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Comment by nine_k | Hacker News Reader

Not arbitrary transcendental numbers, but the e specifically.

Yeah, this was disappointed. Information about e is widely available, but I don't recall ever really reading much about transcendental numbers in general, other than a brief introduction to e.

Definitely a missed opportunity.

OK, here's a fun fact: most numbers are transcendental, which is to say, if you choose a real number uniformly at random in any finite interval the probability of choosing a transcendental number is 1. (The proof is easy and left as an exercise.) But actually describing a transcendental number is very hard. The first transcendental number other than pi and e was not described until the mid 19th century, and even today only a few dozen classes of transcendental numbers are known.

Possible spoiler (I don't know if my proof is correct)--

Algebraic numbers are countable, since their description is a countable combination of rationals which are also countable. The measure of a countable set is zero, so the measure of its complement is 1, and thus the probability of choosing a transcendental is 1.

Exactly right!

Another thing that tingles my mind: it's easy to describe numbers that are likely to be transendential (just define a turing machine that spews out digits by some more or less chaotic algorithm – most would agree that it seems extremely unlikely, that the output would correspond to some algebraic number by chance!), it's just hard to PROVE that they are.

If such a program ever terminates, then the result is rational. If it doesn't terminate, then pausing at any point will always describe an infinite number of irrationals.

Sorry, I caught your comment only a few days later. What do you mean by "infinite number of irrationals"? We are talking about a deterministic machine here, so surely it describes only one number?

But you have to let the Turing machine run forever for this to work! Every number with a terminating decimal representation in rational.

Maybe it's not so easy to describe a transcendental number, at least in our lifetimes...

You don't have to let the Turing machine run forever anymore than you have to actually fully write out every digit of PI. It's possible to determine strictly from the description of the Turing Machine that the value it outputs will converge to a specific number equal to the output of some other process, such the process of writing out 4 - 4/3 + 4/5 - 4/7 + 4/9 + ...

The equivalence class of all such representations is the number we call PI.

So, I was thinking that the Turing machine itself is a description of the number, because a description of the machine uniquely determines the output. If you require the digits to be spelled out, then yeah, that's impossible.

Not just very hard, but actually impossible for almost all transcendentals. Reason being, any description must be finite in length; descriptions therefore form a countable set; but transcendentals are uncountable. The subset of reals which are describable are known as "definable real numbers" [1]. (Notably, all algebraic numbers -- the reals of which form the complement of transcendentals -- are definable by definition.)

[1] https://en.wikipedia.org/wiki/Definable_real_number

Yes, that's true. But what I intended to convey was that it's very hard in the sense that we have only described a few dozen classes of transcendentals. So the actual limiting factor has to be something other than this counting argument. The hard part is finding descriptions of transcendentals within the space of describable numbers.

On this line of thought, the Numberphile YT video titled "All the Numbers" hits on this and other interesting truths. (I'd never learned the term 'normal number' until watching this.) Runtime 14m27s. Highly recommended.

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