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1.01 Multiplying and dividing integers

Lesson

Multiplication of integers

We know  how to multiply and divide  whole numbers when they are all positive. Now let's look at what happens when negative integers are included in questions. The product is the answer when two numbers are multiplied together.

Your knowledge of times tables will really help you through this chapter, we just need to know what to do when one or more of the numbers we are multiplying or dividing are negative.

A positive times a positive equals a positive.

The product of two positive integers is a positive integer.

Examples:

  • 4 \times 5 = 20

  • 16 \times 10 = 160

We just need to learn a couple of extra rules to our existing knowledge, to account for negatives.

A positive times a negative equals a negative. A negative times a positive equals a negative.

The product of a positive integer and a negative integer is a negative integer.

Examples:

  • 16\times-10=-160

  • -4\times5=-20

In both of the above cases, we can take out a factor of -1, giving us -1\times4\times5=-20, and \\-1\times16\times10 respectively.

We can see that it does not matter which term is negative when we are finding the product; the product of a negative number and a positive number will always be negative. But what if both numbers are negative?

A negative times a negative equals a positive.

The product of two negative integers is a positive integer.

Examples:

  • -4\times(-5)=20

  • -16\times(-10)=160

In these cases, we can think of it as taking out two factors of -1.

So (-4)\times(-5)=(-1\times-1)\times(4\times5)=1\times20= 20.

Exploration

Use the following applet to explore multiplication of integers.

Select two integers to multiply by using the sliders to change the value and the check boxes to change the sign.

  • Blue tiles represent positive integers

  • Red tiles represent negative integers

  • The product is shown as an array

Loading interactive...

The product of two positive integers or two negative integers will result in a positive integer. The product of one positive and one negative integer will result in a negative integer.

Examples

Example 1

Evaluate 5 \times (-9).

Worked Solution
Create a strategy

We have the product of negative integer and a positive integer, so the product will be negative.

Apply the idea
\displaystyle 5 \times (-9)\displaystyle =\displaystyle -45Perform the multiplication

Example 2

Evaluate (-6)^{3}.

Worked Solution
Create a strategy

Write the multiplications in expanded form and complete one multiplication at a time.

Apply the rules:

  1. The product of two negative integers is a positive integer.

  2. The product of a positive integer and a negative integer is a negative integer.

Apply the idea
\displaystyle (-6)^{3}\displaystyle =\displaystyle -6\times(-6)\times(-6)Write in expanded form
\displaystyle =\displaystyle 36\times(-6)Multiply -6 by -6
\displaystyle =\displaystyle -216Multiply 36 by -6
Idea summary
A positive times a positive equals a positive.

The product of two positive integers is a positive integer

A positive times a negative equals a negative. A negative times a positive equals a negative.

The product of a positive integer and a negative integer is a negative integer.

A negative times a negative equals a positive.

The product of two negative integers is a positive integer.

Division of integers

As division is just the opposite of multiplication, the rules for dividing by integers are the same as the rules for multiplying integers.

A positive divided by a positive equals a positive.

The quotient of two positive integers is a positive integer.

Examples:

  • 20\div5=4

  • 160\div10=16

To account for negatives, we can follow the same rules as we did when multiplying with integers.

A positive divided by a negative equals a negative. A negative divided by a positive equals a negative.

The quotient of a negative and a positive integer is a negative integer.

Examples:

  • 20\div(-5)=-4

  • -160\div10=-16

Again, in both of the above cases, we can take out a negative factor of -1, giving us \dfrac{1}{-1}\times \dfrac{20}{5}=-4, and \dfrac{-1}{1}\times \dfrac{160}{10}=-16 respectively.

As before, we can see that it does not matter which term is negative when we are finding the quotient; the quotient of two numbers, when one is negative, will always be negative. But what if both numbers are negative?

A negative divided by a negative equals a positive.

The quotient of two negative integers is a positive integer.

Examples:

  • -20\div(-5)=4

  • -160\div(-10)=16

Like with multiplying two negatives, we can think of it as taking out two factors of -1.

So (-20)\div(-5) = \dfrac{-1}{-1}\times\dfrac{20}{5}=1\times4=4.

Examples

Example 3

Evaluate -20 \div 4.

Worked Solution
Create a strategy

We are dividing a positive integer by a negative integer, so the answer will be negative.

Apply the idea
\displaystyle -20 \div 4\displaystyle =\displaystyle -5Perform the division
Idea summary
A positive divided by a positive equals a positive.

The quotient of two positive integers is a positive integer.

A positive divided by a negative equals a negative. A negative divided by a positive equals a negative.

The quotient of a negative and a positive integer is a negative integer.

A negative divided by a negative equals a positive.

The quotient of two negative integers is a positive integer.

Outcomes

VCMNA273

Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies and make estimates for these computations

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