**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**

**For each real number a, except 0,
there is a unique
real number such
that **

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*.

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.

**If a and b are real
numbers and **

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**.

The product of the absolute values 4 x 3 is 12 and they
have opposite
signs, so **our answer is -12.**

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**

Working this problem left to right we get:

The quotient of the absolute values 10/2 is 5 and they
have the same
signs, so **our answer is 5.**

***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**

*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).

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

Dividing 0 by any expression other than 0 **results in
an answer of
0.**

*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.

Dividing by 0 **results in an undefined answer.**

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.**

***Subtract **

***(-)/(-) = +**

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:**

** 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.

1c. (-2)(3)(5)

(answer/discussion to 1c)

(answer/discussion to 1c)

Practice Problems 2a - 2c: Divide.

Practice Problem 3a:Simplify.

Practice Problem 4a:Evaluate the expression whenx= 5 andy= -5.

** Need Extra Help on these Topics?**

**http://www.mathleague.com/help/integers/integers.htm#multiplyingintegers**

This webpage helps you with multiplying integers.

**http://www.mathleague.com/help/posandneg/posandneg.htm#**

**multiplyingpositiveandnegativenumbers**

This webpage goes over multiplying positive and negative numbers
together.** **

**http://www.mathleague.com/help/integers/integers.htm#dividingintegers**

This webpage covers dividing integers.

**http://www.mathleague.com/help/posandneg/posandneg.htm#**

**dividingpositiveandnegativenumbers**

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.**

Last revised on July 25, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.