Adding and Subtracting Integers

Addition as a walk along the line

In the foundations chapter you added whole numbers like $3 + 4$ by counting on: start at $3$, take four more steps, and land on $7$. On the number line that is a walk to the right, because for positive numbers, larger means farther right. Nothing about that walk needs to change when negatives join in.

Watch $3 + 4$ and $3 + (-4)$ side by side. Both start at $3$. The first steps four units right to $7$; the second steps four units left, past $2, 1, 0$, to $-1$.

The starting point is the same; only the direction of the second number’s step is different.

A positive number is a push forward; a negative number is a pull back. This matches the picture you built for signed quantities: a deposit ($+$) raises a balance and a withdrawal ($-$) lowers it. Addition just records the move.

The difference between those two walks, right versus left, is the whole of integer addition:

To compute $a + b$, start at $a$. If $b$ is positive, step $b$ units to the right. If $b$ is negative, step that many units to the left. Where you land is the sum.

Now try the walk yourself. The number line below takes a number to start from and a number to add, and you can change either one.

On the figure the arrow is the step you are adding, and the filled point is where that step lands you.

Three things are worth trying before you read on.

1. Leave the start at $1$ and change the number you add from $-4$ to $4$. The landing point moves from $-3$ to $5$. Both are four units from the start, one to the left and one to the right. So the sign of the second number decided the direction and nothing else.
2. Set the number you add to $0$. The arrow disappears, because adding zero asks you to take no step.
3. Move the start instead, and leave the number you add alone. The arrow keeps its length and its direction, and only its position changes. The step does not care where it begins.

Adding two numbers with the same sign

When both numbers carry the same sign, both steps go the same direction, so they pile up. Two moves the same way cover their combined distance.

Take $(-2) + (-5)$. Start at $-2$, which is already two units left of zero. Adding $-5$ steps five more units left, to $-7$. The two leftward moves add up to seven units left in total:

The same thing happens with two positives: $2 + 5$ is two rightward steps then five more, landing seven units right of zero at $7$. So when the signs match, the distances from zero simply add, and the answer keeps that shared sign:

Same signs: add the two distances from zero, and keep the common sign. Two pushes the same way make a bigger push the same way.

So $(-2) + (-5) = -7$ for exactly the reason $2 + 5 = 7$: in each case you travel $2 + 5 = 7$ units. The only remaining question is which side of zero you end on.

A number plus its opposite is zero

Before mixing signs, look at the most important sum in the whole chapter: a number added to its opposite. Recall from the last lesson that the opposite of a number sits the same distance from zero, on the other side. Zero is the one exception, being its own opposite. So $5$ and $-5$ are opposites, one on each side of zero.

Add them. Start at $5$, five units right of zero. Adding $-5$ steps five units left, which is exactly the distance back to zero. You land on $0$:

This is no coincidence of the numbers $5$ and $-5$. It works for every integer, and the reason is built into what “opposite” means.

Zero pairs are what make the next section work. Whenever a positive and a negative meet, you can cancel as many matched units as possible and read off whatever is left over.

Adding two numbers with different signs

When the signs are different, the two steps go opposite ways, so instead of piling up they partly undo each other. This is where zero pairs do the work.

Take $7 + (-3)$. Start at $7$ and step three units left to $4$. The leftward step of $3$ cancels three of the seven rightward units you already had, leaving four units still to the right of zero:

Now flip which number is bigger: $3 + (-7)$. Start at $3$ and step seven units left. The first three steps bring you to zero (a zero pair cancels), and four more steps carry you past zero to $-4$:

Compare the two results. In $7 + (-3)$ the positive part was larger, so after the cancelling we ended up on the right (positive) side at $4$. In $3 + (-7)$ the negative part was larger, so we ended on the left (negative) side at $-4$. In both cases the leftover distance was the difference of the two sizes, $7$ and $3$ differing by $4$:

Different signs: the numbers cancel in pairs, so subtract the smaller distance from the larger. The answer takes the sign of whichever number was farther from zero, because that side has units left over after the cancelling.

If the two distances are equal, every unit finds a partner and you are left with nothing, which is just the zero-pair fact again: $6 + (-6) = 0$.

Subtraction means adding the opposite

Subtraction looks like a new operation, but it is really addition wearing a disguise. To see why, return to what subtraction has always meant: $7 - 3$ asks “start at $7$ and take away three.” On the line, taking away three is a step of three units to the left, landing on $4$.

Stepping three units left is exactly what adding $-3$ does. So $7 - 3$ and $7 + (-3)$ trace the same walk and reach the same place.

Underneath that is an idea worth stating on its own, because it settles the harder case too. Adding a number makes a step. Subtracting that same number undoes the step, and undoing a step means travelling the same distance back the other way.

Adding $3$ steps three units right, so subtracting $3$ steps three units left. That is the walk from $7$ you just took. Now try it on a negative. Adding $-6$ steps six units left, so subtracting $-6$ has to step six units right:

So $4 - (-6)$ starts at $4$, travels six units right, and lands on $10$. Six units right is exactly what adding $6$ does, and $6$ is the opposite of $-6$. Once again, subtracting a number and adding its opposite are the same walk.

Both cases say one thing. Subtracting a number does the same thing as adding its opposite, so for any integers,

Every subtraction can be rewritten as an addition this way, and then handled with the same-sign and different-sign reasoning you already have. For the case above that reads

Subtracting a negative moving you to the right surprises almost everyone the first time. Money makes it sensible: taking away a debt of six dollars leaves you six dollars better off, the same as being given six dollars.

Worked examples

Common mistakes