# Algebraic explanation of - times - = +

1. Sep 29, 2010

### G037H3

I'm pretty sure this isn't a homework style question.

could someone please explain this, I may be overly tired, but it's confusing me:

"A further illustration of this rule is generally given by algebraists as follows: First, we know that +a multiplied by +b gives the product +ab; and if +a be multiplied by a quantity less than b, as in b-c, the product must necessarily be less than ab; in short, from ab we must subtract the product of a, multiplied by c; hence a times (b-c) must be expressed by ab-ac; therefore it follows that a times -c gives the product -ac. If now we consider the product arising from the multiplication of the two quantities (a-b), and (c-d), we know that it is less than that of (a-b) times c, or of ac-bc; in short, from this product we must subtract that of (a-b) times d; but the product (a-b) times (c-d) becomes ac-bc-ad, together with the product of -b times -d annexed; and the question is only what sign we must employ for this purpose, whether + or -. Now we have seen from that the product ac-bc we must subtract a quantity less than ad, we have therefore subtracted already too much by the quantity bd; this product must therefore be added; that is, it must have the + sign prefixed, hence we see that -b times -d gives +bd for a product; or -minus multiplied by -minus gives +plus."

2. Sep 29, 2010

### JonF

A negative times a negative equaling a positive is best understood from what it means to be negative. If a + b = 0 we call b “-a” or we say that b is a’s additive inverse. Given that:
1 + -1 = 0
-1(1 + -1)=-1*0
-1(1 + -1) = 0
-1*1 + (-1)(-1) =0
So -1 + (-1)(-1) = 0
Thus -1’s additive inverse is (-1)(-1), but -1’s additive inverse is 1. So (-1)(-1) = 1.
From here it gets simple, consider
(-a)(-b) = (-1)a(-1)b = (-1)(-1)ab = 1ab = ab

3. Sep 29, 2010

### G037H3

I understand that completely. What I'm asking for is an explanation of what I posted.

4. Sep 29, 2010

### Tac-Tics

The context part you have quoted isn't very clear. It's all just a bunch of rambling algebra identities.

What is it you are having problems understanding?

5. Sep 29, 2010

### G037H3

I understand why - x - = +, i simply want to understand the argument given

it's in Euler's Elements of Algebra x.x

6. Oct 3, 2010