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Three surprising facts about transcendental numbers: 1: Almost all numbers are transcendental. 2: If you could pick a real number at random, the probability o
by mg 9mo 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.
- deleted 9mo ago[deleted]
- testaccount28 9mo agohow 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?
- mg 9mo agoHow 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?
- BeetleB 9mo agoAll floating point numbers are rational.
- jmgao 9mo agoWell, except for inf, -inf, and nan.
- Someone 9mo agoand, depending on how you define the rationals, -0. https://en.wikipedia.org/wiki/Integer https://en.wikipedia.org/wiki/Integer: “An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...)” According to that definition, -0 isn’t an integer. Combining that with https://en.wikipedia.org/wiki/Rational_number https://en.wikipedia.org/wiki/Rational_number: “a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q” means there’s no way to write -0 as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
- zeroonetwothree 9mo agoAll numbers that actually exist in our finite visible universe are rational.
- deleted 9mo ago[deleted]
- tsimionescu 9mo agoNot really. In all of our physical theories, curved paths are actual curves. So, (assuming circular orbits for a second) the ratio between the length of the Earth's orbit around the Sun and the distance between the Earth and the Sun is Pi - so, either the length of the path or the straight line distance must be an irrational number. While the actual orbit is elliptical instead of circular, the relation still holds. Of course, we can only measure any quantity up to a finite precision. But the fact that we chose to express the measurement outcome as 3.14159 +- 0.00001 instead of expressing it as Pi +- 0.00001 is an arbitrary choice. If the theory predicts that some path has length equal exactly to 2.54, we are in the same situation - we can't confirm with infinite precision that the measurement is exactly 2.54, we'll still get something like 2.54 +- 0.00001, so it could very well be some irrational number in actual reality.
- edanm 9mo agoWhat does "actually exist" mean? Does Pi "actually exist"?
- tantalor 9mo agoPick a digit, repeat, don't stop.
- markusde 9mo agoExactly 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!
- jibal 9mo agoThat's just saying that you can pick and use rational numbers (which are a subset of the reals.)
- skulk 9mo agoNot really. You can simulate a probability of 1/x by expanding 1/x in binary and flipping a coin repeatedly, once for each digit, until the coin matches the digit (assign heads and tails to 0 and 1 consistently). If the match happened on 1, then it's a positive result, otherwise negative. This only requires arbitrary but finite precision but the probability is exactly equal to 1/x which isn't rational.
- jibal 9mo agoNo, it isn't ... an infinite expansion isn't possible.
- markusde 9mo agoKind of, but you're not just picking rationals, you're picking rationals that are known to converge to a real number with some continuous property. You might be interested in this paper [1] which builds on top of this approach to simulate arbitrarily precise samples from the continuous normal distribution. [1] https://dl.acm.org/doi/10.1145/2710016 https://dl.acm.org/doi/10.1145/2710016
- techas 9mo ago
- andrewflnr 9mo agoYou 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.
- teraflop 9mo agoCareful -- that statement is half true. It's true that no matter what symbolic representation format you choose (binary or otherwise) it will never be able to encode all irrational numbers, because there are uncountably many of them. But it's certainly false that computers can only represent rational numbers. Sure, there are certain conventional formats that can only represent rational numbers (e.g. IEEE-754 floating point) but it's easy to come up with other formats that can represent irrationals as well. For instance, the Unicode string "√5" is representable as 4 UTF-8 bytes and unambiguously denotes a particular irrational.
- cozzyd 9mo agoOr use pieee-754 which is the same as iee-754 but everything is mimtipled by pi.
- electroglyph 9mo agoi really wanted "mimtipled" to be a word =)
- cozzyd 9mo agoI guess my phone thinks it might be since it didn't correct it :)
- andrewflnr 9mo agoI was careful. :) > representable in a finite number of digits or bits Implying a digit-based representation.
- 9mo ago
- kridsdale1 9mo agoUse an analog computer. Sample a voltage. Congrats.
- why-o-why 9mo agoSample 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)
- rcxdude 9mo agoWell, sampling is technically an analog operation that is separate from the conversion operation that makes the result digital. But then analog circuits don't ever actually hold a single real number, in practice there is always noise and that in practice limits the precision to less than what can be fairly easily achieved digitally.
- why-o-why 9mo agoSure, but we are talking about generating a random number, not sampling noise: those are two different things, albeit the former can be derived from the latter but not directly and as simply as the parent post claimed. Just sampling analog noise does not generate a "true" random number that can satisfy a set of design parameters to configure the NIST 800-90b entropy assessment (well, one could pick shitty parameters for the probability tests, but let's assume experts at the helm). Hence the need for software whitening. https://en.wikipedia.org/wiki/Hardware_random_number_generator#Software_whitening https://en.wikipedia.org/wiki/Hardware_random_number_generat... https://github.com/usnistgov/SP800-90B_EntropyAssessment https://github.com/usnistgov/SP800-90B_EntropyAssessment (^^^ this is a fun tool, I recommend playing with it to learn how challenging it is to generate "true" random numbers.) An infinite precision ADC couldn't be subject to thermal attack because you could just sample more bits of precision. (Of course, then we'd be down to Planck level precision so obviously there are limits, but my point still stands, at least _I_ think it does. :))
- deleted 9mo ago
- canjobear 9mo ago> 1: Almost all numbers are transcendental. Even crazier than that: almost all numbers cannot be defined with any finite expression.
- dinosaurdynasty 9mo agoLeads 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).
- dwohnitmok 9mo agoThis 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 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.
- canjobear 9mo agoI think you're overthinking it. Define a "number definition system" to be any (maybe partial) mapping from finite-length strings on a finite alphabet to numbers. The string that maps to a number is the number's definition in the system. Then for any number definition system, almost all real numbers have no definition.
- sorokod 9mo agoBy common definition of "almost all", 1 == 2