The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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 

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 
Associated Topics:
Elementary Multiplication
Middle School Negative Numbers

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.