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Topic: Can a fraction have none noneending and nonerepeating decimal representation?
Replies: 108   Last Post: Aug 16, 2013 5:22 PM

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 grei Posts: 132 Registered: 11/27/12
Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted: Aug 3, 2013 12:59 PM

That is, in fact, the exact OPPOSITE of what he said. It seems that everytime someone says "No, you are wrong!" you response is "Ah, so you agree with me."

I haven't yet figured out if you have difficulty understanding English, difficulty understanding mathematics, or are just having fun contradicting everyone.

What you have been told, repeatedly, is that if a number is a repeating "decimal" in one (integer) base then it is a repeating "decimal" in any (integer) base.
A few people have added a caveat that IF we were to allow non-integer bases then we might have something different. For example, if we wrote pi in "base pi" it would be represented as "1.0". But that is NOT what we have been talking about. Your original question was about integer bases.

In fact, your original question was NOT about "bases" at all, but about whether pi was rational or irrational. The DEFINITIONS of "rational" and "irrational" have nothing to do with decimal representations.

You probably will not understand this but there is a very nice theorem that says "Let c be a positive real number. If there exist a function, f, such that f and all of its anti-derivatives can be taken to be integer valued at 0 and c (an "anti-derivative" always involves a choice of additive constant. We can always choose that constant to make the anti-derivative integer valued at 0 OR c. The point here is that we can choose it so the function is integer values at BOTH 0 and c.) then c is irrational".

As an application of that, we note that pi is a positive real number. f(x)= sin(x) has anti-derivatives that can be taken to be either cos(x)+ C (odd anti-derivatives) or sin(x)+ C (even anti-derivatives). Choosing C equal to 0 makes all anti-derivatives -1, 0, or 1 at 0 and at pi, all of which are integer values. Therefore pi is irrational.

No matter how much of that you may have understood, my point is that there is absolutely no mention of pi being expressed as a "decimal" expansion or an any base numeration system. The only property used is the basic definition of "rational" and "irrational" numbers: that a rational number can be expressed as a "fraction", one integer over another, and an irrational number can't.

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7/30/13 JT
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