏
one number 𝑎 such
For any number a,
the product of a and
1 is a.
𝑎
𝑎
that the product of 𝑏
𝑏
and 𝑎 is 1.
Examples
Key Words
For and numbers
a, b, and c,
(a+b)+c = a+(b+c)
and
(a•b)•c = a•(b•c)
- Identity Properties Symbols
a+0=a
0+a=a
Examples
Key Words
Examples
Key Words
a•1=a
1•a=a
- Inverse Properties Symbols
a + (-a) = 0
a–a=0
𝑎
𝑏
•𝑎=1
𝑏
𝑏
𝑎
•𝑏=1
𝑎
Property
Words
Multiplication For any number a,
Property of
the product of a and
Zero
0 is 0.

Name the Property illustrated in each example below.
5. 5 + 0 = 5
6. 8 + 6 = 6 + 8
1
7. 5 ∙ 5 = 1
8. (9 ∙ 5) ∙ 2 = 9 ∙ (5 ∙ 2)
9. 12 ∙ 1 = 12
10. 4 + (−4) = 0
11. 7 ∙ 0 = 0
12.
Property of Zero
Symbols
a•0=0
0•a=0
2 3
∙ =1
3 2
13. 8 ∙ 2 = 2 ∙ 8
14. 0 + 14 = 14
15. 9 + (8 + 7) = (9 + 8) + 7
16. 3 − 3 = 0
Examples
Key Words

Evaluating Expressions using Properties of Numbers
Properties Checklist for Evaluating Expressions
 Identity Property of Addition
 Identity Property of Multiplication
 Inverse Property of Addition
 Inverse Property of Multiplication
 Multiplication Property of Zero
 Substitution Property

Evaluate each expression. Name the property used in each step.
1
17. 7(4 − 3) − 1 + 5 ∙ 5
1
7
18. 7 ∙ + 6(15 ÷ 3 − 5)
19. 7 + (9 − 32 )
1
20. 6 ∙ 6 + 5(12 ÷ 4 − 3)