Proving Irrational Numbers

Date: 05/10/2000 at 22:48:43
From: Robert Dawson
Subject: Irrational numbers

I know that there are only three types of decimals:

  1) Terminating
  2) Repeating
  3) Non-terminating and non-repeating

I know and can prove that terminating and repeating decimals are 
always rational.

I know that non-terminating and non-repeating decimals are irrational. 
But I don't know how to prove that this is the case. How do you prove 
it?

Sincerely,
Rob Dawson

Date: 05/11/2000 at 02:27:25
From: Doctor Floor
Subject: Re: Irrational numbers

Hi, Rob,

Thanks for writing.

To prove that case 3 always yields irrational numbers we use indirect 
reasoning:

We can show that rational numbers - fractions - are always in category 
(1) or (2). This can be done by converting the fraction into a decimal 
by long division. After each step you get a remainder. After a couple 
of initial steps, you have to attach a zero to the remainder to 
continue on. Since the remainder is always smaller than the 
denominator of the fraction, there is only a limited number of 
numbers you can have to work with. The remainder might be 0 at some 
point; then you have case (1). If 0 does not appear, then at some 
point you will get a remainder that you have had before, and 
repetition is born, leading us to case (2).

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/