Imaginary unit...Real question

Apr 2016
244
29
Australia
Since −i is algebraic but not rational, eπ is transcendental
from https://en.wikipedia.org/wiki/Gelfond%27s_constant


if anyone uses, maple, their facility for checking that a number 'x' is algebraic is called by:

type(x, algnum)

and if you do so for x=-i you will note that this Boolean returns false, where as if you check the Euler–Mascheroni constant for this or the same for it's rationality, it will return FAIL since is not known as yet.

So does anyone know why this difference of opinion exists? surely the most basic non-real should have a straight forward proof regarding its algebraicity.
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
Exactly which definition of "algebraic" are you using? Many textbooks define the "algebraic numbers" as a subset of the real numbers.
 
  • Like
Reactions: 1 person
Jun 2015
99
19
Ohio
Using the Wikipedia definition

"An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients"

x^2 + 1 is a polynomial with rational coefficients. -i is a root of this polynomial, so it is algebraic.
 
  • Like
Reactions: 1 person
Apr 2016
244
29
Australia
exactly that is what I call an algebraic number and that is what everyone should call an algebraic number. The real numbers are a subset of the complex and a univariate polynomial has complex roots equal in number to its degree on the reals. Thus this definition being so intimately tied (almost equivalent in some ways to the predicate for a number to belong to the set of algebraic numbers) I hate to be a troll but if I find such a text book that defines them as a subset of the reals, well, then I will be a troll, namely in emails to the author.
 
Apr 2016
244
29
Australia
and so again, I am used to maple and thus it would be a disadvantage to swap elsewhere is guess, this wouldn't be the first time ive called them up on being wrong there is a part of the code that is a "main" module (or a part of the kernel) called evalf which gives gives incorrect output for float approximations to poly and rational functions at transcendental arguments so don't worry... its maples fault despite them being mathematical gods of course, and me just a hobo.