TITLE: Significance of multiplying $-1$ by $-1$
QUESTION [13 upvotes]: Maybe this is a weird question but it's been bugging me. 
In the childhood we were taught that $4 \times 3$ means $4+4+4$ i.e. adding 4, 3 times.
My question is then how would you explain $-1 \times -1$ using some kind of mathematical logic?I want to know the significance in real life.
 It doesn't have a meaning when I say adding $-1$, $-1$ times.

REPLY [2 votes]: I've often seen this proof/explaination
$$\bigg(ab + (-a)b\bigg) + (-a)(-b) = \bigg(ab + (-a)b\bigg) + (-a)(-b)$$
$$\bigg(ab + (-a)b\bigg) + (-a)(-b) = ab + \bigg((-a)b + (-a)(-b)\bigg)$$
$$\bigg(a+(-a)\bigg)b+(-a)(-b) =ab +\bigg(b+(-b)\bigg)(-a)$$
$$0b+(-a)(-b) =ab +0(-a)$$
$$0+(-a)(-b) =ab +0$$
$$(-a)(-b) =ab$$