Beginning Algebra
Tutorial 8: Properties of Real Numbers
Learning Objectives
After completing this tutorial, you should be able to:
 Identify and use the addition and multiplication commutative properties.
 Identify and use the addition and multiplication associative properties.
 Identify and use the distributive property.
 Identify and use the addition and multiplication identity properties.
 Identify and use the addition and multiplication inverse properties.

Introduction
It is important to be familiar with the properties in this tutorial.
They lay the foundation that you need to work with equations, functions,
and formulas all of which are covered in later tutorials, as well as, your
algebra class. In
some cases, it isn't very helpful to rewrite an expression, but
in other cases it helps to write an equivalent expression to be
able to continue with a problem and solve it. An equivalent
expression is one that is written differently, but has the same value.
The properties on this page will get you up to speed as to how you can
write expressions in equivalent forms.

Tutorial
The Commutative Properties of
Addition and Multiplication
a + b = b + a and
ab = ba

The Commutative Property, in general, states that changing the ORDER
of two numbers either being added or multiplied, does NOT change
the value of it.
The two sides are called equivalent expressions because they look different
but have the same value. 
Example
1: Use the commutative property to write an equivalent
expression to 2.5x + 3y. 
Using the commutative property of addition (where changing the order
of a sum does not change the value of it) we get
2.5x + 3y = 3y + 2.5x.

Example
2: Use the commutative property to write an equivalent
expression to . 
Using the commutative property of multiplication (where changing
the order of a product does not change the value of it), we get

The Associative Properties of
Addition and Multiplication
a + (b + c) = (a + b) + c and
a(bc) = (ab)c

The Associative property, in general, states that changing the GROUPING
of numbers that are either being added or multiplied does NOT change the
value of it. Again, the two sides are equivalent to each other.
At this point it is good to remind you that
both the commutative and associative properties do NOT work for subtraction
or division.
Example
3: Use the associative property to write an equivalent
expression to (a + 5b) + 2c. 
Using the associative property of addition (where changing the grouping
of a sum does not change the value of it) we get
(a + 5b) + 2c = a + (5b + 2c).

Example
4: Use the associative property to write an equivalent
expression to (1.5x)y. 
Using the associative property of multiplication (where changing the
grouping of a product does not change the value of it) we get
(1.5x)y = 1.5(xy)

Distributive Properties
a(b + c) = ab + ac
or
(b + c)a = ba + ca

In other words, when you have a term being multiplied times two
or more terms that are being added (or subtracted) in a ( ), multiply the
outside term times EVERY term on the inside.
Remember terms are separated by + and .
This idea can be extended to more than two terms in the ( ).
Example
5: Use the distributive property to write 2(x  y) without parenthesis. 
Multiplying every term on the inside of the ( ) by 2 we get: 

*Distribute 2 to EVERY term inside (
)

Example
6: Use the distributive property to write  (5x + 7) without
parenthesis. 

*A  outside a ( ) is the same as times (1)
*Distribute the (1) to EVERY term inside
( )
*Multiply

Basically, when you have a negative sign in front of a ( ), like this
example, you can think of it as taking a 1 times the ( ). What you
end up doing in the end is taking the opposite of every term in the ( ). 
Example
7: Use the distributive property to find the product
3(2 a + 3 b + 4 c). 
As mentioned above, you can extend the distributive property to as
many terms as are inside the ( ). The basic idea is that you multiply
the outside term times EVERY term on the inside. 

*Distribute the 3 to EVERY term
*Multiply

Addition
The additive identity is 0
a + 0 = 0 + a = a

In other words, when you add 0 to any number, you end up with that
number as a result. 
Multiplication
Multiplication identity is 1
a(1) = 1(a) = a

And when you multiply any number by 1, you wind up with that number
as your answer. 
Additive Inverse (or negative)
For each real number a, there is a unique real number,
denoted a, such that
a + (a) = 0.

In other words, when you add a number to its additive inverse, the
result is 0. Other terms that are synonymous with additive inverse are
negative and opposite. 
Multiplicative Inverse
(or reciprocal)
For each real number a, except 0, there is a unique
real number such that

In other words, when you multiply a number by its multiplicative inverse
the result is 1. A more common term used to indicate a multiplicative
inverse is the reciprocal. A multiplicative inverse or reciprocal
of a real number a (except 0) is found by "flipping" a upside
down. The numerator of a becomes the
denominator of the reciprocal of a and the
denominator of a becomes the numerator of the
reciprocal of a.
These two inverses will come in big time handy
when you go to solve equations later on. Keep them in your memory
bank until that time.
Example
8: Write the opposite (or additive inverse) and
reciprocal (or multiplicative inverse) of 3. 
The opposite of 3 is 3, since 3 + 3 = 0. 
The reciprocal of 3 is 1/3, since 3(1/3) = 1.
When you take the reciprocal, the sign of the original number stays
intact. Remember that you need a number that when you multiply times
the given number you get 1. If you change the sign when you take
the reciprocal, you would get a 1, instead of 1, and that is a no no. 
Example
9: Write the opposite (or additive inverse) and reciprocal
(or multiplicative inverse) of 1/5. 
The opposite of 1/5 is 1/5, since 1/5 + (1/5) = 0. 
The reciprocal of 1/5 is 5, since 5(1/5) = 1. 
Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Use a commutative property to write an equivalent expression.
Practice
Problems 2a  2b: Use an associative property to write an equivalent
expression.
Practice
Problems 3a  3b: Use the distributive property to find the product.
Practice
Problems 4a  4b: Write the opposite (additive inverse) and the reciprocal
(multiplicative inverse) of each number.
Need Extra Help on these Topics?
Last revised on July 24, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
