I've always had a doubt about why a negative multiplied by negative results in a positive.
I could understand why two positives multiplied together result in a positive - you're duplicating an amount by a multiple so you have multiple or increase of that amount. If you had lots and you multiply it by a factor, you are more 'lots'.
The same logic applies to multiplying negative by a positive - if you had less than nothing or a debt and you multiply that debt you have more of nothing or more of that debt.
So that explains why \(+ \times + = +\) or why \(- \times + = -\) but how do people accept that a negative multiplied by a negative is a positive?
I finally found how and why, and it's a testament to algebra's ability to reduce complexity that makes it easier to understand and accept. For example, using letters:
if \(a \times b = ab \tag{1} \) and \(-a \times b = -ab \tag{2} \) then \(-a \times -b\) must be negative or positive ab, i.e \( \pm ab\) - but which one?
Well, intuitively its not easy to know which one (see below), but it can only be 1 or two possible signs, i.e \(-\) or \(+\) right? And if \(-ab\) is already made provision by \(-a \times b\), then it cannot be \(-\), so.... it must be \(+\)!
Simple!
This is purely by the elimination of choices for the signs that we have left, which is only \(+\). So purely because it cannot be anything other than \(+\) or \(-\) and \(-\) is already been made provision for, it has to be \(+\).
So we have no choice but to define \(-a \times -b\) as equalling +ab.
Now, what I find interesting about this is that without algebra to simplify the numbers in terms of letters, and being able to make generalised rules, by those rules, you can determine something that's not intuitive otherwise.
For example, if I intuitively try to reason what I've just determined through elimination, its not intuitive at all:
I did think about multiplying by a negative as successive subtractions instead of successive addition:
\(a \times -b\) as a reducing by itself by b times, so \(8 \times -2\) is:
8 reduce itself(8) by 8 is 0
0 reduce the second time by 8 is -8
this is wrong, so instead if instead of reducing 8 (itself) by itself (8), you reduce 0 by 8:
0 reduced by 8 is -8
-8 reduced by 8 is -16
This is correct, in the same way, that 8 times of -2 is 0 reduced by -2, 8 times. Essentially you start with 0 and perform 8 subtractions of 2.
So incidentally, this shows that all expressions always are 0 to begin with, and then the numbers and operations apply to 0.
So \(-a \times -b\) is reducing -a by -b times. This doesn't intuitively make clear what would happen - its not intuitive.
So \(-8 \times -2\) is
0 reduced by -8 once is 8 (this is not intuitive)
8 reduced by -8 again is 16 (this is not intuitive)
This seems to suggest that reducing by a negative (or a multiple of a negative, i.e. twice -8) is making a 0 less negative and doing so successively will eventually make it positive. In the same way, reducing 0 by a positive is making it 0 more negative and doing so successfully will eventually make it negative.
All of this is not intuitive by default, and I've tried to reason about it before (See Rules then Since then) however, the elimination of possibilities using Albegra at the beginning of this text makes it much more intuitive and acceptable.
Even if it doesn't explain why, it shows what it must be as there was no other choice. That's easier to accept than, "...a negative times a negative is a positive."
This is one pretty useful characteristics of Algebra - that it can reduce complexity so you can reason within the limits of rules that algebra puts in place.