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### 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/
```
Associated Topics:
Middle School Number Sense/About Numbers

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