Beginning Algebra Tutorial 7

Beginning Algebra
Tutorial 7: Multiplying and Dividing Real Numbers

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deskLearning Objectives

After completing this tutorial, you should be able to:
  1. Find the reciprocal of a number.
  2. Multiply positive and negative numbers.
  3. Divide positive and negative numbers.
  4. Multiply by zero.
  5. Know that dividing by zero is undefined.

desk 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.



desk Tutorial


Multiplicative Inverse
(or reciprocal)

For each real number a, except 0,
there is a unique real number inverse 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.

notebook 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.

notebook 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.

notebook 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.

notebook Example 4:  Find the product example 4a.

example 4b
*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

notebook Example 5:  Find the product example 5a

Working this problem left to right we get:

example 5b

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

notebook 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.

notebook Example 7:  Divide example 7a.

example 7b
*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


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).

notebook Example 8:   Multiply  0(½).

0(½) = 0.

Multiplying any expression by 0 results in an answer of 0.

notebook 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. 

notebook Example 10:   Divide 5/0.

5/0 = undefined

Dividing by 0 results in an undefined answer.

notebook Example 11:   Simplify example 11a.

Since we have several operations going on in this problem, we will have to use the order of operations to make sure that we get the correct answer. 

If you need to review the order of operations go to Tutorial 4: Operations of Real Numbers.

example 11b
*Evaluate inside the absolute values


*(-)/(-) = +

notebook Example 12:   Evaluate the expression example 12a   if  x = -2 and y = - 4.

To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations.

Plugging -2 for x and - 4 for y and simplifying we get:

example 12b
*Plug in -2 for x and -4 for y


desk 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.


pencil Practice Problems 1a - 1c: Multiply.


1a.  (-2)(-25)
(answer/discussion to 1a)
1b.    (0)(-100)
(answer/discussion to 1b)


1c.    (-2)(3)(5)
(answer/discussion to 1c)


pencil Practice Problems 2a - 2c: Divide.


2a. problem 2a
(answer/discussion to 2a)

2b. problem 2b
(answer/discussion to 2b)


2c. problem 2c
(answer/discussion to 2c)



pencil Practice Problem 3a: Simplify.


3a. problem 3a
(answer/discussion to 3a)


pencil Practice Problem 4a: Evaluate the expression when x = 5 and y = -5.


4a. problem 4a
(answer/discussion to 4a)





desk Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:
This webpage helps you with multiplying integers.
This webpage goes over multiplying positive and negative numbers together.
This webpage covers dividing integers.
This webpage goes over dividing positive and negative numbers.


Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


WTAMU > Virtual Math Lab > Beginning Algebra

Last revised on July 25, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.