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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.

Does this help?  Write back if you'd like to talk about this some 
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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