Rules for Adding and Subtracting Integers | Grade 7 Math | LevelKite

In the last lesson you added and subtracted integers by moving along a number line. Integers are the counting numbers ( $1, 2, 3, \ldots$ ), their negatives ( $-1, -2, -3, \ldots$ ) and $0$ . Now you'll turn those moves into rules that work without drawing anything.

Let $p$ and $q$ stand for any two integers. The absolute value of $q$ , written $|q|$ , is the distance between $q$ and $0$ on a number line, where one unit is the distance from $0$ to $1$ . A distance is never negative, so $|-9| = 9$ and $|9| = 9$ .

Picture a number line with larger numbers to the right. The sum $p + q$ is the result of adding $q$ to $p$ . It is the number that is $|q|$ units away from $p$ : to the right when $q$ is positive, and to the left when $q$ is negative. When $q = 0$ you don't move, so $p + 0 = p$ . If you start at $0$ , the move lands exactly on $q$ , so $0 + q = q$ .

Same signs. Two integers have the same sign when both are positive or both are negative. Look at $-6 + (-9)$ . The number $-6$ is $6$ units to the left of $0$ . Adding $-9$ moves you $|-9| = 9$ units, and the move goes left because $-9$ is negative. Both distances go the same way, so they combine: you end $6 + 9 = 15$ units to the left of $0$ . That number is $-15$ , so $-6 + (-9) = -15$ .

Two positive integers work the same way, with both distances going right: $6 + 9 = 15$ .

So when two integers have the same sign, add their absolute values. The sum has the sign that the two integers share.

Different signs. Look at $-9 + 6$ . Start at $-9$ , which is $9$ units to the left of $0$ . Adding $6$ moves you $6$ units right, back toward $0$ . That move cancels $6$ of the $9$ units, so you stop $9 - 6 = 3$ units to the left of $0$ . So $-9 + 6 = -3$ .

Now look at $-4 + 10$ . Start at $-4$ , which is $4$ units to the left of $0$ . Adding $10$ moves you $10$ units right. The first $4$ units bring you to $0$ , and the other $10 - 4 = 6$ units carry you to the right of $0$ . So $-4 + 10 = 6$ .

When one integer is positive and the other is negative, their distances work against each other. If the absolute values are different, subtract the smaller absolute value from the larger one. The sum has the sign of the integer with the larger absolute value, because part of that integer's distance is left over.

How to add two integers

The two integers	What to do	Example
Both positive	Add the absolute values. The sum is positive.	$6 + 9 = 15$
Both negative	Add the absolute values. The sum is negative.	$-6 + (-9) = -15$
One positive and one negative, with different absolute values	Subtract the smaller absolute value from the larger. Use the sign of the integer with the larger absolute value.	$-9 + 6 = -3$ and $10 + (-4) = 6$
Additive inverses, such as $-7$ and $7$	The sum is $0$ .	$-7 + 7 = 0$
At least one is $0$	The sum is the other integer.	$-6 + 0 = -6$ and $0 + 9 = 9$

Worked example

Add two integers with different signs

Find the sum $-14 + 9$ .

Look at the signs first, because they tell you which rule fits. The first number, −14, is negative and the second number, 9, is positive. Their distances work against each other.
Find the absolute value of each number, which is its distance from 0.
$|-14| = 14 \qquad |9| = 9$
The absolute values are different, so subtract the smaller one from the larger one. This tells you how much distance is left after the shorter move cancels part of the longer one.
$14 - 9 = 5$
Choose the sign. The number with the larger absolute value is −14, and it is negative. So the 5 units that are left lie to the left of 0, and the sum is negative.
$-14 + 9 = -5$
Check with the meaning of addition. Adding the positive number 9 moves you 9 units to the right of −14. Count the 9 steps: −13, −12, −11, −10, −9, −8, −7, −6, −5. You land on −5.

Answer

The sum $-14 + 9$ is $-5$ .

What subtraction means. You won't need a second set of rules for subtraction, because subtraction undoes addition. For integers $p$ and $q$ , the difference $p - q$ is the number that gives $p$ when you add $q$ to it. You know this from earlier grades: $9 - 4 = 5$ because $5 + 4 = 9$ .

Use that idea on $3 - 5$ . You need the number that gives $3$ when you add $5$ to it. Adding $5$ moves you $5$ units right. So to find where you started, begin at $3$ and move $5$ units left. Adding $-5$ makes exactly that move: the first $3$ units bring you to $0$ , and the other $5 - 3 = 2$ units end at $-2$ . So $3 - 5 = 3 + (-5) = -2$ . Check: $-2 + 5 = 3$ .

Subtracting $5$ gave the same result as adding $-5$ , the additive inverse of $5$ . The same reasoning works for every pair of integers.

Key concept

Subtracting means adding the additive inverse

For all integers $p$ and $q$ , subtracting $q$ gives the same result as adding its additive inverse $-q$ . Here is why. Adding $q$ moves you $|q|$ units from $p$ : right if $q$ is positive, left if $q$ is negative, and not at all if $q = 0$ . Subtracting $q$ undoes that move, so you go $|q|$ units the other way. Adding $-q$ makes exactly that move, because $-q$ is the same distance from $0$ as $q$ and, when $q$ is not $0$ , has the opposite sign.

To subtract, keep $p$ the same and change two things together: the subtraction becomes addition, and $q$ becomes $-q$ . Then add.

p - q = p + (-q)

Rewriting subtraction as adding the additive inverse

Subtraction	Additive inverse of the number subtracted	Rewritten as addition	Result
$7 - 10$	$-10$	$7 + (-10)$	$-3$
$7 - (-10)$	$10$	$7 + 10$	$17$
$-7 - 10$	$-10$	$-7 + (-10)$	$-17$
$-7 - (-10)$	$10$	$-7 + 10$	$3$
$-7 - (-7)$	$7$	$-7 + 7$	$0$

Worked example

Subtract a negative integer

Find the difference $6 - (-8)$ .

Find the number you are subtracting and its additive inverse. You are subtracting −8. Its additive inverse is 8, because −8 and 8 add to 0.
$-8 + 8 = 0$
Rewrite the subtraction as adding that additive inverse, and keep the first number, 6, the same. This is valid because subtracting −8 undoes a move of 8 units left, and adding 8 moves you 8 units right.
$6 - (-8) = 6 + 8$
Add. Both numbers are positive, so add their absolute values. The sum is positive.
$6 + 8 = 14$
Check by undoing. The result plus the number you subtracted must equal the number you started with. Start at 14 and add −8: you move 8 units left and land on 6.
$14 + (-8) = 6$

Answer

The difference $6 - (-8)$ is $14$ .

Worked example

Compare two temperatures

At dawn the temperature in a town is $-13$ degrees Celsius. The town's record low temperature is $-21$ degrees Celsius. How many degrees warmer is the dawn temperature than the record low?

Write the subtraction. To find how much greater one amount is than another, subtract the second amount from the first. Both amounts are temperatures in degrees Celsius, so subtract the record low from the dawn temperature.
$-13 - (-21)$
Rewrite the subtraction as addition, because subtracting a number gives the same result as adding its additive inverse. You are subtracting −21, and its additive inverse is 21. The first number, −13, stays the same.
$-13 - (-21) = -13 + 21$
Add. One number is negative and the other is positive, and their absolute values, 13 and 21, are different. Subtract the smaller absolute value from the larger one. The number with the larger absolute value, 21, is positive, so the sum is positive.
$-13 + 21 = 21 - 13 = 8$
Check by undoing. Add the number you subtracted to the result. Start at 8 and add −21: you move 21 units left. The first 8 units reach 0, and the other 13 units end at −13, the dawn temperature.
$8 + (-21) = -13$

Answer

The dawn temperature is $8$ degrees Celsius warmer than the record low.

Takeaway. For any integers $p$ and $q$ , adding $q$ moves you $|q|$ units from $p$ : right when $q$ is positive, left when $q$ is negative, and nowhere when $q = 0$ . So integers with the same sign combine their distances, a positive and a negative integer cancel part or all of each other, and additive inverses add to $0$ . Subtracting $q$ undoes adding $q$ , which makes every subtraction an addition: $p - q = p + (-q)$ .

Next, you'll add and subtract three or more integers using properties of operations, and you'll complete addition and subtraction equations with integers.