Beginning Algebra
Tutorial 7: Multiplying and Dividing Real Numbers
Learning Objectives
After completing this tutorial, you should be able to:
 Find the reciprocal of a number.
 Multiply positive and negative numbers.
 Divide positive and negative numbers.
 Multiply by zero.
 Know that dividing by zero is undefined.

Introduction
This tutorial reviews multiplying and dividing real
numbers and intertwines
that with some order of operation and evaluation problems. It
also
reminds you that dividing by 0 results in an undefined answer. In
other words, it is a big no, no.
I have the utmost confidence that you are familiar with
multiplication
and division, but sometimes the rules for negative numbers (yuck!) get
a little mixed up from time to time. So, it is good to go over
them
to make sure you have them down.

Tutorial
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. 
Example
1: Write the reciprocal (or multiplicative
inverse)
of 3. 
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
2: Write the reciprocal (or multiplicative
inverse)
of 1/5. 
The reciprocal of 1/5 is 5, since 5(1/5) = 1. 
Quotient of Real Numbers
If a and b are real
numbers and
b is not 0, then

Multiplying or Dividing Real
Numbers

Since dividing is the same as multiplying by the
reciprocal, dividing
and multiplying have the same sign rules.
Step 1: Multiply or
divide their absolute
values.
Step 2: Put the correct
sign.
If the two numbers have the same sign,
the product or quotient
is positive.
If they have opposite signs, the product
or quotient is negative. 

Example
3: Find the product (4)(3). 
(4)(3) = 12.
The product of the absolute values 4 x 3 is 12 and they
have opposite
signs, so our answer is 12. 
Example
4: Find the product . 

*Mult. num. together
*Mult. den. together
*()() = (+)
*Reduce fraction 
The product of the absolute values 2/3 x 9/10 is 18/30
= 3/5 and they
have the same sign, so that is how we get the answer 3/5.
Note that if you need help on fractions go to Tutorial
3: Fractions 
Example
5: Find the product 
Working this problem left to right we get: 

*(3)(2) = 6
*(6)(10) = 60

Example
6: Divide (10)/(2). 
(10)/(2) = 5
The quotient of the absolute values 10/2 is 5 and they
have the same
signs, so our answer is 5. 
Example
7: Divide . 

*Div. is the same as mult. by
reciprocal
*Mult. num. together
*Mult. den. together
*(+)() = 
*Reduce fraction 
The quotient of the absolute values 4/5 and 8 is 4/40 =
1/10 and they
have opposite signs, so our answer is 1/10.
Note that if you need help on fractions go to Tutorial
3: Fractions 
Multiplying by and
Dividing into Zero
a(0) = 0
and
0/a = 0 (when a does not equal 0)

In other words, zero (0) times any real number is zero
(0) and zero
(0) divided by any real number other than zero (0) is zero (0). 
Example
8: Multiply 0(½). 
0(½) = 0.
Multiplying any expression by 0 results in an answer
of 0. 
Example
9: Divide 0/5. 
0/5 = 0.
Dividing 0 by any expression other than 0 results in
an answer of
0. 
Dividing by Zero
a/0 is undefined

Zero (0) does not go into any number, so whenever you
are dividing
by zero (0) your answer is undefined.
Example
10: Divide 5/0. 
5/0 = undefined.
Dividing by 0 results in an undefined answer. 
Example
11: Simplify . 

*Evaluate inside the absolute
values
*Subtract
*()/() = +

Example
12: Evaluate the expression
if x = 2 and y =  4. 
Plugging 2 for x and 
4 for y and
simplifying we get: 

*Plug in 2 for x and 4 for y
*Exponent
*Multiply
*Add 
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  1c: Multiply.
Practice
Problems 2a  2c: Divide.
Practice
Problem 3a: Simplify.
Practice
Problem 4a: Evaluate the expression when x = 5 and y = 5.
Need Extra Help on these Topics?
Last revised on July 25, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
