Comment by mg
6 hours ago
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.
> 1: Almost all numbers are transcendental.
Even crazier than that: almost all numbers cannot be defined with any finite expression.
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).
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?
Pick a digit, repeat, don't stop.
Exactly right. You can pick and use real numbers, as long as they are only queried to finite precision. There are lots of super cool algorithms for doing this!
And don’t die.
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?
All floating point numbers are rational.
2 replies →
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.
Use an analog computer. Sample a voltage. Congrats.
Sample it with what? An infinite precision ADC?
This is how old temperature-noise based TRNGs can be attacked (modern ones use a different technique, usually a ring-oscillater with whitening... although i have heard noise-based is coming back but i've been out of the loop for a while)