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Proving That (-1) * (-1) = 1

Date: 01/30/2006 at 10:59:35
From: Ken
Subject: proof of why -1 x -1 = 1

At the high school level it is difficult to prove that -1 x -1 = 1. 
Is there a published proof somewhere?  Thank you.



Date: 01/30/2006 at 11:14:11
From: Doctor Jerry
Subject: Re: proof of why -1 x -1 = 1

Hello Ken,

Thanks for writing to Dr. Math.  If you scan down a bit in the FAQ

  Negative Times a Negative
    http://mathforum.org/dr.math/faq/faq.negxneg.html 

you will find an explanation.  

A different way of approaching the matter is explained--at great 
length--in the book Foundations of Analysis, written by Edmund Landau.  
Landau starts with Peano's Axioms for the positive integers and from 
this constructs the set of real numbers.  On page 85 of this text, 
Landau proves
 
  Theorem 197: For each pair of real numbers x and y, (-x)(-y) = x*y.

This may be a bit much for high school kids, but it's a nice proof.

Please write back if my comments are not clear.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Multiplication
Middle School Negative Numbers

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