Math212

Applied And Computational Mathematics

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: [3]
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).