Additive inverse
The additive inverse of a number is a number that has the same distance from $0$0 on the number line, but is on the opposite side of $0$0. That sounds a bit confusing but, if you remember when we learnt 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 I add to the number so that my answer is zero.
The picture below shows an example of this using the term $3$3 and its additive inverse, $-3$−3.
Calculating a term's additive inverse
Any term's additive inverse can be calculated by multiplying the term by $-1$−1.
For example, the additive inverse of $8$8 is $-8$−8 ($8\times\left(-1\right)=-8$8×(−1)=−8),
the additive inverse of $-12$−12 is $12$12 ($-12\times\left(-1\right)=12$−12×(−1)=12)
and the additive inverse of $a$a is $-a$−a ($a\times\left(-1\right)=-a$a×(−1)=−a).
A number and its additive inverse should sum to zero. e.g. $7+\left(-7\right)=0$7+(−7)=0.
Using additive inverses to solve equations
It's helpful to imagine the adding or subtracting as moving up or down the number line.
Moving in a positive direction (i.e. if we're adding a positive number) means moving to the right along a number line.
Conversely, moving in a negative direction (i.e. subtracting a positive number) means moving to the left along a number line.
If we're 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+\left(-5\right)=4-5$4+(−5)=4−5$=$=$-1$−1.
- Subtracting a negative number is equivalent to adding its inverse, so we can solve it as an addition problem, e.g. $2-\left(-10\right)=2+10$2−(−10)=2+10$=$=$12$12.
Examples
Question 1
Evaluate: $2-3$2−3
Think: This is a subtraction problem so we are moving to the left down the number line.
Do: $2-2=0$2−2=0. Then we still have $1$1 left to take away. So, $2-3=-1$2−3=−1.
Question 2
Evaluate: $-2-8$−2−8
Think: Like question 1, this is a subtraction problem so we are moving to the left down the number line.
Do: $-2-8$−2−8 will be the same distance away from $0$0 as $2+8$2+8. $2+8=10$2+8=10 so $-2-8=-10$−2−8=−10.
Question 3
Evaluate: $3-\left(-8\right)$3−(−8)
Think: Two negative signs together become a positive.
Do:
| $3-\left(-8\right)$3−(−8) | $=$= | $3+8$3+8 |
| $=$= | $11$11 |
Question 4
What is the additive inverse of $26$26?
Think: the additive inverse of $26$26 is the number that is the same distance from $0$0 on the number line as $26$26, but is on the opposite side of $0$0.
Do: The additive inverse of $26$26 is $-26$−26.
Question 5
In the last financial year, Delicious Donuts had an overall loss of $\$88000$$88000.
a) What integer is used to represent how much the company made?
b) What is the additive inverse of this result?
c) $-88000+88000=\editable{}$−88000+88000=
d) What does the additive inverse represent here?
- A) the amount, in dollars, the company made the year before
- B) the amount, in dollars, the company needs to make to turn a profit of $\$88000$$88000
- C) the amount, in dollars, the company is expected to make this financial year
- D) the amount, in dollars, the company needs to make to break even