Why is minus times minus plus?

At school you learn it by heart: minus times minus is plus, so (−3) × (−2) = 6. It looks like a rule someone made up, who could just as well have chosen −6. But it isn’t that free a choice.

Multiplying is jumping

On the number line, positive numbers lie to the right of zero and negative ones to the left. Three times two is three jumps of two: from zero to 2, 4 and 6. The first number says how many times you jump, the second how long each jump is and in which direction.

Jumping backwards

Three times minus two is three jumps of minus two, so to the left: to −2, −4 and −6. Positive times negative is negative. The order of the numbers doesn’t matter, so negative times positive is negative too: (−3) × 2 = −6.

A pattern that carries on

Take away jumps and watch the answer: with each jump fewer, it goes up by 2. Jumps can’t tell you what “minus one jump” means, but the pattern can: if it carries on the same way, you get (−1) × (−2) = 2 and finally (−3) × (−2) = 6.

A car drives forwards

A car drives forwards at 2 metres a second; each second is one jump of 2 metres. In 3 seconds it will be 3 × 2 = 6 metres ahead of where it is now. 3 seconds ago, at time −3, it was at (−3) × 2 = −6, six metres behind.

The car reverses

Now the car reverses at −2 metres a second. In 3 seconds it will be at 3 × (−2) = −6, six metres behind. But 3 seconds ago it was six metres ahead, so for the maths to work, (−3) × (−2) has to come out as 6.

See for yourself

Film yourself walking backwards on your phone and play the video in reverse. In the video you walk forwards: walking backwards is a negative speed, playing in reverse is negative time, and together they make forward motion.

Multiplying out brackets

You know multiplying out brackets from school: (3 + 4) × 2 = 3 × 2 + 4 × 2. On the number line you can see it: seven jumps of two land where three jumps and then four more do. Almost every rearrangement of an expression relies on this, so we want it to work for negative numbers too.

If it were −6

In (3 + (−3)) × (−2) the bracket is zero, so the answer is zero. Multiplying out gives 3 × (−2) + (−3) × (−2) = −6 + (−3) × (−2), and that is zero only if (−3) × (−2) = 6. With −6, the same expression would come out as 0 one way and −12 the other.

So why is minus times minus plus?

Because any other answer would break the arithmetic that works for positive numbers: the pattern of multiples would break, the reversing car would end up in the wrong place and brackets couldn’t be multiplied out. Minus times minus is plus so that the rules work the same for all numbers.