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### Non-Periodic, Non-Terminating Decimals

```
Date: 06/26/2001 at 17:45:53
From: rebecca rocha
Subject: Math (algebra)

Why is a non-periodic, non-terminal decimal an irrational number?

I have looked in math books at the library.
```

```
Date: 06/27/2001 at 12:13:44
From: Doctor Ian
Subject: Re: Math (algebra)

Hi Rebecca,

When you have a terminating decimal, like 0.234, that is just a
shorthand way of writing 234/1000. So any terminating decimal is a
rational number.

When you have a repeating decimal, like 0.234234..., you can do this
little trick:

x =   0.234234...

1000x = 234.234234...

1000x - x = 234

999x = 234

x = 234/999

So any repeating decimal is a rational number.

In short, to convert a decimal to a rational number, you have to be
able to figure out what the denominator is. But unless the decimal
terminates or repeats, there is no way to do this.

Or to put it another way, if you want to make up a decimal that isn't
a rational number, you'd better make sure that it doesn't terminate or
repeat.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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