Adding and Subtracting Integers

Adding is moving along the line

Start at the first number. A positive moves right, a negative moves left.

$$3 + 5 = 8 \qquad \text{(start at 3, move 5 right)}$$ $$3 + (-8) = -5 \qquad \text{(start at 3, move 8 left)}$$

That is the whole of addition. The size of the number is how far to move, and the sign is which way.

Opposites add to zero

A number and its opposite are additive inverses: they cancel exactly.

$$6 + (-6) = 0 \qquad -14 + 14 = 0$$

Six steps right then six steps left ends where it started.

This shows up constantly in context. A deposit of $50$ and a withdrawal of $50$ leave the balance unchanged. A lift going up four floors then down four is back on the same floor.

Two signs in a row

When a sign meets a sign, they combine into one:

Written	Becomes	Because
$+ (+3)$	$+3$	add a gain
$+ (-3)$	$-3$	add a loss
$- (+3)$	$-3$	take away a gain
$- (-3)$	$+3$	take away a loss

Two like signs make a plus. Two unlike signs make a minus.

$$7 + (-2) = 7 - 2 = 5 \qquad 7 - (-2) = 7 + 2 = 9$$

The last one is the one that surprises people, and the context makes it sensible: removing a debt makes you better off. If a $2$ dollar charge is cancelled, your balance goes up by $2$.

Subtracting is adding the opposite

$$a - b = a + (-b)$$

Every subtraction can be rewritten as an addition:

$$3 - 8 \;=\; 3 + (-8) \;=\; -5$$ $$5 - (-3) \;=\; 5 + 3 \;=\; 8$$ $$-4 - 6 \;=\; -4 + (-6) \;=\; -10$$

Why that rewrite is worth making every time

You could learn one set of rules for adding signed numbers and a second set for subtracting them. Rewriting means you never need the second set.

There is only ever one operation to handle, and one question to ask: which way does each move go? That is why the rewrite is the first step worth taking whenever a subtraction and a negative appear together — $-4 - (-9)$ is confusing to look at and straightforward once written as $-4 + 9$.

It also keeps working past this topic. In algebra, $x - 3$ is treated as $x + (-3)$ so that terms can be reordered freely; $3 + x$ and $x + 3$ are equal, while $3 - x$ and $x - 3$ are not. Turning subtraction into addition is what makes combining like terms safe to do in any order.

Adding two numbers with different signs

Once everything is an addition, one shortcut handles the mixed-sign case.

Subtract the distances from zero, and keep the sign of whichever is further out.

$$-9 + 4$$

$9$ and $4$ are the distances, and their difference is $5$. Since $-9$ is further from zero, the answer is negative:

$$-9 + 4 = -5$$

This is why a large loss and a small gain still leave a loss. The two pull in opposite directions, and the bigger one decides which way the answer falls.

Same signs are simpler — add the distances and keep the shared sign:

$$-6 + (-7) = -13$$

Worked examples

Common mistakes

Practice problems