What are transcendental numbers in layman’s terms?

Answer by Alan Bustany:

Real numbers, [math]\mathbb R[/math], despite their name are a complicated mathematical structure that includes an awful lot of numbers that you can't even define.
By way of illustration here is a collection of subsets of [math]\mathbb R[/math] in strictly included order:
The thing is, every one of those subsets is Countable: it can be put in one-to-one correspondence with the Natural numbers. In a sense there are equally many elements in any of those sets. But the Real numbers are Uncountable: there are a lot more of them than in any of these subsets.
The sets from [math]\mathbb Q[/math] on are Dense: you can approximate any number as closely as you like with a member of the set. Nevertheless almost all Real numbers are not in those sets.
In particular Transcendental numbers are defined as Real numbers that are not Algebraic (not the roots of an algebraic equation with integer coefficients). They are defined by what they are not as the complement of the set [math]\mathbb A[/math] in [math]\mathbb R[/math]. As a result they are rather enigmatic and hard to pin down. The two most well-known Transcendental numbers are [math]\pi[/math] and [math]e[/math], but these two are (of course) Computable.
Are numbers that you can't even compute or define of any use whatsoever? Some would argue not, but the fact is that almost all Real numbers are of exactly that character. I may not have succeeded in giving you a layman's definition of Transcendental numbers but, with any luck, you now appreciate that Real numbers, especially the Transcendental Real numbers may not be so "real" after all…

What are transcendental numbers in layman’s terms?

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