What is your favorite math constant that is NOT a real number?

I have a strong relationship to what you get when you divide by zero.

Good ol' NaN

You only get NaN for division by zero if you divide 0 by 0 in IEEE floating point. For X/0 with X ≠ 0, you get sign(X)•Inf.

And for real numbers, X/0 has to be left undefined (for all real X) or else the remaining field axioms would allow you to derive yourself into contradictions. (And this extends to complex numbers too.)
I like writing swear words into the mantissa of NaN numbers

Negative zero, comes up in comp sci.

ε, the base of the dual numbers.

It’s a nonzero hypercomplex number that squares to zero, enabling automatic differentiation.

Came here to say this, but since it's already here, I'll throw in a bonus mind-melting fact: ε itself has no square root in the dual numbers.
- As a not good at math person, advanced math sounds hella fake sometimes. Like almost a parody of math. I accept that it's real but part of me will always think it's some inside joke or a secret society cult for the lamest god.
Complex numbers 🤝 Split-complex numbers 🤝 Dual numbers

All super rad.

j = √-1

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Set phasors to "confuse"

∞

Which one?

So complex or quaternion I imagine? 'i' it is!

√-4 = 2

It's all fun and games until someone loses an i.

i=sqrt(-1) is nice, but im hoping someone finds a use for the number x where |x| = -1 or some nonsense like that because it looks fun to mess with

Those exist in the split-complex number system which adds the number j, where j^2=1 (but j does not equal 1 or -1). The modulus of j is -1.

I'm a big fan of 10-adics, especially this one.

j is cool too, as is (1+j)/2.

Matrix representations in general, if that counts?

Complex numbers, polynomials, the derivative operator, spinors etc. they're all matrices. Numbers are just shorthand labels for certain classes of matrices, fight me.

Would you elaborate on the last statement?
- I'm just being silly, but I mean that if everything can be represented as a matrix then there's a point of view where things like complex numbers are just "names" of specific matrices and the rules that apply to those "names" are just derived from the relevant matrix operations.
  
  Essentially I'm saying that the normal form is an abstract short hand notation of the matrix representation. The matrices are of course significantly harder and more confusing to work with, but in some cases the richness of that structure is very beautiful and insightful.
  
  (I'm particularly in love with the fact one can derive spinors and their transforms purely from the spacetime/Lorentz transforms. It's a really satisfying exercise and it's some beautiful algebra/group theory.)

Elevendy

I always like seeing π.

Pi is a real number, though. It's irrational, but real.
- Pi written out is a real number, yes, I’m referring to the symbol representing Pi. Does that not count?

Meow

Chaitin's constants.

Not even a number.

They are still real numbers. Specifically uncomputable, normal numbers. Which means their rational expansion contain every natural number.
- Oops, I misunderstood what an uncomputable is...
  
  In that case, I would say Infinity-Infinity. This time, it's truly not a number.

Zero

It's the absence of a number and has all manner of interesting edge cases associated with it.

Zero is the absence of a quantity, but it is still a number.
Zero is a real number, but interestingly, it's also a pure imaginary number. It's the only number that's both things at once.
- It gets deeper. It's also the same as the 0-k-vector, the 0-k-blade, the 0-multivector, the only number that is its own square besides 1, etc...
- As I said .. lots of edge cases :)
For a while 1 wasn't considered to be a number either http://aleph0.clarku.edu/~djoyce/elements/bookVII/defVII1.html

{j|∧} or ĵ just a base vector