Negative times Negative equals Positive

Introduction
Those with some familiarity with secondary school maths will be aware that a negative number multiplied by a negative number gives a positive number.

For example:
-2 X -3 = +6 , it does not equal -6.

I thought that rule was an axiom or assumption. An equivalent in standard English is the double negative. The statement "I don't want nothing" means that you want something because whatever you don't want it isn't nothing. Of cause in colloquial English and other versions of English it can still mean a negative. As well a double negative can mean negative in some other languages, for example Africans and Greek.

The Distributive Law
In mathematics though negative times negative equals positive turns out to be a provable law. The first issue to deal with is the Distributive Law of Mathematics, which can be written as : A x C + B x C = (A + B) x C . Stated in words the distributive law says: to multiply a given number C by the sum of two numbers is the same as multiplying C by each of those numbers and and then adding the result. The Distributive Law links multiplication with addition and is one of the fundamental laws of maths.

Here is a demonstration of how the law works with positive whole numbers.

If A = 2, B = 3 and C = 4:
2 x 4 + 3 x 4 = (2 + 3) x 4
8 + 12 = 5 x 4
20 = 20

There are algebraic proofs of the Distributive Law, but they require considerable explanation. As a visual thinker I find the diagram at right compelling proof.

Applying the Distributive Law to Negative Numbers
Our task is to apply the distributive law to negative numbers.
When we do this two other rules will be resolved: positive x negative = ? and negative x negative = ?.

Positive x Negative = ?
Assume: positive x negative = positive ie -3 x 4 = +12 and -1 x 4 = 4
If A = 2, B = -3, C = 4
By the distributive law:
2 x 4 + -3 x 4 = (2 + -3) x 4
8 + 12 = -1 x 4
20 = 4
This result is clearly incorrect so our assumption (positive X negative = positive) is incorrect.
Assume: positive x negative = negative ie -3 x 4 = -12 and -1 x 4 = -4
By the distributive law:
2 x 4 + -3 x 4 = (2 + -3)x 4
8 + -12 = -1 X 4
-4 = -4
This is clearly correct so our assumption (positive X negative = negative) is correct.

Negative X Negative = ?
Assume: negative x negative = negative ie -3 x -4 = -12 and -1 x -4 = -4
If A = 2, B = -3, C = -4
By the distributive law:
2 x -4 + -3 x -4 = (2 + -3) x -4
-8 + -12 = -1 x -4
-20 = -4
This is clearly incorrect so our assumption (negative x negative = negative) is incorrect.
Assume: negative x negative = positive ie -3 x -4 = 12and -1 x -4 =4
If A = 2, B = -3, C = -4
By the distributive law:
2 x -4 + -3 x -4 = (2 + -3) x -4
-8 + 12 = -1 x -4
4 = 4
This is clearly correct so our assumption (negative x negative = positive) is true which is what we set out to do at the beginning of the post.

A More Concise Proof
The following proof is more shorter than that described above. The only extra mathematical rule that this contains is a number multiplied by zero is equal to zero, ie N x 0 = 0 .

An example is 5 x 0 = 0 which equals 0 + 0 + 0 + 0 + 0
which is of cause 0.

Let a and b be any two real numbers. Consider the number n defined by:
n = ab + (-a)(b) + (-a)(-b)

We can write
n = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a) (0)
= ab + 0
= ab

Also,
n = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 x b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b)

So we have,
n = ab
and
n = (-a)(-b)

Hence ab = (-a)(-b)

UPDATE: 28 / 01/ 2013

Here is an argument from the Khan Academy using the distributive law, demonstrating that a negative times a negative equals a positive:



Here are two more videos from the Khan Academy explaining negative times negative equals positive.

The first one explains why positive times negative equals negative:



The next video uses ideas from the previous video to demonstrate that negative times negative equals positive: