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Section 1.4
Properties of Real Numbers
-Relationships that are ALWAYS true for real
numbers are called properties.
-Equivalent expressions are two algebraic
expressions that have the same value for all values
of the variable(s).
Commutative Property

Changing the order of the
addends/factors does not change the
sum/product

ADDITION:
◦a+b=b+a
 Ex:

8+4=4+8
MULTIPLICATION:
◦a•b=b•a
 Ex:
2•3=3•2
Associative Property

Changing the grouping of the
addends/factors does not change the
sum/product

ADDITION:
◦ (a + b) + c = a + (b + c)
 Ex:

(20 + 3) + 5 = 20 + (3 + 5)
MULTIPLICATION
◦ (a • b) • c = a • (b • c)
 Ex:
(2 • 3) • 4 = 2 • (3 • 4)
Identity Property

ADDITION: The sum of any real
number and 0 is the original number
◦a+0=a
 Ex:

6+0=6
MULTIPLICATION: The product of any
real number and 1 is the original number
◦a•1=a
 Ex:
7•1=7
Zero Property of Multiplication

The product of a and 0 is 0
◦a•0=0
 Ex:
5•0=0
Multiplication Property of -1

The product of -1 and a is –a
◦ -1 • a = -a
 Ex:
-1 • 7 = -7
Identifying Properties

What property is illustrated by each
statement?
1)
42 • 0 = 0
2)
(y + 2.5) + 28 = y + (2.5 + 28)
3)
10x + 0 = 10x
Example
A movie ticket costs $7.75. A drink costs
$2.40. Popcorn costs $1.25. What is the
total cost for a ticket, a drink and a
popcorn? Use mental math!!
7.75 + 2.40 + 1.25 =
Equivalent Expressions
5(3n)
=
=
(4 + 7b) + 8
6xy=
y
(5 • 3)n
15n
=
=
=
6x • y
(7b + 4) + 8
7b + (4 + 8)
7b + 12
=
1 •y
6x • 1 = 6x
Examples - Simplify each expression.
Justify each step.
2.1(4.5x)
6 + (4h + 3)
8m
12mn
Deductive Reasoning

The process of reasoning logically from
given facts to a conclusion

A counterexample is needed to show
that a statement is false.
◦ You need only one counterexample to prove
that a statement is false
Is the statement true or false? If it is
false, give a counterexample.
1) For all real numbers a and b,
a • b = b +a
2) For all real numbers a, b, and c,
(a + b) + c = b + (a + c)