The rule i learned is:
For multiplication or division two likes give a positive answer, two unlikes give a negative answer.
E.g.
-2 * -2 = 4
-2 * 2 = -4
BUT look at this...
2 * 2 = 4 is the same as saying 2 + 2 = 4. Right?So....
-2 * -2 = 4 should be the same as saying -2 + -2 = x
But... -2 + -2 = -2 - 2 = -4 ..... so it isn't the same pattern as above.
Can anyone shed some light?
What kind of answer do you want? You can show from the "field axioms" that the product of two negative numbers is positive. That's very simple, completely correct and very technical- requiring that you know a lot of definitions and abstract concepts you probably don't.
Or you can argue that, just as if you have, say, 5 boxes, each containing 10 dollars, then you have 5*10= 50 dollars, that if you owe $10 to each of 5 people (so - $10), and each of the 5 people cancels ("negates") the debt, you no longer owe any thing- its just as if you had suddenly gained $50 to pay off your debts: (-$10)(-5)= +$50. That's very basic, very simple, and very "hand-waving".
But that's the way mathematics is- you can have technical and precise or non-technical and vague.
The flaw is in that the rule a*b = a+a+a+a+... (and that b times) can't be a applied to negative numbers.
You can't say: I'll add -2 to itself -2 times. You can't do something a negative amount of times.
How HallsofIvy explained it is a better way to think about it.
Imagine Do is +, Don't is -
Imagine Win is +, lose is -
Imagine we a gambling - so we either win or lose.
2 times I do(+) win(+) £10 : so I win(+) £20 : so +x+ = +
2 times I dont(-) win(+) £10: so I lose(-) £20: so -x+ = -
2 times I do(+) lose(-) £10 : so I lose(-) £20: so +x- = -
Here is you tricky bit.
2 times I dont(-) lose(-) £10 : so I win(+) £20: so -x- = +
You are using the * operator as an addition operation.
"2 times 2" maybe ordinarily worded as 2 "two times with a plus between".
2(2) or 2*2 is 2+2.
3(2) or 3*2 is 2+2+2 or 3+3
4(2) or 4*2 is 2+2+2+2 or 4+4
-2*2 is a symbolic way to represent a "double subtraction of 2"
-2-2=2(-2)=2(-1)2=(-1)4
Basically it is "subtract 2 twice" in common language (just one way to describe it).
-2*3=-2-2-2
-2*4=-2-2-2-2
(-2)*(-2)=-[2(-2)]=-(-2-2)=-(-4)
"-" is the opposite of, if you'd like to think of it that way.
So you have the "opposite of subtract 2 twice" which is....
What you're really asking is why is the product of a negative and a positive number defined to be a negative number and why is the product of two negative numbers defined to be a positive number?
Let's do a pattern:
4 x 4 = 16
4 x 3 = 16 - 4 = 12
4 x 2 = 12 - 4 = 8
4 x 1 = 8 - 4 = 4
4 x 0 = 4 - 4 = 0
4 x -1 = 0 - 4 = -4...
let's do another pattern:
-4 x 4 = -16
-4 x 3 = -16 + 4 = -12
-4 x 2 = -12 + 4 = -8
-4 x 1 = -8 + 4 = -4
-4 x 0 = -4 + 4 = 0
-4 x -1 = 0 + 4 = 4...
If you see the patterns and you know a little algebra to generalize the results, then you have your answer.