Anyway...

By virtue of everyday usage, the fact that (-1) x (-1) = 1 has been engraved onto our heads. But, only recently did I actually sit down to explore why, in general negative times negative yields a positive number !

The Intuitive Argument

Let’s play a game called “continue the pattern”. You would be surprised, how intuitive the results are:

2 x 3 = 6

2 x 2 = 4

2 x 1 = 2

2 x 0 = 0

2 x (-1) = ??  (Answer : -2 )

2 x (-2 ) = ?? (Answer : -4 )

2 x ( -3) = ?? (Answer : -6 )

The number on the right-hand side keeps decreasing by 2 !

Therefore positive x negative = negative.

2 x -3 = -6

1 x -3 = -3

0 x -3 = 0

-1 x -3 = ?? (Answer : 3)

-2 x -3 = ?? (Answer : 6)

The number on the right-hand side keeps increasing by 3.

Therefore negative x negative = positive.

Pretty Awesome, right? But, let’s up the ante and compliment our intuition.

The Banker’s Interpretation

We can think of negative number as a debt.

If my bank account contains $ -3 then I owe the bank $3.

Suppose that my debt is multiplied by 2, then surely the debt becomes $6. So it makes perfect sense to insist that 2 x -3= -6.

What then should -2 x -3 be?

Well if the bank kindly writes off ( takes away ) two debts of $3 each. My account balance is now 0. It is exactly as though i had deposited $6 to my account.

So, in banking terms -2 x -3 = +6

The Number Line Approach.

Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y.

Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps.

So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.

Ergo, -1 x -1 means to stand on 0, face in the negative direction, and then take 1 backward step.

This lands us smack right on +1 !

I am saving the best for the last.

Obviously I cannot call -1x -1 as -1 since if that were the case then :

-1 x 1 = -1 and also -1 x -1 = -1. We get,

-1 x 1 = -1 x -1

i.e then -1 = 1 by virtue of cancellation.Which is absurd.

Hence it is +1

Powerful technique!

Concluding remarks.

Hope you enjoyed the post and Pardon me if you found this to be rudimentary for your taste. This post was inspired by Joseph H. Silverman’s Book - A friendly Introduction to Number Theory. If you are passionate about numbers or math, in general it is a must read.

There are several other arithmetic methods that prove the same, if you are interested feel free to explore.

Have a Good Day!