What are the digits of a complex number?

Question

The digits of a natural number like $5782$ can be written as a sequence of natural numbers $$(5,7,8,2)$$ Similarly, the digits of a real number can be written in the same way, e.g. the digits of $\pi$ are $$(3,1,4,1,5,9,2,...)$$ Is there an analogous notion for complex numbers? Writing them in a single sequence of natural numbers doesn't really make much sense to me. I could imagine either a vector of sequences, like the number $\pi + i \mathbb{e}$ being written as $$\begin{pmatrix}(3,1,4,1,5,9,2,...)\\(2,7,1,8,2,8,1,...)\end{pmatrix}$$ or as a sequence of vectors $$ \left( \begin{pmatrix}3\\2\end{pmatrix}, \begin{pmatrix}1\\7\end{pmatrix}, \begin{pmatrix}4\\1\end{pmatrix}, \begin{pmatrix}1\\8\end{pmatrix}, \begin{pmatrix}5\\2\end{pmatrix}, \begin{pmatrix}9\\8\end{pmatrix}, \begin{pmatrix}2\\1\end{pmatrix},...\right) $$ or maybe as a sequence of complex numbers with real/imaginary parts in $\{0,1,...,8,9\}$

$$\left( 3+2i, 1+7i, 4+1i, 1+8i, 5+2i, 9+8i, 2+1i,...\right)$$

The first version seems very wrong, because (to me) the digits of a number can be enumerated one after the other and in that case, there is only one single thing: a vector of two sequences.

The latter versions seem similar to each other in the same way that $\mathbb{R}^2$ and $\mathbb{C}$ are similar.

All of my ideas seem pretty clunky and I wasn't able to find anything online. Is there even such a thing as the digits of a complex number, that people generally agree on? Does this question even make sense?

Answer 1

Some different point of views:

$\bullet$ Suppose they are comparable. Complex numbers form a super set of real numbers i.e., a real number $a$ is also a complex number and can be written in the general form of $a+0.i$. With this assumption, we proceed to compare $i$ and $0=0+0.i$. Cleaely, $i\ne0$. Hence, $i>0$ or $i<0$.
Case 1. $i>0\Rightarrow i.i>0.i\Rightarrow-1>(0\times0)+(0\times1)i=0~\rightarrow\leftarrow$
Case 2. $i<0\Rightarrow i.i>0.i\Rightarrow-1>(0\times0)+(0\times1)i=0~\rightarrow\leftarrow$

$\bullet$ The set of complex numbers are unlike, the set of real numbers or its subsets, "unordered". So, the concepts of comparibility do not apply at all i.e., it is not possible to enumerate them.

$\bullet$ Geometrically, the complex numbers are viewed on a plane, NOT a straight line which is why, your second way of writing them is correct: a set of 2 dimensional vectors. All the numbers lying on either axes are real numbers, a subset of complex numbers. By similar, if you mean you can write a bijective function $f:\mathbb{R}^2\rightarrow\mathbb{C}$ then, yes. This is correct.

Now, if you want to talk about digits, it can only be done separately for the real parts and the imaginary parts. So, your first visualization is OK in such a case. Although usually, the preferred representation for a single number is using one vector to represent it. In case of a real number it is 1 dimensional; for an imaginary number it is a 2-dimensional representation.