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Multiplying and Dividing Positive and Negative Numbers

Author: Sophia

what's covered
In this lesson, you will learn how to determine a product or quotient with positive and negative numbers. Specifically, this lesson will cover:

Table of Contents

1. Multiplying and Dividing Positive and Negative Numbers

Multiplication and division of integers both work in a very similar pattern to adding and subtracting. The short description of the process is that we multiply or divide like normal, and if the signs match (both positive or both negative) the answer is positive. If the signs don't match (one positive and one negative), the answer is negative.

1a. Matching Signs

Here are examples of multiplication and division of integers with matching signs:

short dash 2 open parentheses short dash 6 close parentheses Signs match, answer is positive
12 Our Solution

fraction numerator short dash 36 over denominator short dash 9 end fraction Signs match, answer is positive
4 Our Solution

In each example, since the two integers have matching signs, we multiply or divide the two numbers, and write the product or quotient as a positive number.

big idea
The product or quotient of two positive numbers is positive. The product or quotient of two negative numbers is also positive.

1b. Opposite Signs

Here are some examples of multiplication and division of integers with opposite signs:

open parentheses 4 close parentheses open parentheses short dash 6 close parentheses Signs do not match, answer is negative
short dash 24 Our Solution

fraction numerator 15 over denominator short dash 3 end fraction Signs do not match, answer is negative
short dash 5 Our Solution

In these examples, the two integers have opposite signs: one is positive and the other is negative. We multiply or divide the two numbers, and write the product or quotient as a negative numbers.

big idea
The product or quotient of a positive and negative number is negative.


2. Considerations

There are a few things to be careful of when working with integers. First, be sure not to confuse a problem like short dash 3 minus 8 with short dash 3 left parenthesis short dash 8 right parenthesis.

  • The first problem, short dash 3 minus 8 comma is subtraction, because the subtraction separates -3 from what comes after it.
  • The second problem, short dash 3 left parenthesis short dash 8 right parenthesis comma is a multiplication problem, because there is nothing between the 3 and the parenthesis. If there is no operation written in between the parts, then we assume that means we are multiplying.
Another item to watch out for is to be careful not to mix up the pattern for adding and subtracting integers with the pattern for multiplying and dividing integers. They can look very similar, for example if the signs match on addition, we keep the sign, even if it is negative: short dash 3 plus open parentheses short dash 7 close parentheses equals short dash 10. However, if the signs match in multiplication, the answer is always positive: open parentheses short dash 3 close parentheses open parentheses short dash 7 close parentheses equals 21.

hint
As you work through multiplication problems in this course, you will see many different ways to express multiplication. Some problems may use the times sign (4 cross times 7), a dot (5 times 6), or sometimes even parentheses (2 open parentheses 3 close parentheses). They all mean the same thing, that you will multiply the values together. Here are a few more instances:

Times Dot Parentheses
table attributes columnalign left end attributes row cell 4 cross times 2 equals 8 end cell row cell 3 cross times 7 equals 21 end cell row cell 5 cross times 9 equals 45 end cell end table table attributes columnalign left end attributes row cell 4 times 2 equals 8 end cell row cell 3 times 7 equals 21 end cell row cell 5 times 9 equals 45 end cell end table table attributes columnalign left end attributes row cell 4 open parentheses 2 close parentheses equals 8 end cell row cell 3 open parentheses 7 close parentheses equals 21 end cell row cell 5 open parentheses 9 close parentheses equals 45 end cell end table

summary
The big idea that multiplying or dividing by a negative number changes the sign of the product or quotient. When the signs match, for instance, when we multiply or divide two positive numbers, the answer is positive. When we multiply or divide by two negative numbers, the answer is also positive. However, when the signs are opposite, for instance, when we multiply or divide a positive number and a negative number, the answer is negative. Some considerations when adding, subtracting, multiplying, or dividing numbers include making sure you don't confuse an addition problem with a multiplication problem. Also that you don't mix up the patterns for adding and subtracting with the patterns for multiplying and dividing.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License