I just watched the Numberphile video on the irrationality of the golden ratio and Ben talks about the continued fraction for the golden ratio $\phi$: $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+…}}}}$$
versus for pi $\pi$: $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+…}}}}$$
He makes the point that large denominators (such as $292$) mean that the next fraction is very small, and therefore the preceding fraction is a good rational approximation of the irrational number: $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+0}}}=\frac{355}{113}\approx3.14159$$
He says that the golden ratio is the "most rational" because it is constructed to have the worst rational approximations, as it has only ones in the denominators.
My question is whether the size of the fractions change when represented in different base systems. If represented in base 12, a $1$ is only a $12^{th}$ and not a $10^{th}$ of the preceding fraction. On the other hand, all of the other $1$'s are also in base 12, so maybe nothing changes.
I found an answer to a different question that says that a continued fraction is not "in" any base system at all, and that makes some sense, since numbers like $292$ are not restricted by the base they are in, but $1$ can't get any smaller, can it?
I'm a mathematical layman, and I don't understand a lot of what is said in the above answer, I guess I'm just looking for some hint of an intuitive understanding of how base systems tango with continued fractions.
Best Answer
The point here is to recognize that $292_{10}$ represents some integer. There are many other ways to write this integer, like $204_{12}$, but this is the most common one.
Basically, in a continued fraction, the entries are not digits, they are integers. So if you do the continued fraction of $\pi$ in base-12, the numbers in the denominators will look like $3, 7, 13, 1, 204,\ldots$ instead of $3, 7, 15, 1, 292, \ldots$. It's the same integers, you just write them in a different way. This is what they mean when they say that continued frations are not in any base: the integers we use are the same no matter what base we are in, they just look different.