# Is Pi a rational number in any other base besides base Pi?

I'm wondering if Pi is a rational number in any other base besides base Pi. Also is there a formula or function to figure this out? I'm not what the relevance would be if we could find one since the integers might be irrational if we did but I am just curious if indeed Pi is only a rational in base Pi or if it is (or possibly is but not found yet) also rational with some other base. Although I suppose in base Tao Pi is a rational number, so maybe a better question is would Pi be rational number in any other RATIONAL base.

dextercioby
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There's no <base pi> afaik, because there's no number you can write in that 'base' (how could you divide 7 by pi ?).

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

Base pi does exist. You can even have a base of imaginary numbers.

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.

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

phinds
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Interesting. Thanks.

I was thinking about how some binary fractions appear as irrational that would be rational in base 10. Then I wondered if there was any case of the opposite where an irrational number could appear rational in another base, specifically Pi.

I was thinking about how some binary fractions appear as irrational that would be rational in base 10. Then I wondered if there was any case of the opposite where an irrational number could appear rational in another base, specifically Pi.
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}##.

Last edited:
Mark44
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I was thinking about how some binary fractions appear as irrational that would be rational in base 10.
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.

With regard to binary fractions, the rational number 1/2 can be represented as .510 or .12. For this number, both representations terminate. For the rational number 1/10, the decimal representation is 0.110, but the binary representation repeats a pattern endlessly. That doesn't make the number irrational, though. Off the top of my head I don't know what the pattern is.
Then I wondered if there was any case of the opposite where an irrational number could appear rational in another base, specifically Pi.

Yeah I see what you mean. Base Pi though is using a symbol to represent an irrational number it isn't really a rational base is it? I mean sure in base Pi, Pi == 1. I was wondering if there was a number, finite number in like base 1024 or something. (Bad example because probably not).

Yeah I see what you mean. Base Pi though is using a symbol to represent an irrational number it isn't really a rational base is it? I mean sure in base Pi, Pi == 1. I was wondering if there was a number, finite number in like base 1024 or something. (Bad example because probably not).
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.

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?

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

$$\pi = d_n \beta^n + ... + d_1\beta + d_0$$

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.

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

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.

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.

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

phinds
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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.

symbolipoint
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