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Does the word "integer" only make sense in base 10?

I've always wondered this and have never seen it really discussed anywhere.

We all understand the typical definition of an irrational number, where a number can't be expressed as a ratio a/b where a and b are integers and b is nonzero.

So what if we were working in base pi? Then we could simply write pi as 10/1 or something.

However I don't think we can remove the irrationality of a number by simply changing bases, but I wanted to see if there were maybe other definitions that might help here.

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As you found, 10 base $\pi$ is not an integer. Definition "integer" does not mention base at all. Look it up. – GEdgar May 5 '12 at 0:07
This question might arise after learning that our familiar "base 10" is rather arbitrary: base 2 or 7 or 3976 are in principle equivalent. But base 1/7 or $\pi$... not so much. – leonbloy May 5 '12 at 0:17
@leonbloy: you can do base $\frac 17$ OK. It flips things around the "decimal" point, so your infinite strings go to the left instead of the right, but you can express all the numbers just fine. You have to work out the mechanics of arithmetic, but it is basically carrying to the right instead of left. You can also do base $-5$. One of my math teachers got me to explore this long ago. – Ross Millikan May 5 '12 at 0:38
@RossMillikan: Sure you can! But it's not "equivalent" to our 10 base system in the strong sense that bases 2 or 7 are. – leonbloy May 5 '12 at 0:44
This question is a clear syntax versus semantics confusion. – Kaz May 5 '12 at 4:25
up vote 10 down vote accepted

The naturals come from adding 1 a finite number of times to 0. The integers include these and their negatives. Neither of these statements refer to the number base needed to express them as numerals. Then the rationals again are ratios of integers without reference to base.

The equivalence of this definition of rational to terminating or repeating numeral expansions is dependent upon the base being a rational. If you want to write numbers in base $\pi$, it would be natural to write $\pi=10_\pi$, but then $4$ and $9$ which will have "irrational looking" expansions. This is an artifact of using an irrational base.

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That $\pi$ is irrational means that $\pi$ is not rational, i.e. $\pi \not\in \mathbb Q.\:$ This is a set-theoretical statement whose truth depends only on the sets of rationals and reals and the membership relation in the ambient set-theoretical universe. Its truth does not depend upon syntactic information, such as how the elements of the sets are represented or notated. It remains true in every such representation of reals: radix representation in some base, continued fractions, Cauchy sequences, Dedekind cuts, etc.

Perhaps what you are contemplating is a more general notion of (ir)rational, such as a notion of being rational relative to a ground domain, i.e. say r is Z-rational if r lies in the fraction field of Z. One might then go on to study analogous criteria for Z-irrationality, etc. This more general notion of irrationality generally does not coincide with the classical special case Z $= \mathbb Z = $ integers.

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While this does get at the general mistake, I'm not sure it's exactly helpful; the questioner is clearly confused as to just what "integer" means, and this fails to answer that. – Harry Altman May 5 '12 at 1:55
@Harry That was alread addressed in the prior answer, so I did not see the need to repeat it here. I don't think that deserves a downvote. – Bill Dubuque May 5 '12 at 2:00

As an aside: a nice example of a number system with an irrational base appears in:

A number system with an irrational base by George M. Bergman. Mathematics Magazine 31 (1957/58), pp. 98-110. JSTOR link

which describes how to do arithmetic base $\phi$ (the golden ratio). But one should not confuse the numbers with their representations, as Ross notes above.

I'd like to point out that the above paper is the peer-reviewed paper with youngest author that I am aware of: it was submitted when Bergman was 12, and the accompanying note by his mother is printed in page 91 of the journal:

Dear Mr. James:

"The paper presented is the work of my twelve year old boy who took more than a year to gather courage to submit it for editorial scrutiny.

You may be interested to know that when my son first received a subscription to Mathematics Magazine about two and a half years ago, he was aghast to note that he couldn't understand a single thing in it. With each successive issue, however, his understanding unfolded (he is a self-taught mathematician) until now he awaits each issue with eagerness, and recently was able to submit his solution to one of the Proposals published in your last issue. He considers his subscription to Mathematics Magazine one of the finest presents he ever received!"

Sincerely yours,

Sylvia Bergman

To quote Tom Lehrer: "It's people like this who make you realize how little you've accomplished..."

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Does the word "integer" only make sense in base 10?

Nope. A number's membership in the set $\mathbb{Z}$ (or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$) is a property of the abstract number and has nothing to do with the numeral representing it. The number we call “forty-two” is an integer whether it's written as 42 or 0x2A or 101010 or *36 or XLII or מב or +---0.

So what if we were working in base pi? Then we could simply write pi as 10/1 or something.

It works just like any other positional number system. Recall that in base $r$, the number denoted by $d_n \dots d_3 d_2 d_1 d_0 . d_{-1} d_{-2} d_{-3} \dots$ is defined as $d_nr^n + \dots + d_3r^3 + d_2r^2 + d_1r + d + d_{-1}r^{-1} + d_{-2} r^{-2} + d_{-3}r^{-3} + \dots$

In every base, the numeral “10” denotes the base itself. So π in base π is simply “10”.

But 10 in base π is $100.010221... = \pi^2 + \pi^{-2} + 2\pi^{-4} + 2\pi^{-5} + \pi^{-6} + \dots$, and in general integers will tend to have non-terminating representations in irrational bases, just like irrational numbers have non-terminating representations in integer bases.

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