Multiplying and Dividing Integers Examples
By Kathleen Cantor, 23 Sep 2020
Integers are perhaps one of the easiest things to work with in mathematics. We start working with integers from the time we start school, maybe even sooner. It begins with basic counting, then addition and subtraction, multiplication, division, and beyond. Without integers, we would not have math as we know it.
Multiplying and dividing integers is a stepping stone when working with these numbers. It opens the door to more advanced functions and shows us how complex an integer is! So let's dive into integers, what they are, and how to multiply and divide them.
Properties of Integers
An integer is a whole number. It can be either positive or negative. All integers less than 0 have a minus sign ( - ) before the numbers. Positive integers have a plus sign ( + ). However, for positive numbers, the plus sign is usually omitted.
An integer is not a fraction, and it is not a decimal. Integers are a subset of all rational numbers, Q, and rational numbers are a subset of all real numbers, R. When you want to represent a set of integers, we use the symbol, Z.
Using set notation, we write integers as {…-3, -2, -1, 0, 1, 2, 3, …}. The ellipsis means that the numbers continue infinitely in both directions.
Properties of integers are presented using algebraic symbols.
Closure
Closure states that the sum or product of two or more integers will always be an integer.
a + b is an integer
a * b is an integer.
Distributive Law
Multiplying a group of added numbers is like adding each number in the group with the multiplier.
a ( b + c ) = ( a * b ) + ( a * c )
( a + b ) * c = ( a * c ) + ( b * c )
Associative Law
When you multiply or add, the order doesn't affect the sum or result.
f + ( b + c ) = ( f + b ) + c
( f * b ) * c = f * ( b * c )
Commutative Law
Like associative law, when you multiply or add, the order doesn't matter.
g + b = b + g
g * b = b * g
Rules for Multiplying and Dividing Integers
Issues arise when multiplying or dividing integers when the numbers used are not natural numbers, i.e., not positive whole numbers. The following rules show you how to multiply and divide these integers.
Multiplying Two Integers with the Same Sign
The result is always a positive integer when you multiply two integers with the same sign.
a * b = ab
( -a ) * ( -b ) = ab
OR
+ * + = +
( - ) * ( - ) = +
So when multiplying two negative integers, multiply the numbers as usual and remove the minus sign.
Multiplying a Positive and a Negative
When you multiply two integers, and the signs are not the same, the result is always a negative integer.
( -a ) * b = -ab
a * ( -b ) = -ab
OR
( - ) * + = ( - )
+ * ( - ) = ( - )
Dividing Two Integers with the Same Sign
You will always get a positive quotient when you divide two integers with the same sign.
a / b = ab
( -a ) / ( -b ) = ab
OR
+ / + = +
( - ) / ( - ) = +
So when dividing two negative integers, the quotient will be positive.
Dividing a Positive and a Negative
When you divide two integers, and the signs are not the same, the result is always a negative integer.
( -a ) / b = -ab
a / ( -b ) = -ab
OR
( - ) / + = ( - )
+ / ( - ) = ( - )
Now that we know the rules of multiplying and dividing integers, let us learn how to use them in the following examples.
Example 1
Multiply the numbers below.
6 * 4
The signs for the two numbers are not there, which means that they are positive integers.
6 * 4 = 24.
The answer is positive because both numbers are positive.
Example 2
Multiply the numbers below.
4 * ( - 5 )
The first number is positive, while the second is negative.
4 * ( - 5 ) = -20
The result is negative because the signs of the two numbers are not the same.
Example 3
Multiply the numbers below.
( - 6 ) * ( - 4 )
Both numbers are negative. They have the same sign.
( - 6 ) * ( - 4 ) = 24
The result is positive because both numbers have the same sign.
Example 4
Divide the numbers below.
12 / 3
Both numbers have the same sign.
12 / 3 = 4
The result is positive, the same as it was for multiplication.
Example 5
Divide the numbers below.
( - 12 ) / 3
The signs are not the same.
( - 12 ) / 3 = - 4
The result is negative.
Example 6
Divide the numbers below.
( - 2 ) / ( - 2 )
Both numbers have the same sign.
( - 2 ) / ( - 2 ) = 1
The result is positive because both signs of the numbers are the same.
All the examples above involve only two numbers. Let's see what happens when there's more than two numbers.
Example 7
Multiply the numbers below.
5 * 2 * ( - 3 )
The first two numbers are positive, while the last one is negative. When there are more than two numbers, the best approach is to multiply two numbers at a time.
Let's start with the first two numbers.
5 * 2 = 10
Both numbers are positive.
The next step is to multiply the answer with the last number remaining ( - 3 ).
10 * ( - 3 ) = - 30
The answer is negative because the signs of 10 and 3 are not the same.
Example 8
Look at the expression below.
7 * ( - 2 ) * 3 * ( - 1 )
We have four numbers now so we can group them into two groups. We will handle the first two, and then the last two. We will then multiply the results.
7 * ( - 2 ) = - 14
3 * ( - 1 ) = - 3
Multiply the two answers.
( - 14 ) * ( - 3 ) = 42
When there are more than two numbers in a problem, group them and multiply. Let's see how to handle division when there are multiple numbers.
Example 9
Divide the numbers below.
12 / ( - 2 ) / 3
As with multiplication, let's first look at the first two numbers.
12 / ( - 2 ) = - 6
We can now divide the answer with the remaining number.
( - 6 ) / 3 = - 2
The final result is negative.
No matter where you are in your mathematics education, whether you're just starting with multiplication and division or need a refresher course, these rules stand the test of time. No educational standard changes can change these integer facts. Memorizing them takes practice, but they are hard to forget.
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