   SEARCH HOME Math Central Quandaries & Queries  Question from priya, a student: why is minus into minus plus? Hi Priya,

I'm going to show you two arguments to try to convince you that a negative times a negative is positive. The first comes from the mathematicians' desire to have the operations of addition and multiplication on the real numbers satisfy certain properties. For example for any numbers a, b and c we want a × b = b × a, a × 0 = 0 and a × (b + c) = a × b + a × c.

Let's look at (-a) × (b - b). Clearly b - b = 0 so (-a) × (b - b) = (-a) × 0 = 0. On the other hand

(-a) × (b - b) = (-a) × [b + (- b)] = (-a) × b + (-a) × (-b) = -(a × b) + (-a) × (-b).

But we know this is zero so we have

-(a × b) + (-a) × (-b) = 0.

Now add a × b to each side to get

(a × b) -(a × b) + (-a) × (-b) = 0 + (a × b)

so

(-a) × (-b) = a × b.

Hence if a and b are both positive we have that a negative times a negative is positive.

Somehow this is not very satisfying. Better questions are "Is there a situation where we use multiplication and it is natural to multiply a negative times a negative?" and "In this situation what is a negative times a negative?" Let me show you an example.

Think of a number as a point on the number line. The size of the number tells you how far it is from zero and the sign of the number tells you if it is right of zero (a positive number) or left of zero (a negative number). So 3 is three units to the right of zero and -3 is three units to the left of zero. Multiplication by a positive number is a scaling factor so

2 time 3 is 6 units to the right of zero, [2 × 3 = 6]
1/2 times 3 is 3/2 = 1 1/2 units to the right of zero, [1/2 × 3 = 1 1/2]
2 times -3 is 6 units to the left of zero [2 × (-3) = -6],
and so on.

Multiplication by a negative number is a scaling factor also but switches to the other side of zero so

-2 time 3 is 6 units to the left of zero, [(-2) × 3 = 6]
-(1/2) times 3 is 3/2 = 1 1/2 units to the left of zero, [(-1/2) × 3 = -1 1/2]

Thus using this interpretation of multiplication, since -3 is to the left of zero,

-2 time -3 is 6 units to the right of zero, [(-2) × (-3) = 6].

I hope this helps,
Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.