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We classify the real numbers into subsets, so I thought is it fruitful or even used to classify the imaginary or complex numbers as well?

I mean for example something like this:

$$\pi+i\pi \qquad(\text{real-real})$$ $$3+i\frac{1}{245} \qquad(\text{natural-rational})$$

Or is this a useless thoughts since there are some fancy complex properties who make it a useless decision?

MarcE
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  • Sure, you can do it, but what's the point? – Arkady Jun 25 '17 at 02:48
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    I'm sure some aspects of what you're asking about have already been described. For instance: https://en.m.wikipedia.org/wiki/Gaussian_integer – Deepak Jun 25 '17 at 02:50
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    If you're gonna classify $\pi+i\pi$ as real-real, I have doubts about how interesting anything you prove will be. – Rocket Man Jun 25 '17 at 02:50
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    Is it possible to have (odd + i odd)^2 = (even + i even) ? in naturals it is easy to see that odd^2 is always odd, maybe this classification can be used to classify results like that? – jimjim Jun 25 '17 at 03:48
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    @Arjang Not only is it possible, but it is so in every case. – Deepak Jun 25 '17 at 04:51

1 Answers1

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We classify the real numbers because it is natural to do so from an algebraic and analytic standpoint. Algebraically, the integers form a ring, the rational numbers form the field of fractions of the whole numbers, and from an analytic standpoint, the irrational numbers are what you thrown in to get a complete field. This is done to some point with complex numbers; for instance, the Gaussian integers are those of the form $a+bi$, where a and b are integers, and the algebraic numbers are those which satisfy a polynomial with $\mathbb{Z}$ coefficients. However to classify them more than that does not really help out so much. But yes, you could do it!

Edit: I am linking to this answer https://math.stackexchange.com/a/199688/223701 by Clive Newstead because it deals with this categorification in a much better way than I could do so.

TomGrubb
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