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# Why does a negative times a negative equal a positive?

## Why minus multiply minus equal to positive number ?

This simple mathematical question has indeed a profound thinking, since it is hardly related to real life calculations.

For instance, the equation  3 X 7 can be interpreted in real life as;  if I eat 3 apples daily then in one week I have been eaten 3 X 7 apples i.e. 21 Apples.
In negative-positive terms, the equation -3 X 4 can be expressed like I dig 3 meters hole daily, thus, in four days the depth of this hole is four times the  first day which is -12.
However, no such clear interpretation can be found for the case of negative times negative. Therefore, here comes the mathematical proofs to prove that negative multiplied by negative equal positive.

### Two simple proofs why minus multiply minus equal to positive number:

The first Proof:
without loss of generality, assume that A, B and C in R+
0 = A - A  multiply the upper equation with -B
0 = -AB -A(-B)
multiply the upper equation with C
0= -C(AB)  -A(-B)(C)

CAB= -A(-B)C divide it by C
AB =  -A(-B)

The Second proof: 
Let a and b be any two real numbers. Consider the number x defined by
x = ab + (-a)(b) + (-a)(-b).
We can write
x = ab + (-a)[ (b) + (-b) ]       (factor out -a)
= ab + (-a)(0)
= ab + 0
= ab.
Also,
x = [ a + (-a) ]b + (-a)(-b)      (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).

So we have
x = ab
and
x = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b).
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