why does (-5)*(-4)=20 ?, eom

[Norfolk And Chance]

The easier "proof" of 4 * -5 = -20 is:

4 * -5 = -5 + -5 + -5 + -5 = -20 (it follows from the axiom a + -a = 0)

From that, you can prove that -5 * -4 = -20 by using the following (stolen) 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)

Therefore

ab = (-a)(-b).

Fucking hell.

It's easy to understand that 5 * -4 = -20, because it's just 5 lots of -4, so for simpletons like me -5 * -4 must give an opposite answer of 20.