Sometimes mathematicians go a little overboard with their naming.
Bibliography:
There's a billion simple looking expressions which are not known to be transcendental numbers or not. It's cute simple to state but hard to prove at its best.
Open as of 2020:
Bibliography:
- www.quantamagazine.org/recounting-the-history-of-maths-transcendental-numbers-20230627/ How Math Achieved Transcendence by David S. Richeson (2023).
Articles by others on the same topic
A transcendental number is a complex number that is not an algebraic number. No nonzero polynomial with coefficients in the integers vanishes at it. Familiar examples are and ; proving their transcendence requires substantially more than proving their irrationality.
There are only countably many integer-coefficient polynomials, and each nonzero one has finitely many roots. Consequently the algebraic numbers are countable, while the real numbers are uncountable; transcendental real numbers therefore exist in abundance. This counting proof establishes existence without identifying a particular example. Every real transcendental number is an irrational number, but shows that the converse fails.
