I believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.

For every point on your mapimg I define two points X + 1 and x + 1i. You can assign infinity to one of them but not both.

I'm afraid you have some basic confusion with mathematical concepts, including "mapping", "irrational", "complex", and "infinity".

So, after you define x+1 and x+i, now what? Also please bear in mind that "infinity" is neither real nor complex: if you have a well-defined mapping into complex numbers, then by definition, it never maps to infinity.

(Yes, there are some "functions" like y = 1/x that "maps to infinity", but it's simply mathematicians being lazy and abusing notations because everybody around them understands what's going on.)

O, I get it.

There is a basic contradiction in set theory. Called Russell's paradox, there are two ways around it. First is ignoring it, aka everything builds from it's self nothing can become recursive. Or Zer's something or other that basically removed membership and equality and hides in the corner crying.

As such infinity is generally assumed not to exit in R. And causes all this all infinite set's map able to each other are equivalent size crap. It's also why real mathematicians laugh at the set guys.

But, sorry the way R was initially defined it included infinity and you only get to put it into 1 place on your mapping. Or as a math professor said, what angle is the highest number in R mapping to.

Where did you get all this idea? You know enough terms, yet somehow you have an incorrect understanding of pretty much everything.

If you are interested, please read an actual math textbook. (Yes, they can be a giant time sink, but at least you'll learn the correct meanings of sets and functions.)

I have taken plenty of high level math classes for fun and easy A's, I even had a department head yell at me for not getting a PHD. So, I can speak the lingo.

But, the absurd results are not generalizable outside of their assumptions sorry Axioms. Set theory being one of the most obvious cases.

It's sadly like a religion in many ways, follow enough false statements and you can prove anything. Yet, if you find a contradiction then don't actually accept at least one of your assumptions are false.

e is irrational, e+ei can't be irrational because it isn't real (all irrationals are reals).

Sorry, imaginary number composed of two irrational numbers.