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- Thread starter p1l0t
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- #1

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- #2

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- #3

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Base Pi is already used for a number of applications but that isn't what I am looking for anyway.

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But the point is that the property of being rational is independent of the base. A number is rational if it can be written as the quotient of two integers. All of these concepts can be developed without any reference to what base you work in. And as such, rational is indepent of the base. Thus, ##\pi## is irrational in any base you work in. Regardless whether you can write it as ##10## in base ##\pi##.

So your question should not be written as "when is ##\pi## rational", but rather a bit different.

- #5

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Base Pi is already used for a number of applications but that isn't what I am looking for anyway.

Name one, please. I could not find any using Google.

- #6

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Name one, please. I could not find any using Google.

http://en.wikipedia.org/wiki/Non-integer_representation

Quote:

There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998).

- #7

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Interesting. Thanks.

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- #9

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OK, I got you, but I'm saying that you shouldn't be using the word "rational" here. It is not what you mean anyway.

What you mean is are there numbers ##\beta## such that ##\pi## has a finite or eventually repeating expansion in the base ##\beta##.

The answer is yes. As you mentioned, ##\pi## in base ##\pi## is just ##10##. Another base is ##\sqrt{\pi}##. In that base, we have ##\pi## written as ##100##. More generally, take ##\sqrt[n]{\pi}##.

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- #10

Mark44

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I think you might be confused. The representation of a number in whatever base doesn't have anything to do with it being rational or irrational.I was thinking about how some binary fractions appear as irrational that would be rational in base 10.

With regard to binary fractions, the rational number 1/2 can be represented as .5

Then I wondered if there was any case of the opposite where an irrational number could appear rational in another base, specifically Pi.

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In base ##\pi##, we have that ##\pi## is ##10##, not ##1##.

So, you're asking whether ##\pi## has a finite or eventually repeating ##\beta##-expansion in base ##\beta##, where ##\beta## is a natural number larger or equal than ##2##?

The answer is no. In integer bases, ##\pi## will always have a nonrepeating expansion. You'll have to look at irrational bases.

- #13

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In base ##\pi##, we have that ##\pi## is ##10##, not ##1##.

So, you're asking whether ##\pi## has a finite or eventually repeating ##\beta##-expansion in base ##\beta##, where ##\beta## is a natural number larger or equal than ##2##?

The answer is no. In integer bases, ##\pi## will always have a nonrepeating expansion. You'll have to look at irrational bases.

Yes, 10 not 1, my mistake. So I guess that makes my follow-up question, "why?" Is this a known property of irrational numbers or something?

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Yes, 10 not 1, my mistake. So I guess that makes my follow-up question, "why?" Is this a known property of irrational numbers or something?

You can easily prove it. For example, let's say that

[tex]\pi = d_n \beta^n + ... + d_1\beta + d_0[/tex]

So we have a finite expansion. Then if ##\beta## is rational, then the sum is rational. Thus ##\pi## is rational. This is obviously false. So in order for the equality to be true, we must have that ##\beta## is irrational (of course, not all irrational bases will work).

A slightly more complicated proof works in the case that the expansion is eventually repeating, but I think you got the idea.

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So is an irrational number always irrational no matter what the base or is Pi just a special case?

- #16

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So is an irrational number always irrational no matter what the base or is Pi just a special case?

The term "irrational" is independent of what base we use. So yes, an irrational number is always irrational.

- #17

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The term "irrational" is independent of what base we use. So yes, an irrational number is always irrational.

That was the flaw in my thinking then. I was thinking about .1 in binary being an irrational number but it isn't. It is just a repeating number. Infinite maybe, but not irrational.

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And to think all this time we never talked about it being a transcendental number on top of it.

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That was the flaw in my thinking then. I was thinking about .1 in binary being an irrational number but it isn't. It is just a repeating number. Infinite maybe, but not irrational.

Correct. In decimal we can represent .1 as 1/10. In binary we can represent that quantity as 1/ 1010, so clearly in neither case is it irrational.

- #20

symbolipoint

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