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Saturday, September 20, 2003

A simple proof that -1*-1 = 1

Carlos Scheidegger sent this proof which is better than my attempt at why -1 * -1 = 1...

The definition of multiplication for whole numbers is:

x * y = y + y + y + ... + y + y, where y appears x times.

Using this, it is easy to prove that, being (succ x) the successor of x, 

if x * y = z, then (succ x) * y = z + y, and vice-versa.

By definition, 0 is the successor of -1. Also by definition,

0 * x = 0, 

and so, 0 * -1 = 0.

(succ -1) * -1 = 0
(succ -1) * -1 = 1 + -1

Now, we apply the property:

(succ -1) * -1 = 1 + -1 ->

-1 * -1 = 1

----

This proof's only assumption is that -n + n = 0, which is easily
provable. (Very easy using peano arithmetic)

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Portland, Oregon, United States
I'm usually writing from my favorite location on the planet, the pacific northwest of the u.s. I write for myself only and unless otherwise specified my posts here should not be taken as representing an official position of my employer. Contact me at my gee mail account, username patrickdlogan.