[mg]

Three surprising facts about transcendental numbers:

1: Almost all numbers are transcendental.

2: If you could pick a real number at random, the probability of it being transcendental is 1.

3: Finding new transcendental numbers is trivial. Just add 1 to any other transcendental number and you have a new transcendental number.

Most of our lives we deal with non-transcendental numbers, even though those are infinitely rare.

[canjobear]

> 1: Almost all numbers are transcendental.

Even crazier than that: almost all numbers cannot be defined with any finite expression.

[dwohnitmok]

This is not necessarily true. It is possible for all real numbers (and indeed all mathematical objects) to be definable under ZFC. It is also possible for that not to be the case. ZFC is mum on the issue.

I've commented on this several times. Here's the most recent one: https://news.ycombinator.com/item?id=44366342

Basically you can't do a standard countability argument because you can't enumerate definable objects because you can't uniformly define "definability." The naive definition falls prey to Liar's Paradox type problems.

[zeroonetwothree]

Maybe it would be better to say almost all numbers are not computable.

[dinosaurdynasty]

Leads to really fun statements like "there exists a proof that all reals are equal to themselves" and "there does not exist a proof for every real number that it is equal to itself" (because `x=x`, for most real numbers, can't even be written down, there are more numbers than proofs).

[bjourne]

Really? Which number can't be defined with a finite expression?

[sorokod]

By common definition of "almost all", 1 == 2

[testaccount28]

how can i pick a real number at random though?

i tried Math.random(), but that gave a rational number. i'm very lucky i guess?

[andrewflnr]

You can't actually pick real numbers at random. You especially can't do it on a computer, since all numbers representable in a finite number of digits or bits are rational.

[tantalor]

Pick a digit, repeat, don't stop.

[mg]

How did you test the output of Math.random() for transcendence?

When you apply the same test to the output of Math.PI, does it pass?

[kridsdale1]

Use an analog computer. Sample a voltage. Congrats.