Is $0$ an imaginary number?

Question

My question is due to an edit to the Wikipedia article: Imaginary number.

The funny thing is, I couldn't find (in three of my old textbooks) a clear definition of an "imaginary number". (Though they were pretty good at defining "imaginary component", etc.)

I understand that the number zero lies on both the real and imaginary axes.
But is $\it 0$ both a real number and an imaginary number?

We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. But I've always previously considered, that a purely imaginary number had to have a square that is a real and negative number (not just non-positive).

Clearly we can (re)define a real number as a complex number with an imaginary component that is zero (meaning that $0$ is a real number), but if one were to define an imaginary number as a complex number with real component zero, then that would also include $0$ among the pure imaginaries.

What is the complete and formal definition of an "imaginary number" (outside of the Wikipedia reference or anything derived from it)?

Answer 1

The Wikipedia article cites a textbook that manages to confuse the issue further:

Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. $R(z) = 0$.

Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$.

This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!

Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?

I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. But $0$ clearly has this property, so we should consider it purely imaginary.

(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time.)

Answer 2

I don't think there is a

complete and formal definition of "imaginary number"

It's a useful term sometimes. It's an author's responsibility to make clear what he or she means in any particular context where precision matters. If $0$ should count, or not, then the text must say so.

Your question shows clearly that you understand the structure of the complex numbers, so you should be able to make sense of any passage you encounter.

Answer 3

The meaning of the term pure imaginary changed over the years and became absurd. Currently, $0$ is both real and pure imaginary, but originally only the numbers of the form $bi$ with a real $b\ne 0$ were pure imaginary.

Moreover, currently, the imaginary part of $a + bi$, with $a$ and $b$ real, is $b$, which is absurd, since $b$ is real. Originally, the imaginary part of $a + bi$ was $bi$.

See Eduard Study and Élie Cartan. Nombres complexes. French. In : Arithmétique et algèbre. T. 1 : Arithmétique. Sous la dir. de Jules Molk. Encyclopédie des sciences mathématiques pures et appliquées Tome I. Paris : Gauthier-Villars, avr. 1908, p. 329-468.

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I've duplicated my answer to Is $0$ pure imaginary, real, or both?