Repeating Decimals - Rational or Irrational?

Date: 09/11/2001 at 19:26:09
From: Mirko l.
Subject: Rational or irrational numbers?

In my 7th grade class we are working on rational and irrational 
numbers. My teacher wants an  opinion about whether these numbers are 
rational or irrational:

0.252252225 
0.125126127 

Date: 09/11/2001 at 21:57:22
From: Doctor Peterson
Subject: Re: Rational or irrational numbers?

Hi, Mirko.

I assume you mean decimals that continue these patterns forever, not 
the terminating decimals you show.

In each case, althought there is a pattern, it is not the special kind 
of pattern that rational numbers have, namely an exact repetition of a 
given group of digits forever. It is only in that case that the 
decimal can be converted to a fraction, which is the true definition 
of a rational number. So these numbers are irrational; the first has a 
"repetition" of a changing number of digits, and the second has 
changing numbers within the "repeating" pattern.

There is a way you could interpret the second number that would make 
it rational. Since both numbers are defined only by example, and 
without clearly stating the rule, we can't be sure. Here's how 
to make it rational:

    0.125 126 127 ... 998 999 000 001 002 ... 124 125 ...
      \_________________________________________/

If we think of the three-digit numbers as wrapping around within 
themselves, so that 999 is followed by 000, rather than somehow 
carrying over into the previous section, then the indicated section is 
in fact a repetend, and this is rational.

The irrational version I had in mind is something like this:

      0.125
    + 0.000 126
    + 0.000 000 127
    ...
    + 0.000 000 000 ... 997 998 999
    + 0.000 000 000 ... 000 000 001 000
    + 0.000 000 000 ... 000 000 000 001 001
    ...
    ---------------------------------------
      0.125 126 127 ... 997 999 000 001 002

This way there is no true repetition. But even that is hard to be sure 
of.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/12/2001 at 10:03:27
From: Doctor Peterson
Subject: Re: Rational or irrational numbers?

Hi again, Mirko!

I thought some more about your problem, and realized it is even more 
subtle than I realized. The "irrational" interpretation I suggested 
for the second number turns out also to be rational, using some 
special tricks.

I suggested taking the number as

      0.125
    + 0.000 126
    + 0.000 000 127
    ...
    + 0.000 000 000 ... 997 998 999
    + 0.000 000 000 ... 000 000 001 000
    + 0.000 000 000 ... 000 000 000 001 001
    ...
    ---------------------------------------
      0.125 126 127 ... 997 999 000 001 002

But this is the same as

      0.125 125 125 ... 125 125 ...
    + 0.000 001 002 ... 872 873 ...

which in turn can be broken up as

      0.125 125 125 ... 125 125 ... =   125/999
    + 0.000 001 001 ... 001 001 ...   + 1/999 * 1/1000
    + 0.000 000 001 ... 001 001 ...   + 1/999 * 1/1000^2
    ...
    + 0.000 000 000 ... 001 001 ...   + 1/999 * 1/1000^872
    + 0.000 000 000 ... 000 001 ...   + 1/999 * 1/1000^873
    ...                               + ...

This gives 125/999 + 1/999 * 1/999, or 124876/998001. That's rational.

The lesson here is that, though 0.125126127... can not be shown to be 
rational BASED ON THE OBVIOUS THREE-DIGIT PERIOD, that doesn't mean it 
might not be a rational number with a longer period. It's very hard to 
show that a given number is irrational!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/