Re: "-1*-1 = 1": when was this first noticed?

From: Adam (addam_at_rogers.com)
Date: 08/05/04 

Date: Thu, 05 Aug 2004 14:32:14 GMT

"Mitch Harris" <harrisq@tcs.inf.tu-dresden.de> wrote in message
news:2neltsF27rbU1@uni-berlin.de...
> So that is a way to convince someone. But how was the rule
> discovered/developed historically?
>
    It wasn't discovered. Rather, it was created. The integers were
conceptualized as being points on a line. Each whole number was given its
negative symmetric partner. Multiplying by minus one was conceptualized as a
rotation of the number line, replacing symmetric partners with one another.

> Frankly, when were the negative numbers recognized as such (comparable to
> our modern conception).
>
    Negative numbers were a human conceptual extension of the whole numbers,
which were motivated by the desires of mathematicians for numbers to have
certain properties. e.g. closure.

> When was a formal proof of -1*-1 = 1 first given?
>
    That depends on what you term as formal. I am sure that (-1)*(-1) = 1
was used much much earlier before any formal proof was given.

    These things are humans creations, Adam.