Math Core

The temperature is $-4^\circ$ F at sunrise and climbs $9$ degrees by noon. A submarine at $-120$ feet dives another $35$ feet. To answer questions like these you need to add and subtract numbers that can be negative. In this lesson you'll learn to do it with a number line first, and then with rules you can use in your head.

Opposites add to zero

Every integer has an opposite: the number the same distance from $0$ on the other side. The opposite of $6$ is $-6$ , and the opposite of $-6$ is $6$ .

If you gain $6 and then spend $6, you are back where you started. In math, $6 + (-6) = 0$ .

Definition

Additive inverses

Two numbers are additive inverses (opposites) if their sum is $0$ . For any number $a$ ,

a + (-a) = 0

This one fact explains almost everything in this lesson. For example, $-5 + 3$ is really $-2 + (-3) + 3$ . The $-3$ and $3$ cancel, leaving $-2$ .

Adding on a number line

Start at the first number. Adding a positive number moves you right. Adding a negative number moves you left.

Here is $3 + (-5)$ : start at $3$ and move $5$ units left. You land on $-2$ .

3 + (−5) = −2

And here is $-4 + 6$ : start at $-4$ and move $6$ units right. You land on $2$ .

−4 + 6 = 2

Rules for adding integers

Number lines get slow with big numbers, so look for the pattern. The absolute value of a number, written $|a|$ , is its distance from $0$ . So $|-7| = 7$ and $|7| = 7$ .

Adding integers

Same signs: add the absolute values and keep the common sign. $\;-8 + (-5) = -13$
Different signs: subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value. $\;-8 + 5 = -3$

Why does the second rule work? In $-8 + 5$ , the $5$ cancels $5$ of the $8$ negatives, and $3$ negatives are left over.

Subtracting means adding the opposite

Compare these two lists:

subtraction	addition
$9 - 4 = 5$	$9 + (-4) = 5$
$2 - 7 = -5$	$2 + (-7) = -5$
$6 - (-3) = 9$	$6 + 3 = 9$

Subtracting a number always gives the same result as adding its opposite. That turns every subtraction problem into an addition problem you already know how to do.

Subtracting integers

a - b = a + (-b)

Keep the first number, change subtraction to addition, and change the second number to its opposite.

Here is $5 - (-3)$ . It equals $5 + 3$ , so you move $3$ units right and land on $8$ .

5 − (−3) = 5 + 3 = 8

Subtraction also measures distance. The distance between two numbers $a$ and $b$ on a number line is $|a - b|$ . The distance from $-3$ to $4$ is $|-3 - 4| = |-7| = 7$ .

Worked example: Adding integers

$-9 + (-6)$
$-14 + 20$
$11 + (-17)$

Solutions.

Same signs. Add $9 + 6 = 15$ and keep the negative sign: $-15$ .
Different signs. $20 - 14 = 6$ . The number with the larger absolute value, $20$ , is positive, so the answer is $6$ .
Different signs. $17 - 11 = 6$ . The number with the larger absolute value, $-17$ , is negative, so the answer is $-6$ .

Worked example: Subtracting integers

$-2 - 5$
$-7 - (-10)$

Solutions.

Add the opposite: $-2 + (-5)$ . Same signs, so the answer is $-7$ .

Add the opposite: $-7 + 10$ . Different signs: $10 - 7 = 3$ , and $10$ is positive, so the answer is $3$ .

Worked example: A word problem

The temperature at 6 a.m. was $-4^\circ$ F. By 3 p.m. it was $13^\circ$ F. How many degrees did the temperature rise?

The change is the final value minus the starting value:

13 - (-4) = 13 + 4 = 17

The temperature rose $17^\circ$ F. Check on a number line: from $-4$ up to $0$ is $4$ degrees, and from $0$ up to $13$ is $13$ more, for $17$ in all.

Common mistake

Don't mix up the two rules. When you subtract, change only the second number to its opposite, and do it before you add. In $-3 - 8$ , the answer is $-3 + (-8) = -11$ , not $5$ or $-5$ .

Tip

Check a subtraction by adding back. If $-7 - (-10) = 3$ , then $3 + (-10)$ should give $-7$ . It does.