Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?

UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.

Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.


closed as unclear what you're asking by Peter, Clayton, Henning Makholm, Daniel Fischer Aug 15 '18 at 14:42

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  • $\begingroup$ @Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly. $\endgroup$ – Ryder Rude Aug 15 '18 at 13:13
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    $\begingroup$ What do you say about $\pi = 4 \;\arctan(1)?$ $\endgroup$ – gammatester Aug 15 '18 at 13:21
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    $\begingroup$ @RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable. $\endgroup$ – Henning Makholm Aug 15 '18 at 13:24
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    $\begingroup$ Since $\pi$ is transcendental it cannot be represented by a finite expression $\sum_{k=0}^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number. $\endgroup$ – gammatester Aug 15 '18 at 13:37
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    $\begingroup$ @RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you. $\endgroup$ – Henning Makholm Aug 15 '18 at 15:21

$\pi$ can be represented by a finite formula in standard mathematical notation, such as: $$ \pi = 4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} $$ This contains all information there is to get about the value of $\pi$.

(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).

More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $\pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.

(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $\pi$ itself. The latter is because we know $\pi$ is not rational, which is not obvious but definitely known).


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