$\pi$ when not in base 10 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...
 A: 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 $\pi$.
A: 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. 
