Are all real and imaginary numbers complex numbers? 
This image makes it seem like real and imaginary numbers are all complex numbers. I thought complex numbers were numbers composed of a real and imaginary part. This got me thinking, and I think I might see it now. All real numbers can be written as $$r \times i^{4k}$$ and all imaginary numbers can be written as $$i^k \times 1$$ or $$i^k + 0$$ There are other ways to combine them in ways that does nothing to the value. So, is it these combinations that make all real and imaginary numbers into complex numbers?
 A: Perhaps we could modify the Venn diagram to let $\mathbb I$ intersect with $\mathbb Z$  (and consequently $\mathbb R$). (Which integer can be considered imaginary? Which imaginary number can be considered an integer?)
rschwieb: It would make me happy to just eliminate the blob with  altogether. $\quad$ Yes that’s true, actually; the set of prime numbers is arguably more deserving of being singled out in this Venn diagram.
And yes, I would certainly classify all real and imaginary numbers as complex; the complex plane includes both the real and imaginary axes, after all.
P.S. I’d never encountered the symbol $\mathbb I$ (which here denotes the set of imaginary numbers) prior to this, but apparently, in the days of yore, it sometimes denoted the set of integers...
A: The actual fact of the matter is that different definitions can lead to different answers on whether the real numbers are complex numbers. (In case you’ve never worried about definitions before, then…What, exactly, is a real number, anyway? A complex number?)
What is most important is that, no matter what your answer to that question is, one can always see the real numbers inside the complex numbers by associating the real numbers $a$ to the complex number $a+0i.$ The possibility of such an association, which is the same as looking at the “copy” of the usual number line given by the real axis in the complex plane, is the only issue at hand that has serious mathematician significance, not whether real numbers literally are complex numbers, or are something quite different.
