“i” is not a Constant

People will sometimes call i, together with numbers like pi or e, a “constant”. This is a misunderstanding of the concept. A constant is something that has a definition from the abstract field.

For example, you can define e as follows: “take a complete ordered field, and find the least upper bound of (1+1/n)**n for all natural numbers n”.

It does not matter how the real numbers are represented: cauchy sequences or Eudoxus reals, it’s all the same. The least upper bound is unique, and points to a specific element in the field.

However, this is not true for i. “In the unique algebraic completion of a complete ordered field, the number that, when squared, is equal to -1” does not identify i, because (-i)**2=-1.

Indeed, there is no theoretical way to identify i. Complex number conjugation is a continuous automorphism of the complex number field, so there is no property of i that -i does not share.

The real number field has no non-trivial automorphisms, so this problem is unique to i. If you need a complex constant, you can indicate the real and imaginary parts, both of them real numbers, and the only “uncertainty” is, once again, what i is.