Beginning Algebra
Tutorial 7: Multiplying and Dividing Real Numbers

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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

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.

*Evaluate inside the absolute values

*Subtract

*(-)/(-) = +

Example 12: Evaluate the expression

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:

*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.