i is not a real number, is not an integer, is not a rational etc.

You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number

0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers

▲ chongli 3 hours ago | parent [-]

The simple fact you wanted to write this:

i=0+1*i

Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.

Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.

Comparing i with a unary operator like - shows the difference:

i*i=-1 makes perfect sense

-*-=???? does not make sense

▲ ttoinou 2 hours ago | parent [-]

i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1

  i*i=-1 makes perfect sense

This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles

	▲	chongli 41 minutes ago \| parent [-]
		There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1]. [1] https://en.wikipedia.org/wiki/Peano_axioms