Why minus times minus equals plus?

Mike Savage gives a brilliant explanation on the excellent British website of mathtutor. I can't but recommend this website heartily.

So, why "minus times minus equals plus"?

(-1)x(-1)= 1

We have to remember two arithmetics rules:

1. The rule of precedence (or order of operations, also called BODMAS from Brackets, Orders - squares roots or powers -, Division and Multiplication, Addition and Subtraction)

2. The rule of the distributive law of multiplication (in other words, the multiplication is distributive over addition -> e.g. 3x(5+2)=3x5 + 3x2 )

 

Then we can now prove that minus x minus = plus.

Let's consider

(-1)x(2+1)

(read "negative one"," times", *two plus one")

Now, for rule 2 (above) the operation can be rewritten as

= (-1)x2 + (-1)x(1)

so, similarly, we can take another operation such as

(-1)x(-1+1)

and by expanding the brackets (rule 2) we obtain

= (-1)x(-1)+(-1)x1

=(-1)x(-1) -1

But (-1)x(-1+1) can also be formulated with the rule 1 above (the precedence law, brackets first) as

(-1)x(-1+1)

= (-1)x(0)

Which simply gives

=0

Now, we combine the two

(-1)x(-1) -1 = 0

which is the same to say

(-1)x(-1) = 0 +1

and so

(-1)x(-1) = 1

Q.E.D.

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