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Irrational Decimals

Date: 08/04/98 at 08:54:13
From: Scott Bennett
Subject: Decimal representations of rational numbers

I was recently shown a proof stating that 0.9 (repeating) is actually 
equal to one. I have come to accept this proof, but I'm wondering from 
a representation standpoint whether they are actually considered to be 
mathematically equal.

The proof involves using an sum of 9/(10^n) for n = 1 to infinity, 
which is an expression for 0.9 repeating. I was actually able to 
justify this by pulling the 9 out of the sum and multiplying it by the 
sum of 1/(10^n) for n = 1 to infinity.

This opens up a whole new can of worms, specifically that any 
irrational number could possibly be expressed as an infinite sum of 
rational numbers. And since the sum of two or more rational numbers is 
another rational number, there may not be any irrational numbers as the 
term is currently defined.

My question is, are the decimal representations of rational numbers and 
the rational numbers themselves mathematically equal, or would they 
more appropriately be called equivalent?

Date: 08/04/98 at 16:59:58
From: Doctor Rob
Subject: Re: Decimal representations of rational numbers

You are beginning to discover some of the beauties and mysteries of

It is true that the sum of any finite number of rational numbers is 
again a rational number, but the sum of an infinite number of rational 
numbers may be irrational. After all, any decimal expansion, such as:

   Pi = 3.14159265358979323846...
   sqrt(2) = 1.414213562...

and so on, are actually infinite sums of rational numbers:

   3.14159... = 3 + 1/10 + 4/10^2 + 1/10^3 + 5/10^4 + 9/10^5 + ...

Yes, every irrational number is the sum of an infinite number of 
rational numbers. That does not preclude them from existing. The 
rational numbers are just those whose decimal expansions are 
eventually periodic. All the others are irrational.

In answer to your last question, the decimal representations and the
rational numbers are mathematically equal. Some rational numbers have 
more than one decimal expansion, as you have come to accept:

   1 = 1.00000000... = 0.99999999...

for example. That is not an impediment.

Keep asking good questions!

- Doctor Rob, The Math Forum
Check out our web site!   

Date: 08/05/98 at 11:39:15
From: Anonymous
Subject: Re: Decimal representations of rational

How can you say that an infinite sum of rational numbers can be an   
irrational number? If you can add two rational numbers and get a   
rational number, then by the associative property of addition, you 
could add up any number of rational numbers, even an infinite number, 
and get a rational number.

It defies logic to say that you can show irrationality by demonstrating 
that a number cannot be shown to be the root of Qx+P = 0. There are an 
infinite number of integers, and no one can know whether there are 
numbers out there for which this method will or will not work. Maybe 
our computing power just is not enough yet to find these numbers, but 
this does not mean they don't exist, they're just beyond our current 

To me, you could prove irrationality by showing that there is not a   
repeating pattern. But what if the repeating pattern is infinitely 
long? And what if the pattern is not necessarily (I don't know the 
right word for it) linear? Maybe something like:

   1.32435465768798109111012 ...   

There is a repeating pattern, but it's not linear, like:

   1.35246135246 ... 

(like I said, I don't know the right word for it, but linear seems 

Thanks, this has been fun and educational, and has definitely made me   
think. This revelation has only come to me in the past few days through 
some research and just plain thinking about things, and it has shaken 
me up a little bit, but I think that's good now and then.


Date: 08/05/98 at 16:09:16
From: Doctor Rob
Subject: Re: Decimal representations of rational

When you pass from a finite number of terms to an infinite number of 
terms, your proof breaks down, and no substitute proof is available. 
You may get for your sum either a rational or an irrational number. 
See below.

An irrational number is defined as one which cannot be written in the 
form a/b for integers a and b with b nonzero. (You might as well assume 
that b > 0, since a/[-b] = [-a]/b.) Those are precisely the solutions 
of b*x + (-a) = 0, so any number which is not the solution of such a 
linear equation with integer coefficients is by the definition an 
irrational number.

Every rational number a/b has a finite repeating sequence of digits 
in its decimal expansion. The proof goes like this. Use long division 
to divide b into a. Look at the sequence of remainders you get. 
All of them are smaller than b (if you have done the long division 
correctly!). After at most b steps, you will either get a remainder 
of 0 (and the decimal expansion terminates) or you will get a nonzero 
remainder you have already seen previously. This is an application of 
the Pigeonhole Principle. (If you put n+1 things in n boxes, at least 
one box must contain two of them.) From that point on, the division 
will exactly duplicate the steps following that previously seen point, 
including getting that same remainder yet again, after the same number 
of steps. If you are getting the same sequence of remainders, you will 
be getting the same sequence of quotient digits, so that will repeat 
with the same period as the remainder sequence.

The above argument is independent of the size of a and b, so is valid 
for every integer choices for those numbers. If b is large, the period 
may be long, but it is finite, since its length is at most b-1. (Why 
can I use "b-1" here, instead of "b"?  Think about it.)

Here is an example:  16/37.

  37 ) 16.0000000000000
       14 8
        1 20
        1 11

Sequence of remainders:  16, 12, 9, 16, 12, 9, 16, 12, 9, 16, ...
                         ---------  ---------  ---------  --------

If two remainders are equal, then when you bring down the 0, the result 
will also be the same, and the quotient digit will be the same, and the 
new remainder will be the same, so all the steps afterwards will look 
like a repetition of earlier steps.  

Another example:  93/176.

   176 ) 93.000000000000...
         88 0
          5 00
          3 52
          1 480
          1 408

Sequence of remainders: 93, 50, 148, 72, 16, 160, 16, 160, 16, ...
                                         -------  -------  -------
There is a certain pattern to the decimal digits, but they do not form 
a periodic sequence of digits. All those decimals cannot be numbers of 
the form a/b, for any a or b, by the above argument. Thus they are 
irrational. Here is another kind:

 1/10^1 + 1/10^4 + 1/10^9 + 1/10^16 + 1/10^25 + ... + 1/10^(n^2) + ...
   = 0.1001000010000001000000001000000000010000000000001000...

The gaps between the 1's get longer and longer, so it is not a periodic 
repeating decimal. That means it must be irrational.

Keep thinking and asking good questions!

- Doctor Rob, The Math Forum
Check out our web site!   
Associated Topics:
High School Sequences, Series

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