Transcendental Numbers

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Obviously I was being a bit tongue-in-cheek about "complex".

Your answer to my question doesn't answer my question.

What I am asking for is: examples of numbers that are...NOT complex numbers.

A "precise mathematical definition" you gave seems to includes all numbers. So in effect it IS a "synonym for 'number'".

If reals, imaginaries, and numbers at the intersection of real and imaginaries on the Cartesian plain are all complex numbers, then what other kinda numbers are there remaining to be outside of the category? Aint that all the kinda numbers that there are?

I misunderstood. Yes, if you can define a set of numbers outside of the complex number set, then they will not be complex numbers. There is an extension to the complex numbers that include the infinities and infinitesimals which are not complex numbers. That set is called the surreal numbers. Additionally, you can define many other kinds of algebraic objects that can be thought of as numbers but are not complex numbers, for example quaternions,

https://en.wikipedia.org/wiki/Quaternion

or the anti commuting Grassmann numbers,

https://en.wikipedia.org/wiki/Grassmann_number.

So complex numbers have a real and an imaginary part. But are there different "types" of imaginary numbers? I'm not quite sure about the actual definition, but a quarternion has three imaginary numbers. Are these of different types of imaginary numbers? I'm not sure if I'm making sense here. But if all I've said up until now is correct, is there then an infinite amount of types of imaginary numbers? Or is there some limit?

Thanks.

Wow. Apparently it gets quite mind bending!

But yes-the label IS confusing to the layman.

It would seem to the layman that regular old counting numbers (1,2,3,...) would be "simple numbers", and that "complex numbers" would be-more complex entities (like the square roots of negative numbers etc). But counting numbers are lumped with more complex entities as "complex numbers". And apparently (from titles of the topics in those links you gave) the entities outside the realm of the "complex" numbers are things even MORE complex, and esoteric, and (some are even officially labeled as) "surreal". Lol!

I just wanted to say something that is only slightly related but: the cardinality of the set of trancendentals has not been proven to be aleph-one. aleph one is the cardinality that is "next" after aleph null. The cardinality of the trancendentals (and reals) is actually 2^(aleph null)- or "beth one". The continuum hypothesis is the unproven (and if I remember correctly: unprovable) assertion that there is no set with a cardinality between that of the naturals and of the reals (that beth one equals aleph one). The continuum hypothesis is a pretty interesting thing to think about. You can continue taking elements out of the set of reals (or some uncountable subset thereof) and keep getting uncountable sets of the same cardinality- or you can remove whole uncountable sets and possible end up with a countable set. But what you cannot do is take away a certain number of points and end up with a set that is intermediate in cardinality. It may seem "obvious" that the set of the reals is the "next largest" size after the naturals (that there is no cardinality in between them) but this has never been proven.

Cantor believed that the Continuum hypothesis was true but could never prove it. Godel believed that it was false - but proved that it is actually independent of ZFC set theory (neither the CF hypothesis nor its negation can be proven).