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Study Guide and Intervention

Properties of Numbers
Identity and Equality Properties The identity and equality properties in the chart
below can help you solve algebraic equations and evaluate mathematical expressions.
Additive Identity For any number a, a + 0 = a.
Additive Inverse For any number a, a + (-a) = 0.
Multiplicative Identity For any number a, a . 1 = a.
Multiplicative Property of 0 For any number a, a . 0 = 0.
a b
Multiplicative Inverse For every number − , where a, b ≠ 0, there is exactly one number −
a such that
a . b b
Property − − = 1.
b a
Reflexive Property For any number a, a = a.
Symmetric Property For any numbers a and b, if a = b, then b = a.
Transitive Property For any numbers a, b, and c, if a = b and b = c, then a = c.
Substitution Property If a = b, then a may be replaced by b in any expression.

Example Evaluate 24  1 - 8 + 5(9 ÷ 3 - 3). Name the property used in each step.
24 ․ 1 - 8 + 5(9 ÷ 3 - 3) = 24 ․ 1 - 8 + 5(3 - 3) Substitution; 9 ÷ 3 = 3
= 24 ․ 1 - 8 + 5(0) Substitution; 3 - 3 = 0
= 24 - 8 + 5(0) Multiplicative Identity; 24 ․ 1 = 24
= 24 - 8 + 0 Multiplicative Property of Zero; 5(0) = 0
= 16 + 0 Substitution; 24 - 8 = 16
= 16 Additive Identity; 16 + 0 = 16

Exercises

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Evaluate each expression. Name the property used in each step.

4 (2) 
2
1 1
1. 2 − + − 2. 15 ․ 1 - 9 + 2(15 ÷ 3 - 5)
= 15  1 - 9 +2(5 - 5) Substitution
1
=2 −
4 ( )
1
+−
4
Substitution = 15  1 - 9 +2(0) Substitution
= 15  1 - 9 + 0 Mult. Prop. Zero
1
=2 −
2 () Substitution = 15 - 9 + 0 Mult. Identity
=1 Mult. Inverse = 6-0 Substitution
= 6 Substitution

1
3. 2(3 ․ 5 ․ 1 - 14) - 4 ․ − 4. 18 ․ 1 - 3 ․ 2 + 2(6 ÷ 3 - 2)
4
1
= 2(15  1 - 14) - 4  − Subst. = 18  1 - 3  2 + 2(2 - 2) Subst.
4
1
= 2(15 - 14) - 4  − Mult. Identity = 18  1 - 3  2 + 2(0) Substitution
4
1 = 18 - 3  2 + 2(0) Mult. Identity
= 2(1) - 4  − Substitution
4
1 = 18 - 6 + 2(0) Substitution
=2-4 −  Mult. Identity
4 = 18 - 6 + 0 Mult. Prop. Zero
=2-1 Mult. Inverse = 12 + 0 Substitution
=1 Substitution = 12 Add. Identity

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Study Guide and Intervention (continued)

Properties of Numbers
Commutative and Associative Properties The Commutative and Associative
Properties can be used to simplify expressions. The Commutative Properties state that the
order in which you add or multiply numbers does not change their sum or product. The
Associative Properties state that the way you group three or more numbers when adding or
multiplying does not change their sum or product.

Commutative Properties For any numbers a and b, a + b = b + a and a  b = b  a.

Associative Properties For any numbers a, b, and c, (a + b) + c = a + (b + c ) and (ab)c = a(bc).

Example 1 Evaluate 6  2  3  5 Example 2 Evaluate

using properties of numbers. Name the 8.2 + 2.5 + 2.5 + 1.8 using properties of
property used in each step. numbers. Name the property used in
6․2․3․5=6․3․2․5 Commutative Property each step.
= (6 ․ 3)(2 ․ 5) Associative Property 8.2 + 2.5 + 2.5 + 1.8
= 18 ․ 10 Multiply. = 8.2 + 1.8 + 2.5 + 2.5 Commutative Prop.
= 180 Multiply. = (8.2 + 1.8) + (2.5 + 2.5) Associative Prop.
= 10 + 5 Add.
The product is 180.
= 15 Add.

The sum is 15.

Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Evaluate each expression using properties of numbers. Name the property used in
each step. Properties will vary. See students’ work.

1. 12 + 10 + 8 + 5 35 2. 16 + 8 + 22 + 12 58 3. 10 ․ 7 ․ 2.5 175

4. 4 ․ 8 ․ 5 ․ 3 480 5. 12 + 20 + 10 + 5 47 6. 26 + 8 + 4 + 22 60

1 1 3 ․
7. 3 − +4+2− + 3 13 8. − 12 ․ 4 ․ 2 72 9. 3.5 + 2.4 + 3.6 + 4.2 13.7
2 2 4

1 1
10. 4 − +5+− + 3 13 11. 0.5 ․ 2.8 ․ 4 5.6 12. 2.5 + 2.4 + 2.5 + 3.6 11
2 2

4 ․ 2 1 ․ 1 ․ 1 ․ ․ 1
13. − 18 ․ 25 ․ − 80 14. 32 ․ − − 10 32 15. − 7 16 ․ − 4
5 9 5 2 4 7

1 ․ 1 3 ․ 1
16. 3.5 + 8 + 2.5 + 2 16 17. 18 ․ 8 ․ − − 8 18. − 10 ․ 16 ․ − 60
2 9 4 2

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