Math 154 Peacemaker
Angel/1.10
Properties of the Real Number System
1.
The Commutative Properties (Order does not matter,)
A. The Commutative Property for Addition:
a+b=b+a
B. The Commutative Property for Multiplication: ab = ba
Example 1 Use the commutative properties for addition and multiplication to write an
expression equivalent to each of the following:
a) a + 2 =
b) 4x =
C. The commutative property DOES NOT hold for subtraction or division.
2.
The Associative Properties (Grouping does not matter.)
A. The Associative Property for Addition:
(a + b) + c = a + (b + c)
B. The Associative Property for Multiplication:
(ab)c = a(bc)
Example 2 Use the associative properties for addition and multiplication to write an
expression equivalent to each of the following:
a) x + (2 + y) =
b) 9(rp) =
C. The associative property DOES NOT hold for subtraction or division.
We often use the commutative and associative properties unconsciously when we add numbers, as
in the example below:
(27 + 35) + (3 + 25) =
=
=
=
=
27 + (35 + 3) + 25
27 + (3 + 35) + 25
(27 + 3) + (35 + 25)
30 + 60
90
associative property
commutative property
associative property
addition
addition
3.
The Distributive Property of Multiplication over Addition
A. The Distributive Property states that for real numbers a, b, and c
a(b + c) = ab + ac
For example, 4(5 + 7) = (4 · 5) + (4 · 7)
4 · 12 = 20 + 28
48 = 48
So you can add first and then multiply or you can multiply first and then add and you’ll get
the same answer.
B. This is the MOST IMPORTANT property for simplifying algebraic expressions.
For example, 4(x + 7) = 4 · x + 4 · 7 = 4x + 28
This is the only way to simplify this expression because we don’t know what value x
represents.
Example 3
Rewrite each of the following using the distributive property.
a) 4(5 + 8) =
b) 4(x + 8) =
c) 3(a + b + c) =
d) (y – 3)4 =
Example 4
Name each property illustrated.
a) –3 + (–5) = –5 + (–3)
b) 6(x – 9) = 6x – 54
c) rg = gr
d) 6 + (w + 1) = (6 + w) + 1
Can you tell the difference between 2(5 + c) vs 2(5 · c)?
4.
The Identity Properties
A. Since adding 0 to any number makes the number remain unchanged or identical, we call 0
the additive identity or, for any real number a, a + 0 = 0 + a = a
B. Since multiplying any number by 1 makes the number remain unchanged or identical, we
call 1 the multiplicative identity or, for any real number a, a · 1 = 1 · a = a
Math 154 – Angel – 1.10
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We often use the identity properties without realizing it. For example when building a fraction to
make it have a common denominator as follows
1
1
1 2
1 3
+
=
⋅ + ⋅
3
2
3 2
2 3
=
=
5.
2
3
+
6
6
5
6
The Inverse Properties
A. Any two numbers that add to 0 (the additive identity) are called additive inverses, so,
additive inverses are opposites.
B. The Additive Inverse Property states that for any real number a,
a + (–a) = (–a) + a = 0
C. Any two numbers that multiply to 1(the multiplicative identity) are called multiplicative
inverses, so multiplicative inverses are reciprocals.
D. The Multiplicative Inverse Property states that for any real number a ≠ 0,
1
1
a⋅ = ⋅a = 1
a
a
Example 5
Name each property illustrated.
a) 5y + (–5y) = 0
b) 2z · 1 = 2z
c) 4(x – 7)
d) 3x ⋅
= (4 · x) + ( 4 · –7)
1
= 1
3x
= 4x – 28
Example 6 In parts a –c, the name of a property is given followed by part of an equation.
Complete the equation, to the right of the equal sign, to illustrate the given property.
a) Associative property of multiplication
(3 · 6) · 9 =
b) Distributive property
7(x – 4) =
c) Identity property of addition
6b + 0 =
Math 154 – Angel – 1.10
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