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I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that question.

This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful. It is a question about what makes them numbers. It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.

Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number.

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Too simple for a proper answer: because they do most of the things we want numbers to do. Particularly, because they form a field: we can add, multiply, and divide them. You could add in the topological properties (specifically closure and the archimedean property) and start to get really close to the core of what makes them IMHO numbers for some definition thereof. – Steven Stadnicki Dec 30 '13 at 23:09
The rational functions form a field, but nobody would call them numbers? Or would they? I don't think I would. – Harald Hanche-Olsen Dec 30 '13 at 23:10
You may be interested in this question – mrf Dec 30 '13 at 23:11
@Harald: Rational numbers are not numbers? Don't be so irrational. And remember to keep it real! gets stricken down by a lightning for telling the worst joke in human history – Asaf Karagila Dec 30 '13 at 23:13
@Harald Functions can be numbers! Note that the ring of polynomial "functions" with integer coefficients $\,\Bbb Z[x]\,$ is isomorphic to $\,\Bbb Z[\pi] =$ polynomials in $\pi$ with integer coefficients. Passing to fraction fields yields that the rational function field $\,\Bbb Q(x)$ is isomorphic to a subfield of real numbers, viz. $\Bbb Q(\pi)$. Because of things like this, it is nontrivial to give a precise algebraic definition of "number". – Bill Dubuque Dec 31 '13 at 4:13

You are right, the question is foremost "what is a number", then you will see that complex numbers are numbers. Mostly because the word "number" doesn't have a clear meaning in mathematics, or even in the natural language of people.

Numbers are notions representing quantity. So the question is, what sort of quantity do complex numbers measure? They have more than a handful of applications where they are used to measure quantities, which seems intrinsic enough to begin with.

I suppose that there is probably a historical-sociological reason as well, but if you are looking for an "intrinsic property" then that would be the best definition of "number" I can come up with, and indeed the complex numbers satisfy it.

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I'm going to have to look more deeply about how complex numbers are used for measuring quantities. That may be a way I can explain it to students just learning about complex numbers, but I've got to find applications they can understand. – David Joyce Dec 31 '13 at 0:48
One standard example of using complex numbers to do real-world calculations is understanding analog circuit behavior. – keshlam Dec 31 '13 at 1:54

My best understanding is that it is just a historical accident that complex numbers ended up being called "numbers".

Historically, each extension of the "number" concept -- from naturals to rationals to reals, and to negative versions of either -- has been met with resistance that needed to be overcome. (It even used to be contentious whether or not $1$ is a "number", because the relevant passages of Euclid can be read as saying that only a multitude of units is a number).

And the driving force that overcame the resistance towards new kinds of numbers doesn't seem to have been that they meet any pre-existing standard for what a "number" ought to be, but simply that there was a perception that they needed to be called "numbers" before one was allowed to do mathematics with them.

Some time after complex numbers won acceptance, it became commonly accepted for new branches of mathematics to consider things that were neither numbers nor geometric objects. Once that happened, there was no pressure to insist that new things you want to reason about are necessarily "numbers".

This might well have happened at some other time in history, and we might have ended up considering, for example, vectors and matrices to be special kinds of "number".

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Interesting point that they needed to be called numbers so that they would be accepted. It was a long process, so that their acceptance went along with them being considered numbers. The name reinforced their acceptance, and their acceptance reinforced the name. – David Joyce Dec 31 '13 at 0:46

Numbers count or measure, but complex numbers don't really seem to do either, so why are they called numbers ? Is this what you're asking ? For starters, they arose in a numerical context, as roots to polynomial equations. Obviously, since all the other roots are numbers, why wouldn't they also be as such ? Furthermore, all operations between two real numbers can also be done between a real number and a complex one, or between two complex ones. Now let us revisit the first sentence: Do complex numbers really not measure anything ? Perhaps not age or distance, but aren't there so many other physical concepts to which their existence fits like a glove ? Like mechanical energy, which is in a constant state of flux, oscillating between two polar opposites, kinetic and potential. Or wave forms, like sound, or alternating electrical current, etc. So if they do measure after all, and if the same laws and operations apply to them as they do to the other numbers, or, in other words, if they share so many characteristics in common, why shouldn't they be called by the same name ?

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I like your point, "they arose in a numerical context", that is, they're solutions to questions about numbers. Infinity also fits that condition, since it answers the question what times 0 equals 1. The idea of measurement is also important. – David Joyce Dec 31 '13 at 0:43

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