# “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.