This is a page associated with the blog post “Why Algebra Needs Imaginary Numbers” discussing algebraic number theory.
I was thinking about including this clause in the main post, but, as is a recurring trait of the transcendental numbers, the explanation felt out-of-place in the context of algebraic closure, so I decided to move it here as an aside for those interested in more information.
Using the simple idea of algebraic closure we’ve been able to rebuild our number system from the ground up and find a home for some of its most inscrutable members in the form of polynomial roots, the set of which form the algebraic numbers. This worked for all typical irrational numbers in the form of radicals like \( \sqrt{31} \). Could we do the same with a number like Euler’s constant e, previously defined with a cumbersome infinite series? Well, in a groundbreaking proof by Joseph Louisville in the mid-1800s it was shown that a group of real numbers could be constructed that cannot be expressed as the root of a polynomial, the most famous of which today is known as Louisville’s constant.
\( L=\sum_{n=1}^{\infty}10^{-n!}=0.11000100000000000000001… \)
Louisville’s Constant, the first proven transcendental number. Contains a 1 at every decimal place coresponding to subsquent values of n.
And in a crushing blow to math nerds, the mathematician Charles Hermite showed that no such polynomial exists for e either: In other words, e is not an algebraic number, and any algebraic power of the constant is similarly outside the realm of classical algebra, a property known as the Lindemann-Weirstrauss Theorem. The proof of these ideas is lengthy and involves advanced calculus so we won’t show it here, but we can use that last fact to demonstrate yet another number that is lost to the algebraic set:
\( e^x \). is transcendental for all algebraic values of x.
\( e^{i\pi}=-1 \), an algebraic number.
By statement (1), \( x=i\pi - \) is not algebraic.
We know i is algebraic, and the algebraic numbers are closed under multiplication.
Therefore, \( \pi \) is transcendental!
So, as it turns out, our Venn diagram is actually incomplete: Nested in the woodworks of the real number line even more sneakily than the irrational numbers are the transcendentals, another infinite continuum that includes treasured constants like e and \( \pi \), and is actually a “larger” number set than their algebraic counterparts (how can one infinite set be larger than another? Further reading here). All our logic for the construction of the algebraic numbers still stands, but for the sake of completeness (and because they’re as cool as they are frustrating), I thought they were worthy of a shoutout. There’s lots more to unpack with this strange group of numbers, so do some exploring!
