9 ms·
The most famous transcendental numbers
- zkmon 10mo agoIf a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?
- moefh 10mo agoYes. A number is transcendental if it's not the root of a polynomial with integer coefficients; that's completely independent of how you represent it.
- frutiger 10mo agoI think the elements of the base need to be enumerable (proof needed but it feels natural), and transcendental numbers are not enumerable (proof also needed).
- kinkyusa 10mo ago[dead]
- tocs3 10mo agoBase pi: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_%CF%80 https://en.wikipedia.org/wiki/Non-integer_base_of_numeration... Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_e https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
- JadeNB 10mo agoI think your parent comment was speaking of a "base-$\alpha$ representation", where $\alpha$ is a single transcendental number—no concerns about countability, though one must be quite careful about the "digits" in this base. (I'm not sure what "the elements of the base need to be enumerable" means—usually, as above, one speaks of a single base; while mixed-radix systems exist, the usual definition still has only one base per position, and only countably many positions. But the proof of countability of transcendental numbers is easy, since each is a root of a polynomial over $\mathbb Q$, there are only countably many such polynomials, and every polynomial has only finitely many roots.)
- jibal 10mo ago> I think the elements of the base need to be enumerable (proof needed but it feels natural) Proof of what? Needed for what? The elements of the number system are the base raised to non-negative integer powers, which of course is an enumerable set. > transcendental numbers are not enumerable Category mistake ... sets can be enumerable or not; numbers are not the sort of thing that can be enumerable or not. (The set of transcendental numbers is of course not enumerable [per Georg Cantor], but that doesn't seem to be what you're talking about.)
- kinkyusa 10mo ago[dead]
- gizmo686 10mo agoThe notion of transcendental is not related to how we right numbers. However, in abstract algebra, we generalize the notion of algebraic/transental to arbitrary fields. In such a framework, a number is only transental relative to a particular field. For instance, the standard statement that pi us transcendental would become the pi is transcendental in Q (the rational numbers). However, pi is trivially not transcendental over Q(pi), which is the smallest field possible after adding pi to the rational numbers. A more interesting question is if e is transcendental over Q(pi); as far as I am aware that is still an open problem.
- senfiaj 10mo ago> Euler's constant, gamma = 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.) So why bring some numbers here as transcendental if not proven?
- auggierose 10mo agoBecause it still might be transcendental. Just because you don't know if the list is correct, doesn't mean it isn't.
- loloquwowndueo 10mo agoSo it’s like “15 oldest actors to win an Oscar” and including someone who’s nominated this year but hasn’t actually won. But he might, right? No, my dudes. Just no. If it’s not proven transcendental, it’s not to be considered such.
- chvid 10mo agoI think the Oscars should go to the algebraic numbers - think about it - they are far less common ...
- deleted 10mo ago[deleted]
- senfiaj 10mo agoYes it's "likely" to be transcendental, maybe there are some evidences that support this, but this is not a proof (keep in mind that it isn't even proven to be irrational yet). Similarly, most mathematicians/computer scientist bet that P ≠ NP, but it doesn't make it proven and no one should claim that P ≠ NP in some article just because "it's most likely to be true" (even though some empirical real life evidence supports this hypothesis). In mathematics, some things may turn out to be contrary to our intuition and experience.
- barishnamazov 10mo agoDon't want to be "that guy," but Euler's constant and Catalan's constant aren't proven to be transcendental yet. For context, a number is transcendental if it's not the root of any non-zero polynomial with rational coefficients. Essentially, it means the number cannot be constructed using a finite combination of integers and standard algebraic operations (addition, subtraction, multiplication, division, and integer roots). sqrt(2) is irrational but algebraic (it solves x^2 - 2 = 0); pi is transcendental. The reason we haven't been able to prove this for constants like Euler-Mascheroni (gamma) is that we currently lack the tools to even prove they are irrational. With numbers like e or pi, we found infinite series or continued fraction representations that allowed us to prove they cannot be expressed as a ratio of two integers. With gamma, we have no such "hook." It appears in many places (harmonics, gamma function derivatives), but we haven't found a relationship that forces a contradiction if we assume it is algebraic. For all we know right now, gamma could technically be a rational fraction with a denominator larger than the number of atoms in the universe, though most mathematicians would bet the house against it.
- joshgachnang 10mo agoBoth Euler's and Catalan's list "(Not proven to be transcendental, but generally believed to be by mathematicians.)". Maybe updated after your comment?
- gizmo686 10mo ago> Essentially, it means the number cannot be constructed using a finite combination of integers and standard algebraic operations (addition, subtraction, multiplication, division, and integer roots) Slight clarification, but standard operations are not sufficient to construct all algebraic numbers. Once you get to 5th degree polynomials, there is no guarantee that their roots can be found through standard operations.
- hidroto 10mo agoI am no mathematician, but i think you may be overstating Galois result. it says that you cant write a single closed form expression for the roots of any quintic using only (+,-,*,/,nth roots). This does not necessarily stop you from expressing each root individually with the standard algebraic operations.
- brianberns 10mo agoI read this with pleasure, right up until the bit about the ants. Then I saw the note from myself at the end, which I had totally forgot writing seven years ago. I probably first encountered the article via HN back then as well. Thanks for publishing my thoughts!
- xnorswap 10mo agoThe ants argument feels rather like a retelling of Zeno's Paradoxes. https://en.wikipedia.org/wiki/Zeno%27s_paradoxes https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
- nuancebydefault 10mo agoI would have expected more numbers originating from physics, like Reynolds number (bad example since it is not really constant though). The human-invented ones seem to be just a grasp of dozens man can come up with. i to the power of i is one I never heard of but is fascinating though!
- SOTGO 10mo agoTo prove something is transcendental we would need to know how to compute it exactly, and I’m struggling to see how that would come up frequently in a physics context. In physics most constants are not arbitrary real numbers derived from a formula, they’re a measured relationship, which sort of inherently can’t be proved to be transcendental
- longemen3000 10mo agosome constants that may or may not be transcendental: - Percolation Thresholds: https://en.wikipedia.org/wiki/Percolation_threshold https://en.wikipedia.org/wiki/Percolation_threshold - Critical scalings in 3d: https://en.wikipedia.org/wiki/Universality_class#Ising_model https://en.wikipedia.org/wiki/Universality_class#Ising_model
- cozzyd 10mo agoYeah I'd expect Bessel function zeroes and such
- mg 10mo agoThree 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 10mo ago[deleted]
- testaccount28 10mo 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 10mo 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 10mo agoAll floating point numbers are rational.
- jmgao 10mo agoWell, except for inf, -inf, and nan.
- Someone 10mo 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.
- adrian_b 10mo agoIt should be noted that the number e = 2.71828 ... does not have any importance in practice, its value just satisfies the curiosity to know it, but there is no need to use it in any application. The transcendental number whose value matters (being the second most important transcendental number after 2*pi = 6.283 ...) is ln 2 = 0.693 ... (and the value of its inverse log2(e), in order to avoid divisions). Also for pi, there is no need to ever use it in computer applications, using only 2*pi everywhere is much simpler and 2*pi is the most important transcendental number, not pi.
- d-us-vb 10mo agoThis comment is quite strange to me. e is the base of the natural logarithm. so ln 2 is actually log_e (2). If we take the natural log of 2, we are literally using its value as the base of a logarithm. Does a number not matter "in practice" even if it's used to compute a more commonly use constant? Very odd framing.
- adrian_b 10mo agoThe number "e" itself is never needed in any application. It is not used for computing the value of ln(2) or of log2(e), which are computed directly as limits of some convergent series. As I have said, there is no reason whatsoever for knowing the value of e. Moreover, there is almost never a good choice to use the exponential function or the hyperbolic logarithm function (a.k.a. natural logarithm, but it does not really deserve the name "natural"). For any numeric computations, it is preferably to use everywhere the exponential 2^x and the binary logarithm. With this choice, the constant ln 2 or its inverse appears in formulae that compute derivatives or integrals. People are brainwashed in school into using the exponential e^x and the hyperbolic logarithm, because this choice was more convenient for symbolic computations done with pen on paper, like in the 19th century. In reality, choosing to have the proportionality factor in the derivative formula as "1" instead of "ln 2" is a bad choice. The reason is that removing the constant from the derivative formula does not make it disappear, but it moves it into the evaluation of the function and in any application much more evaluations of the functions must be done than computations of derivative or integral formulae. The only case when using e^x may bring simplifications is in symbolic computations with complex exponentials and complex logarithms, which may be needed in the development of mathematical models for some linear systems that can be described by linear systems of ordinary differential equations or of linear equations with partial derivatives. Even then, after the symbolic computation produces a mathematical model suitable for numeric computations it is more efficient to convert all exponential or logarithmic functions to use only 2^x and binary logarithms.
- drob518 10mo agoSome of these seem forced. For instance, does Chapernowne's number (number 7 on the list, 0.12345678910111213141516171819202122232425...) occur in nature, or was it just manufactured in a mathematical laboratory somewhere?
- zeeboo 10mo agoIt is indeed manufactured specifically to show the existence of "normal" numbers, which are, loosely, numbers where every finite sequence of digits is equally likely to appear. This property is both ubiquitous (almost every number is normal in a specific sense) and difficult to prove for numbers not specifically cooked up to be so.
- drob518 10mo agoOkay, fair. It just seemed to me to have pretty limited utility.
- kaffekaka 10mo agoHm who cares about utility in this case?
- drob518 10mo agoWell, if we don’t care about utility I could define infinitely many transcendental numbers with no utility other than I just made them up. The number that is the concatenation of the digits of all prime numbers in sequence, for instance: 0.23571113171923… I christen this Dave’s Number. (It probably already has a name, but I’m stealing it.) Let’s add it to the list. Now we can define Dave’s Second Number as the first prime added to Dave’s Number: 2.235711131723… Dave’s Third Number is the second prime added to Dave’s Number: 3.235711131723… Since we’re cataloguing numbers with no utility, let’s add them all to the list.
- kaffekaka 10mo ago
- keepamovin 10mo agoThis guy's books sounds fascinating, Keys to Infinity and Wonder of Numbers. Definitely going to add to Kindle. pi transcends the power of algebra to display it in its totality what an entrace I think I read a book by this guy as a kid: it was an illustrated mostly black and white book about Chaitin's constant, halting problema and various ways of counting over infinite sets.
- tshaddox 10mo ago> Did you know that there are "more" transcendental numbers than the more familiar algebraic ones? Indeed. And by similar arguments, there are more uncomputable real numbers than computable real numbers. (And almost all transcendental numbers are uncomputable).
- why-o-why 10mo agoI can't believe Champerowne's constant was only analyzed as of 1933. Seems like Cantor would have been all over this. https://en.wikipedia.org/wiki/Champernowne_constant https://en.wikipedia.org/wiki/Champernowne_constant
- tzs 10mo agoIf we are including numbers that aren't actually proven to be transcendental but that most mathematicians think are, I'd put Lévy's constant on the list. It is e^(pi^2/(12 log 2)) Here's where it comes from. For almost all real numbers if you take their continued fraction expansion and compute the sequence of convergents, P1/Q1, P2/Q2, ..., Pn/Qn, ..., it turns out that the sequence Q1^(1/1), Q2^(1/2), ..., Qn^(1/n) converges to a limit and that limit is Lévy's constant.
- globular-toast 10mo agoRelated Numberphile video: https://www.youtube.com/watch?v=5TkIe60y2GI https://www.youtube.com/watch?v=5TkIe60y2GI One of my favourites which I happened to look up just yesterday.
- phyzome 10mo agoYes, but the most famous ones are boring, we already know these! Let's get a list of the least famous transcendental numbers.
- Syzygies 10mo agoMathematicians get enamored with particular ways of looking at things, and fall into believing this is gospel. I should know: I am one, and I fight this tendency at every turn. On one hand, "rational" and "algebraic" are far more pervasive concepts than mathematicians are ever taught to believe. The key here is formal power series in non-commuting variables, as pioneered by Marcel-Paul Schützenberger. "Rational" corresponds to finite state machines, and "Algebraic" corresponds to pushdown automata, the context-free grammars that describe most programming languages. On the other hand, "Concrete Mathematics" by Donald Knuth, Oren Patashnik, and Ronald Graham (I never met Oren) popularizes another way to organize numbers: The "endpoints" of positive reals are 0/1 and 1/0. Subdivide this interval (any such interval) by taking the center of a/b and c/d as (a+c)/(b+d). Here, the first center is 1/1 = 1. Iterate. Given any number, its coordinates in this system is the sequence of L, R symbols to locate it in successive subdivisions. Any computer scientist should be chomping at the bit here: What is the complexity of the L, R sequence that locates a given number? From this perspective, the natural number "e" is one of the simpler numbers known, not lost in the unwashed multitude of "transcendental" numbers. Most mathematicians don't know this. The idea generalizes to barycentric subdivision in any dimension, but the real line is already interesting.
- afiori 10mo agoUsing this representation can one "efficiently" sum or multiply numbers? I was under the impression that this was its main drawback
- divbzero 10mo agoiⁱ is a fun one.
- sriku 10mo agoi^i isn't unique right? The "let x = π/2" could very well have been "let x = π(4k+1)/2" for any integer k.