"i" is not a Constant
=====================

People will sometimes call
:code:`i`,
together with numbers like
:code:`pi` or :code:`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
:code:`e`
as follows:
"take a complete ordered field,
and find the least upper bound of
:code:`(1+1/n)**n`
for all natural numbers
:code:`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
:code:`i`.
"In the unique algebraic completion
of a complete ordered field,
the number that,
when squared,
is equal to
:code:`-1`"
does not identify
:code:`i`,
because
:code:`(-i)**2=-1`.

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

The real number field has no non-trivial automorphisms,
so this problem is unique to
:code:`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
:code:`i`
is.