when multiplying negatives, i find the following equivalent expressions of a series of addition a more direct explanation since multiplication is just a repeated addition.
what do mathematicians call the addition progressions from #4 to #1?
on -1:
1. -1 * -3 = -1 + 1 + 1 + 1 + 1 = 3
2. -1 * -2 = -1 + 1 + 1 + 1 = 2
3. -1 * -1 = -1 + 1 + 1 = 1
4. -1 * 0 = -1 + 1 = 0
5. -1 * 1 = -1 + 0 = -1
6. -1 * 2 = -1 + -1 = -2
well, of course, definitions are better because their short, accurate, and diverse, but from a formally generalized rule their always exist a rudimentary series from which it's based upon. what is the proof that encompass -a*-b=ab without using negative numbers to prove that negative numbers is equal to a positive number since we are to prove mathematically that multiplying negative numbers results to a positive number?
if we wish the distributive law to hold for negative, as well as positive integers:
then 0 = a*0 = a(b+(-b)) = ab + a(-b).
evidently, we will need a(-b) = -(ab) for this to be true.
we would also like for 0 = (-a)*0 = (-a)(b+(-b)) = (-a)b + (-a)(-b) = -(ab) + (-a)(-b).
this then, forces us to set (-a)(-b) = ab, unless we don't want that silly distributive law anymore.