Cardinality of number of digits

Question

What is the cardinality of the number of digits (in decimal form) of an irrational number like $ \pi $?

Answer 1

Digits in a decimal (or any other positional) expansion of any real number are arranged in a sequence: each (except the first one) follows some preceding digit. So there is the same number of digits as natural numbers.

Same applies to $\pi$: the cardinality of digits in the decimal expansion of $\pi$ is $|\mathbb N|$, commonly denoted with $\aleph_0$.

Answer 2

Decimal representation means assigning to each real number $x$ the one and only function $f_x:\{\diamond\}\cup\Bbb Z\to\Bbb \{0,\cdots,9\}$ such that:

1. $(-1)^{f_x(\diamond)}\sum_{k\in\Bbb Z} 10^kf_x(k)=x$

2. there is some $m\in\Bbb Z$ such that $f_x(i)\ne 9$ for all $i\le m$

3. $f_x(\diamond)=1$ or $f_x(\diamond)=0$

This establishes a bijection between $\Bbb R$ and the set of functions $f:\{\diamond\}\cup\Bbb Z\to\{0,\cdots,9\}$ such that $(2) \land (3)$ holds and there is some $n\in\Bbb Z$ such that $f(i)=0$ for all $i\ge n$.

If $x$ is either an irrational number or a rational number such that $2^h5^kx\notin\Bbb Z$ for any $h,k\in\Bbb N$, then $\lvert\{n\in\Bbb Z\,:\, f_x(n)\ne 0\}\rvert=\aleph_0$.