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Will the terms complex and imaginary ever be replaced? At least within beginning classes?

I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse students (the implication is that if terminology is too much of a problem for a student, then the student is just not "good" at math.) But would it not be better to use such terms as "irregular numbers" instead of "imaginary numbers", and "complete field" instead of "complex field"? The original terms are only convention after all (albeit a very established one.)

I remember when first learning about $\sqrt{-1}$ and musing about what it really is. Now that I am more experienced I know it is just a concept within a consistent system of rules. Why label it as complex along with the connotations of the term, or as imaginary insinuating the non-existence of it. Of course it exists--even if only as a concept or a "completion". It exists just as much as any "real" number exists. It just has a different place in the field, the field that includes $\sqrt{-1}$.

My suggested replacements may step on other established terms that I do not use regularly. Does anyone have any other suggestions? At least for the beginning students?

Maybe the question can be answered by voting. My suggestion can still use $\mathbb{C}$ for the field, and $i=\sqrt{-1}$, which are some of the reasons I like it. Can we have other suggestions?

(I originally thought this would be a community wiki, where is that option?)

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A "complete field"? So that when they learn about completeness, they will think only $\mathbb{C}$ is complete? – Nameless Dec 28 '12 at 19:32
@Nameless Which is why I mentioned maybe terminology gets stepped on. But maybe that will be a good way to start considering what completeness means? – adam W Dec 28 '12 at 19:36
I don't agree. If the name "complex" were to be replaced by something, I would prefer that to be "elementary algebraically closed field" or something along these lines – Nameless Dec 28 '12 at 19:38
I'd prefer Peter, which also - to my knowledge - doesn't overlap with any other mathematical concept. – gnometorule Dec 28 '12 at 19:55
@Nameless: You've focused so much on avoiding connotation that you've forgotten the purpose of communication is to communicate. Nobody would ever say "every elementary algebraically closed field polynomial of degree $n$ has exactly $n$ elementary algebraically closed field roots, counting multiplicity"; it's just too much work. – Hurkyl Dec 28 '12 at 20:30
up vote 3 down vote accepted

Gauss tried and failed to get people to say "lateral part" instead of "imaginary part". I think we're lucky that use of "complex number" instead of "imaginary number" is as widespread as it is.

Changing terminology is a major struggle; not only is there's a lot of inertia to continue establish terminology, but there will be active opposition from people who opine that imaginary numbers are imaginary whereas real numbers are real.

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I entirely agree about the inertia, though still wonder if it may be done just for beginning students. That way the concepts have time to sink in without such contrary connotations. I'm thinking the terms could be just elementary school terms, but still carry some weight. – adam W Dec 28 '12 at 19:49
I was hoping to see some more naming suggestions just for the fun of the possibilities, rather than a discussion about if it would ever happen. But I suppose that that is my fault for the wording of my question. I like your Gauss example though, so accept this answer. – adam W Jan 5 '13 at 6:07

It's a good question. The connotations of the existing terminology don't reflect modern understanding, and this can be a barrier for students. (Another confusing case is the terms "open" and "closed" in topology... a set could be neither open nor closed, and a set could be both open and closed!)

The big problems are: (1) the teachers are already used to the old terminology; (2) if you teach students new terminology, they will have trouble reading older literature (they will need to know the mapping between old and new terminology). Of course, once everyone gets used to it, there won't be a problem, but the transition is difficult. This would have to be a community-wide change, and if the benefit is mostly for students, it's hard to build up momentum for this because students don't have that big a voice.

In a way, it's like changing the alphabet for a language, or changing to a different calendar system. Existing people are already used to the old system, the benefits are in the future, and the transition would be painful. Changes like these are usually imposed by an authority such as a government. But there is no authority that can impose math terminology.

So, unfortunately, I don't see this happening any time soon.

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Good observation. And I agree that having multiple terms is generally not desireable. I guess I was hoping that the mere effort would communicate to students that the concept is just a kind of completion. – adam W Dec 28 '12 at 19:53

My suggestion (after short considerations inspired by @Nameless):

$\mathbb{C}$ the Compound field.

$\sqrt{-1}$ the irregular unit.

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I would say the symbol $i$, rather than the terms "imaginary" or "complex", is the real nuisance. The two terms either have a clear historical background ("imaginary") or explain themselves very well ("complex" means an aggregate of different parts; a complex number means a number consisting of a real part and an imaginary part). Also, they are not as overloaded as other mathematical terms such as "normal" or "regular". So, I don't see any needs to replace them.

The symbol $i$ (or $j$ in some engineering literature), however, deprives us of the symbol that is commonly used as a running integer variable. I would be glad if someone manages to replace it by something that can get widely accepted.

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The historical background is the problem. Long ago, people thought real numbers were more "real" than imaginary numbers. The modern problem the OP seeks to solve is that the names predispose people to repeat history. – Hurkyl Dec 29 '12 at 4:13

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