They look similar because (per [1]) they both come from Latin ratio ("reckoning, accounting"), which derives from reri (think, judge).
[1] http://www.dictionary.com/browse/ratio?s=t
[2] They're the numbers expressible as one integer divided by another.
Anyway, all the people who said stuff like "When are we going to use X math topic in real life" were even more flummoxed by random math historical facts that weren't going to be on the test.
Anyway, at least for me, I think I passed a point of maturity where I don't feel "bombarded" when I encounter additional information in regards to a math topic. I did poorly in math in high school, and, at the time, I felt that the less information, the better-- I could only integrate so much. But now at my age (almost 40) learning additional historical facts seems to help me understand the topic better.
Maybe as "unreal as anything else" would be a more accurate way to put it. Mathematical realism and even any kind of Platonism have a lot of detractors. I think the attitude change was more due to what is now known as Logicism (https://en.wikipedia.org/wiki/Logicism) and Formalism ( https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe... ) when people realized that you did not need to appeal to the physical world to come up with useful things.
I would be tempted to say "pictured", "in a picture".
https://en.wiktionary.org/wiki/imago#Latin
But imago also has a lot of other meanings, and so "ghostly" and "conceptual" are also possibilities. :-)
> early 16th cent. (as an adjective meaning ‘entire, whole’): from Latin, ‘intact, whole,’ from in- (expressing negation) + the root of tangere ‘to touch.’ Compare with entire, also with integral,integrate, and integrity.
Indeed, whole numbers is another word for integers, dated in English, but that's what they are called in some other languages like German and Russian.
I also suspect natural is like in natural philosophy: occurring in nature, not like the other meaning of innate, although the two meanings are closely related.
Imaginary is due to Descartes (https://en.wikipedia.org/wiki/Complex_number#History):
> [...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
It says Descartes didn't like them, but imaginary literally means not occurring in nature, and this was presumably before people accepted mathematical concepts as being as real as anything else.
Transcendental also literally means beyond the algebraic numbers, since that's the definition, that's what they transcend.
Surreal numbers are surreal because somebody (Conway?) needed to attach a prefix to real, and my dictionary says sur- is the same as super-. My dictionary says it is surreal that is the odd word here: a backformation for surrealism, which is some kind of art movement (I wouldn't trust some French painters to get English etymology right, eh? ;)).
Also, what about dyadic, and p-adic, numbers? Ordinal numbers? Hyperreal?