mathsGrade 6-82 min read

The rules of signs, and why two negatives make a positive

Negative numbers trip up more middle-schoolers than anything else. Here are the four rules, the number-line picture behind them, and a reason 'minus times minus' isn't just something to memorise.

Negative numbers are where careful students suddenly start losing marks, because the rules feel arbitrary. They aren't — each one has a picture. Learn the picture and you stop guessing.

Adding and subtracting: walk the number line

Think of a number line. Adding moves you right; subtracting moves you left. A negative number just flips the direction of the move.

$5 + (-3)$ means start at $5$ , move $3$ left $\rightarrow 2$ .
$5 - (-3)$ means start at $5$ , but subtracting a negative flips it to moving right $\rightarrow 8$ .

"Subtracting a negative" is the same as adding — taking away a debt makes you richer.

Multiplying and dividing: count the negatives

For $\times$ and $\div$ , ignore the signs, do the numbers, then decide the sign by a single rule: an even number of negatives gives a positive; an odd number gives a negative.

Signs	Result
$(+)\times(+)$	$+$
$(+)\times(-)$	$-$
$(-)\times(+)$	$-$
$(-)\times(-)$	$+$

Why minus times minus is plus

This is the one nobody explains, so here's the reason. A negative sign means "the opposite of." So $-(-3)$ is "the opposite of negative three," which is $+3$ . Multiplying by a negative reverses direction; multiplying by two negatives reverses it twice — and turning around twice leaves you facing the way you started.

(-4)\times(-3) = 12, \qquad (-12)\div(-4) = 3

The most common slip

Watch a squared negative carefully — the brackets decide everything:

(-3)^2 = (-3)\times(-3) = 9 \qquad\text{but}\qquad -3^2 = -(3\times3) = -9

Do the number first, then the sign. Work out the digits as if everything were positive, then count the minus signs at the end and apply one rule. Splitting it into two steps stops the panic.

Last revised 15 October 2024.