1. Integers

Lesson

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 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 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.

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`).

Remember!

A number and its additive inverse should sum to zero. e.g. $7+\left(-7\right)=0$7+(−7)=0.

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.

Remember!

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.

**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.

**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.

**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 |

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.

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?

**Think:** The key word here is loss. What type of number would represent losing money?

**Do:** We need to use a negative number to represent the loss of money, so the integer $-88000$−88000 represents how much the company made.

b) What is the additive inverse of this result?

**Think:** $-88000$−88000 lies to the left of $0$0 on the number line, What number would be the same distance from zero on the number line, but to the right instead?

**Do:** The additive inverse of $-88000$−88000 is $88000$88000.

c) $-88000+88000=\editable{}$−88000+88000=

**Think:** What do additive inverses always add up to?

**Do:** Additive inverses always add to $0$0 so $-88000+88000=0$−88000+88000=0

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

**Think:** The additive inverse is $88000$88000. Which scenario could a positive $88000$88000 represent?

**Do:** Since we know that adding $-88000+88000=0$−88000+88000=0 what would $0$0 represent financially? $0$0 would represent breaking even because the company has not lost money or earned money. So the correct answer here is **D)** the amount, in dollars, the company needs to make to break even.

What is the additive inverse of $-23$−23?

Fill in the blank to make the statement true.

$8-13=8+\editable{}$8−13=8+

You have two numbered cards in your hand. The sum of the numbers is $0$0. If the number on one of the cards is $6$6, what is the number on the other card?

Nanga Parbat is a mountain that is $8126$8126 meters above sea level.

What integer is used to represent the elevation?

What is the additive inverse of this elevation?

$8126+\left(-8126\right)$8126+(−8126) $=$= $\editable{}$

What does the additive inverse represent here?

The amount we would have to travel to get back to sea level.

AThe number of meters above sea level.

BThe number of meters the mountain is below Mt Everest.

CThe width of the mountain.

DThe amount we would have to travel to get back to sea level.

AThe number of meters above sea level.

BThe number of meters the mountain is below Mt Everest.

CThe width of the mountain.

D

On the number line attached, the numbers $x$`x` and $y$`y` are the same distance from $0$0. What is the sum $x+y$`x`+`y`?

(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.