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The opposite of a number is also called its additive inverse. The additive inverse is the number you have to add to it so that the sum is zero. So the additive inverse of 5 is -5. This is sometimes called the property of opposites.

If you add any number to its opposite, or additive inverse, the answer is always zero. For example:

$-989+989=0$

$6.5+\left(-6.5\right)=0$

$1+\left(-1\right)=0$

Once you know this, there are several ways you can think about adding and subtracting negative numbers.

## Adding and subtracting negatives: The algebra tile method

Let the yellow tiles represent positive numbers and the red tiles represent negative numbers.

Example 1

The addition problem $5+\left(-2\right)$ can be represented as:

Group the two negative tiles with two of the positive tiles, zeroing them out.

Since $2+\left(-2\right)=0$ , these tiles disappear. We are left with 3 positive tiles.

Therefore, $5+\left(-2\right)=3$ .

When both numbers are negative, we have only negative tiles.

Example 2

The addition problem $-3+\left(-4\right)$ can be represented as

The result is simply 7 negative tiles.

Therefore, $-3+\left(-4\right)=-7$

## Adding and subtracting negatives: The number line method

When you add a positive number, you move to the right on the number line.

When you add a negative number, you move to the left on the number line.

Example 3

Add $6+\left(-8\right)$ using a number line.

Start at 6 and move 8 units to the left.

So $6+\left(-8\right)=-2$

Subtracting a number is the same as adding its opposite. So subtracting a positive number is like adding a negative number - you move to the left on the number line.

Subtracting a negative number is like adding a positive number - you move to the right on the number line.

Example 4

Subtract $-4-\left(-7\right)$ .

Start at -4 and move 7 units to the right.

So $-4-\left(-7\right)=3$ .

## Adding negative numbers without visual aids

Once you've become used to adding negative numbers using the algebra tile method or the number line method, you'll notice that adding a negative number is basically the same as performing a subtraction operation.

So these two problems $5+\left(-4\right)$ and $5-4$ are going to have the same answer.

Example 5

Solve the following addition problems involving negative numbers using your understanding of subtraction.

$8+\left(-3\right)$

Because we know that $8-3=5$ , you know that

$8+\left(-3\right)=5$

$259+\left(-384\right)$

Since we can work out that $259-384=-125$ , you know that $259+\left(-384\right)=-125$

What about when the negative number is larger than the positive number we are adding it to? Well, we know how to subtract a number that is larger than the number we are subtracting it from - we end up with a negative number.

Example 6

Solve the following addition problems involving negative numbers using our understanding of subtraction.

$8+\left(-15\right)$

Since we know that $8-15=-7$ , you know that

$8+\left(-15\right)=-7$

## Subtracting negative numbers without visual aids

Once you've become used to subtracting negative numbers using the algebra tile method or the number line method, you'll notice that subtracting a negative number is basically the same as performing an addition operation. So the two problems $10-\left(-5\right)$ and $10+5$ are going to have the same answer.

Knowing this can help us perform the subtraction of negative numbers in your head.

Example 7

Solve the following subtraction problems involving negative numbers using your understanding of addition.

1. $7-\left(-3\right)$

Because we know that $7+3=10$ , we know that

$7-\left(-3\right)=10$

2. $157-\left(-90\right)$

Since we can work out that $157+90=247$ , we know that

$157-\left(-90\right)=247$