$\pi$ 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

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 $\pi$.

Answer 2

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 3

In base two, there must be an infinite number of 0's and 1's. If there is a finite number of either, then pi would be rational, which it is not.

And since they are infinite, the number of 0's and 1's is equal. You can prove this with a one-to-one correspendence: match the first occurrance of 0 with the first occurrance of 1, the second 0 with the second 1, etc.

By the same token, in any base there must be at least two numerals that occur the same number of times, that is infinitely, or else again pi would be rational.