$\pi$ when not in base 10

Question

Very novice amateur mathematician here. My daughter (8 yo) is a math junkie and is trying to wrap her head around irrational numbers. We were talking about $\pi$, and I rambled on about how folks have put a lot of energy into researching the 'following digits' of $\pi$ and their properties.

Then it occurred to me that we usually discuss $\pi$ in Base 10, simply because humans have 10 fingers and toes, etc. Would any properties ascribed to qualities of digits of $\pi$ vanish in other bases / (base 2, 12, etc.)

Please forgive my naivety...

Answer 1

You might try learning continued fractions with your daughter. https://en.wikipedia.org/wiki/Continued_fraction and PIIIIIIII

One aspect is immediate: a rational number has a finite (simple) continued fraction, an irrational number has an infinite one. Meanwhile, as far as history, the approximation of $\pi$ by Archimedes is a continued fraction convergent. Let me look that up...Hmmm. The things I found say Archimedes gave upper and lower bounds. Anyway, just before the 292, the convergent $\frac{355}{113}$ is a very good approximation, relative to the size of the numerator and denominator.

Simple continued fraction tableau:
$$ \begin{array}{cccccccccccccccccccccc} & & 3 & & 7 & & 15 & & 1 & & 292 & & 1 & & 1 & & 1 & & 2 & \\ \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 3 }{ 1 } & & \frac{ 22 }{ 7 } & & \frac{ 333 }{ 106 } & & \frac{ 355 }{ 113 } & & \frac{ 103993 }{ 33102 } & & \frac{ 104348 }{ 33215 } & & \frac{ 208341 }{ 66317 } & & \frac{ 312689 }{ 99532 } \end{array} $$

https://oeis.org/A002485

https://oeis.org/A002486

This seems a good idea to me as many of the students on this site cannot work out what to do with them; for a variety of reasons, continued fractions are no longer in the curriculum at any level, but then show up in number theory classes at college level. The result is a large dose of jargon with subscripts all at once.

Answer 2

Excellent question! There's two things going on here - properties ascribed to $\pi$ itself, and properties of the digits of $\pi$. In particular, properties of the number itself can't change when the base changes, (the ratio of circumference to diameter of a circle doesn't change if you lose a finger), so $\pi$ itself stays as it is. Properties like irrationality remain, since we know $\pi$ can't be written as a fraction of whole numbers, regardless of the base we're in. On the other hand, properties of the digits of $\pi$ can change! In other bases, $\pi$ wouldn't start off like the familiar $3.14\dots$, and interesting coincidences like the Feynman point won't exist any more.

On a deeper level, we don't know if the digits of $\pi$ appear "uniformly" in base 10, nor in any other base, related to the idea of a normal number. Talking about other bases specifically, the BBP algorithm gives a convenient way of computing the digits of $\pi$ in base 16, and as a spigot algorithm it doesn't rely on previous digits to find the next one, unlike most familiar algorithms for calculating $\pi$.