Does $\pi$ have countably or uncountably many decimal digits?

Question

I think I know the answer - countably many, and intuitively it does make sense i.e. it wouldn't make sense that a number has uncountably many decimal digits (is that even possible). However, I've been trying to prove it in set theory and couldn't do it so please help. Thank you!

Answer 1

You can count the decimal digits of any real number. Notice that any real number can be written in the form $x = \sum_{k=-\infty}^{+\infty} a_k \cdot10^k$ with $a_k \in \{0,...,9\}$.

The set $(a_k)$ refering to the digits of $x$ is countable and therefore the decimal digits of $x$ are also countable.