The additive inverse of a number is a number that has the same distance from 0 on the number line, but is on the opposite side of 0. That sounds a bit confusing but, if you remember when we learned about absolute value , you'll know that there is a positive value and a negative value that are equal distances from zero.

Another way to think about an additive inverse is what value do we add to the number so that the answer is zero.

The picture below shows an example of this using the term 3 and its additive inverse,-3.

A number line showing the distance between 0 and negative 3 and the distance between 0 and 3 is 3 units.

The additive inverse

Any term's additive inverse can be calculated by multiplying the term by -1.

For example:

8 \times (-1) = -8, so the additive inverse of 8 is -8
-12 \times (-1) = -12, so the additive inverse of -12 is 12
a \times (-1) = -a, so the additive inverse of a is -a

A number and its additive inverse should sum to zero, e.g. 7 + (-7) = 0.

Examples

Example 1

What is the additive inverse of 26?

Worked Solution

Create a strategy

To find the additive inverse, multiply by -1.

Apply the idea

The additive inverse of 26 is 26\times (-1) = -26.

Reflect and check

We can check our answer by adding 26 and -26. Does that sum to zero?

Example 2

What is the additive inverse of -23?

Worked Solution

Create a strategy

To find the additive inverse multiply by -1.

Apply the idea

The additive inverse of -23 is -23\times (-1) = 23.

Reflect and check

We can check our answer by adding -23 and 23. Does that sum to zero?

Idea summary

Any term's additive inverse can be calculated by multiplying the term by -1.

A number and its additive inverse should sum to zero, e.g. 7 + (-7) = 0.

Additive inverses to solve equations

It is helpful to imagine the adding or subtracting as moving left or right the number line.

Moving in a positive direction (e.g. if we're adding a positive number or subtracting a negative number) means moving to the right along a number line.

Conversely, moving in a negative direction (e.g. subtracting a positive number or adding a negative number) means moving to the left along a number line.

If we are solving these kinds of questions mentally, using the jump strategy for example, using additive inverses can help.

There are important rules to following when adding or subtracting negative terms:

Adding a negative number is the same as subtracting its inverse, so we can solve it as a subtraction problem, e.g. 4 + (-5) = 4 - 5 = -1.
Subtracting a negative number is equivalent to adding its inverse, so we can solve it as an addition problem, e.g. 2 - (-10) = 2 + 10 = 12.

Examples

Example 3

Find the value of:

2+(-3)

Worked Solution

Create a strategy

This is an addition problem involving numbers with opposite signs. Instead on adding -3 in one step, we can divide it into two steps with the same effect: adding the additive inverse of 2 first then adding the remaining -1 from the result.

Apply the idea

First, rewrite the expression to break apart the -3 to give the additive inverse of 2:

2 + (- 3) = 2 + (-2 + (-1))

Then, we use the associative property to rearrange the expression: (2 + (-2)) + -1

\displaystyle (2 + (-2)) + (-1)	\displaystyle =	\displaystyle 0 + (-1)	Add the additive inverse of 2 first
	\displaystyle =	\displaystyle -1	Evaluate

This means 2 + (- 3) = -1.

Reflect and check

Will the strategy still work when adding numbers with the same sign? How do we apply the strategy when we change the given to 3+(-2)?

Example 4

Find the value of:

-2-\left(-8\right)

Worked Solution

Create a strategy

Subtracting a negative number is equivalent to adding its inverse so we can get the additive inverse of -8 then add it to -2.

Apply the idea

The additive inverse of -8 is 8. We add the additive inverse of -8 to -2:

-2 + 8

Then, instead on adding 8 in one step, we can divide it into two steps with the same effect: adding the additive inverse of -2 first then adding the remaining 6 from the result.

First, rewrite the expression to break apart the 8 to give the additive inverse of -2:

-2 + 8 = -2 + (2 + 6)

Then, we use the associative property to rearrange the expression: (-2 + 2) + 6

\displaystyle (-2 + 2) + 6	\displaystyle =	\displaystyle 0 + 6	Add the additive inverse of -2 first
	\displaystyle =	\displaystyle 6	Evaluate

This means -2 -(-8) = 6.

Example 5

Fill in the blank to make the statement true:

8 - 13 = 8 + ⬚

Worked Solution

Create a strategy

Find the opposite of -13.

Apply the idea

Subtracting 13 from 8 is equal to adding the opposite of 13, its additive inverse, to 8.8 - 13 = 8 + (-13)

Idea summary

There are important rules to following when adding or subtracting negative terms:

Adding a negative number is the same as subtracting its inverse, so we can solve it as a subtraction problem, e.g. 4 + (-5) = 4 - 5 = -1.
Subtracting a negative number is equivalent to adding its inverse, so we can solve it as an addition problem, e.g. 2 - (-10) = 2 + 10 = 12.

Outcomes

7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1.A

Describe situations in which opposite quantities combine to make 0.

7.NS.A.1.C

Understand subtraction of rational numbers as adding the additive inverse, p-q=p+(-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

1.03 Additive inverses

Introduction

Ideas

The additive inverse

Examples

Example 1

Example 2

Additive inverses to solve equations

Examples

Example 3

Example 4

Example 5

Outcomes

7.NS.A.1

7.NS.A.1.A

7.NS.A.1.C

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