[HN Gopher] The most famous transcendental numbers
___________________________________________________________________
The most famous transcendental numbers
Author : vismit2000
Score : 126 points
Date : 2025-12-31 12:32 UTC (10 hours ago)
(HTM) web link (sprott.physics.wisc.edu)
(TXT) w3m dump (sprott.physics.wisc.edu)
| senfiaj wrote:
| > 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 wrote:
| Because it still might be transcendental. Just because you
| don't know if the list is correct, doesn't mean it isn't.
| loloquwowndueo wrote:
| So 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 wrote:
| I think the Oscars should go to the algebraic numbers -
| think about it - they are far less common ...
| senfiaj wrote:
| Yes 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.
| auggierose wrote:
| It comes with the explicit comment "Not proven to be
| transcendental, but generally believed to be by
| mathematicians."
|
| That's really all you can do, given that 3 and 4 are really
| famous. At this point it is therefore just not possible to
| write a list of the "Fifteen Most Famous Transcendental
| Numbers", because this is quite possibly a different list
| than "Fifteen Most Famous Numbers that are known to be
| transcendental".
| senfiaj wrote:
| So "Fifteen Most Famous Transcendental Numbers" isn't the
| same as "Fifteen Most Famous Numbers that are known to be
| transcendental"?
|
| I might be OK with title "Fifteen Most Famous Numbers
| that are believed to be transcendental" (however, some of
| them have been proven to be transcendental) but "Fifteen
| Most Famous Transcendental Numbers" is implying that all
| the listed numbers are transcendental. Math assumes that
| a claim is proven. Math is much stricter compared to most
| natural (especially empirical) sciences where everything
| is based on evidence and some small level of uncertainty
| might be OK (evidence is always probabilistic).
|
| Yes, in math mistakes happen too (can happen in complex
| proofs, human minds are not perfect), but in this case
| the transcendence is obviously not proven. If you say "A
| list of 15 transcendental numbers" a mathematician will
| assume all 15 are proven to be transcendental. Will you
| be OK with claim "P [?] NP" just because most professors
| think it's likely to be true without proof? There are
| tons of mathematical conjectures (such as Goldbach's)
| that intuitively seem to be true, yet it doesn't make
| them proven.
|
| Sorry for being picky here, I just have never seen such
| low standards in real math.
| auggierose wrote:
| You are not picky, you just don't understand my point.
|
| "Fifteen Most Famous Transcendental Numbers" is indeed
| not the same as "Fifteen Most Famous Numbers that _are
| known to be_ transcendental ". It is also not the same as
| "Fifteen Most Famous Numbers that _have been proven to
| be_ transcendental ". Instead, it is the same as "Fifteen
| Most Famous Numbers that _are_ transcendental ".
|
| That's math for you.
| senfiaj wrote:
| Again, it seems we are arguing because of our subjective
| differences in the title correctness and rigor.
| Personally, I would not expect such title even from a
| pop-math type article. At least it should be more obvious
| from the title.
|
| "Transcendental" or even "irrational" isn't a vibesy
| category like "mysterious" or "beautiful", it's a hard
| mathematical property. So a headline that flatly labels a
| number "transcendental" while simultaneously admitting
| "not even proven" inside the article, looks more like a
| clickbait.
| rkowalick wrote:
| As far I know, Euler's constant hasn't even been proven to be
| irrational.
| barishnamazov wrote:
| Don'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.
| servercobra wrote:
| Both Euler's and Catalan's list "(Not proven to be
| transcendental, but generally believed to be by
| mathematicians.)". Maybe updated after your comment?
| gizmo686 wrote:
| > 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.
| zkmon wrote:
| If a number system has a transcendental number as its base, would
| these numbers still be called transcendental in that number
| system?
| moefh wrote:
| Yes. 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 wrote:
| I 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).
| tocs3 wrote:
| Base pi: https://en.wikipedia.org/wiki/Non-
| integer_base_of_numeration...
|
| Base e: https://en.wikipedia.org/wiki/Non-
| integer_base_of_numeration...
| JadeNB wrote:
| I 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.)
| gizmo686 wrote:
| The 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.
| brianberns wrote:
| I 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 wrote:
| The ants argument feels rather like a retelling of Zeno's
| Paradoxes.
|
| https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
| nuancebydefault wrote:
| I 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!
| mg wrote:
| 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.
| testaccount28 wrote:
| 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?
| mg wrote:
| 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?
| BeetleB wrote:
| All floating point numbers are rational.
| jmgao wrote:
| Well, except for inf, -inf, and nan.
| Someone wrote:
| and, depending on how you define the rationals, -0.
|
| 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: _"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.
| tantalor wrote:
| Pick a digit, repeat, don't stop.
| markusde wrote:
| 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!
| jibal wrote:
| That's just saying that you can pick and use rational
| numbers (which are a subset of the reals.)
| techas wrote:
| And don't die.
| jibal wrote:
| At no point will your number be transcendental (or even
| irrational).
| andrewflnr wrote:
| 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.
| kridsdale1 wrote:
| Use an analog computer. Sample a voltage. Congrats.
| why-o-why wrote:
| 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)
| jibal wrote:
| Use an analog computer how, to do what? An analog computer
| can do analog operations on analog signals, but you can't
| get an irrational number out of it ... this can be viewed
| as a sort of monad.
| canjobear wrote:
| > 1: Almost all numbers are transcendental.
|
| Even crazier than that: almost all numbers cannot be defined
| with any finite expression.
| dinosaurdynasty wrote:
| 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).
| dwohnitmok wrote:
| 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.
| sorokod wrote:
| By common definition of "almost all", 1 == 2
| adrian_b wrote:
| It 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 wrote:
| This 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.
| qnleigh wrote:
| Uuuuuum no?
|
| e^(i _x) = cos(x) + i_ sin(x). In particular e^(i _pi) = -1
|
| (1 + 1/n)^n = e. This is part of what makes e such a uniquely
| useful exponent base.
|
| Not applied enough? What about:
|
| d/dx e^x = e^x. This makes e show up in the solutions of all
| kinds of differential equations, which are used in physics,
| engineering, chemistry...
|
| The Fourier transform is defined as integral e^(i_omega*t) f(t)
| dt.
|
| And you can't just get rid of e by changing base, because you
| would have to use log base e to do so.
|
| Edit: how do you escape equations here? Lots of the text in my
| comment is getting formatted as italics.
| lutusp wrote:
| > Edit: how do you escape equations here? Lots of the text in
| my comment is getting formatted as italics.
|
| Just escape any asterisks in your post that you want rendered
| as asterisks: this: \\* gives: *.
| selecsosi wrote:
| Guessing the original comment hasn't taken complex analysis
| or has some other oriented view point into geometry that
| gives them satisfaction but these expressions are one of the
| most incredible and useful tools in all of mathematics (IMO).
| Hadn't seen another comment reinforcing this so thank you for
| dropping these.
|
| Cauchy path integration feels like a cheat code once you
| fully imbibe it.
|
| Got me through many problems that involves seemingly
| impossible to memorize identities and re-derivation of
| complex relations become essentially trivial
| lutusp wrote:
| > It 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.
|
| In calculations like compound financial interest, radioactive
| decay and population growth (and many others), e is either
| applied directly or derived implicitly.
|
| > ... 2*pi is the most important transcendental number, not pi.
|
| Gotta agree with this one.
| BigTTYGothGF wrote:
| What an odd thing to say. I find that it shows up all the time
| (and don't find myself using 2pi any more than pi).
| jcranmer wrote:
| It took me quite a bit to figure out what you're trying to say
| here.
|
| The importance of e is that it's the natural base of exponents
| and logarithms, the one that makes an otherwise constant factor
| disappear. If you're using a different base b, you generally
| need to adjust by exp(b) or ln(b), neither of which requires
| computing or using e itself (instead requiring a function call
| that's using minimax-generated polynomial coefficients for
| approximation).
|
| The importance of p or 2p is that the natural periodicity of
| trigonometric functions is 2p or p (for tan/cot). If you're
| using a different period, you consequently need to multiply or
| divide by 2p, which means you actually have to use the value of
| the constant, as opposed to calling a library function with the
| constant itself.
|
| Nevertheless, I would say that despite the fact that you would
| _directly_ use e only relatively rarely, it is still the more
| important constant.
| constantcrying wrote:
| >but there is no need to use it in any application.
|
| Applications such as planes flying, sending data through wires,
| medical imaging (or any of a million different direct
| applications) do not count, I assume?
|
| Your naivety about what makes the world function is not an
| argument for something being useless. The number appearing in
| one of the most important algorithms should give you a hint
| about how relevant it is
| https://en.wikipedia.org/wiki/Fast_Fourier_transform
| drob518 wrote:
| Some 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 wrote:
| It 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 wrote:
| Okay, fair. It just seemed to me to have pretty limited
| utility.
| kaffekaka wrote:
| Hm who cares about utility in this case?
| Strilanc wrote:
| It's fame comes from the simplicity of its construction rather
| than its utility elsewhere in mathematics.
|
| For example, Graham's number is pretty famous but it's more of
| a historical artifact rather than a foundational building
| block. Other examples of non-foundational fame would be the
| famous integers 42, 69, and 420.
| tantalor wrote:
| Yes, it occurs in the nature of the mathematician's mind.
| jerf wrote:
| All the transcendental numbers are "manufactured in a
| mathematical laboratory somewhere".
|
| In fact we can tighten that to all _irrational_ numbers are
| manufactured in a mathematical laboratory somewhere. You 'll
| never come across a number in reality that you can prove is
| irrational.
|
| That's not necessarily because all numbers in reality "really
| are" rational. It is because you can't get the infinite
| precision necessary to have a number "in hand" that is
| irrational. Even if you had a quadrillion digits of precision
| on some number in [0, 1] in the real universe you'd still not
| be able to prove that it isn't simply that number over a
| quadrillion no matter how much it may seem to resemble some
| other interesting irrational/transcendental/normal/whatever
| number. A quadrillion digits of precision is still a flat 0% of
| what you'd need to have a provably irrational number "in hand".
| 5ver wrote:
| It appears quantum phenomena are accurately described using
| mathematics involving trig functions. As such we do
| encounters numbers in reality that involve transcendental
| numbers, right?
| kevin_thibedeau wrote:
| They're accurately _modeled_. Just as Newtownian phenomena
| are accurately modeled, until they aren 't. Reality is not
| necessarily reflective of any model.
| kergonath wrote:
| You don't need quantum mechanics. Trigonometric functions
| are everywhere in classical mechanics. Gaussians,
| exponential, and logs are everywhere in statistical
| physics. You cannot do much if you don't use transcendental
| numbers. Hell, you just need a circle to come across pi.
| It's rational numbers that are special.
| jerf wrote:
| Consider the ideal gas law: pV=nRT
|
| Five continuous quantities related to each other, where by
| default when not specified we can safely assume real
| values, right? So we must have real values in reality,
| right?
|
| But we know that gas is not continuous. The "real" ideal
| gas law that relates those quantities really needs you to
| input every gas molecule, every velocity of every gas
| molecule, every detail of each gas molecule, and if you
| really want to get precise, everything down to every
| neutrino passing through the volume. Such a real formula
| would need to include terms for things like the self-
| gravitation of the gas affecting all those parameters. We
| use a simple real-valued formula because it is good enough
| to capture what we're interested in. None of the five
| quantities in that formula "actually" exist, in the sense
| of being a single number that fully captures the _exact_
| details of what is going on. It 's a model, not reality.
|
| Similarly, all those things using trig and such are models,
| not reality.
|
| But while true, those in some sense miss something even
| more important, which I alluded to strongly but will spell
| out clearly here: What would it mean to have a provably
| irrational value _in hand_? In the real universe? Not
| metaphorically, but some sort of real value fully in your
| hand, such that you fully and completely know it is an
| irrational value? Some measure of some quantity that you
| have to that detail? It means that if you tell me the value
| is X, but I challenge you that where you say the Graham 's
| Number-th digit of your number is a 7, I say it is actually
| a 4, you can _prove_ me wrong. Not by math; by measurement,
| by observation of the value that you have "in hand".
|
| You can never gather that much information about any
| quantity in the real universe. You will always have finite
| information about it. Any such quantity will be
| indistinguishable from a rational number by any real test
| you could possibly run. You can never tell me with
| confidence that you have an irrational number _in hand_.
|
| Another way of looking at it: Consider the Taylor expansion
| of the sine function. To be the transcendental function it
| is in math, it must use _all_ the terms of the series. Any
| finite number of terms is still a polynomial, no matter how
| large. Now, again, I tell you that by the Graham 's Number
| term, the universe is no longer using those terms. How do
| you prove me wrong _by measurement_?
|
| All you can give me is that some value in hand sure does
| seem to bear a strong resemblance to this particular
| irrational value, pi or e perhaps, but that's all. You
| can't go out the infinite number of digits necessary to
| prove that you have _exactly_ pi or e.
|
| Many candidates for the Theory of Everything don't even
| have the infinite granularity in the universe in them
| necessary to have that detailed an object in reality,
| containing some sort of "smallest thing" in them and
| minimum granularity. Even the ones that do still have the
| Planck size limit that they don't claim to be able to
| meaningfully see beyond with real measurements.
| tshaddox wrote:
| > You'll never come across a number in reality that you can
| prove is irrational.
|
| If a square with sides of rational (and non-zero) length can
| exist in reality, then the length of its diagonal is
| irrational. So which step along the way isn't possible in
| reality? Is the rational side length possible? Is the right
| angle possible?
| 613style wrote:
| They're saying you can't find a ruler accurate enough to be
| sure the number you measure is sqrt(2) and not sqrt(2) for
| the first 1000 digits then something else. And eventually,
| as you build better and better rulers, it will turn out
| that physical reality doesn't encode enough information to
| be sure. Anything you can measure is rational.
| eichin wrote:
| > mathematical laboratory
|
| Love the image of mathematicians laboring over flasks and test
| tubes, mixing things and extracting numbers... would have far
| more explosions than day-to-day mathematics usually does...
| keepamovin wrote:
| This 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 wrote:
| > 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 wrote:
| I 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
| tzs wrote:
| If we are including numbers that aren't actually proven to be
| transcendental but that most mathematicians think are, I'd put
| Levy'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 Levy's constant.
| globular-toast wrote:
| Related Numberphile video:
| https://www.youtube.com/watch?v=5TkIe60y2GI One of my favourites
| which I happened to look up just yesterday.
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