The most famous transcendental numbers
sprott.physics.wisc.edu145 points by vismit2000 16 hours ago
145 points by vismit2000 16 hours ago
Mathematicians 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.
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!
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.
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.
I 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.
Maybe it would be better to say almost all numbers are not computable.
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?
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.
Careful -- 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.
I was careful. :)
> representable in a finite number of digits or bits
Implying a digit-based representation.
Or use pieee-754 which is the same as iee-754 but everything is mimtipled by pi.
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!
That's just saying that you can pick and use rational numbers (which are a subset of the reals.)
Not 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.
At no point will your number be transcendental (or even irrational).
That's why you can't stop.
That's irrelevant. It's like saying that you can count to infinity if you never stop counting ... but no, every number in the count is finite.
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.
All numbers that actually exist in our finite visible universe are rational.
Not 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.
Well, except for inf, -inf, and nan.
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.
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)
Well, 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.
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.
If 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.
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.
> 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.
I 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.
I think you are thinking of the Abel–Ruffini impossibility theorum, which states that there is no general solution to polynomials of degree 5 or greater using only standard operations and radicals.
Galois went a step further and proved that there existed polynomials whose specific roots could not be so expressed. His proof also provided a relatively straightforward way to determine if a given polynomial qualified.
Both Euler's and Catalan's list "(Not proven to be transcendental, but generally believed to be by mathematicians.)". Maybe updated after your comment?
If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?
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.
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.
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).
> 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.)
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.)
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!
To 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
some constants that may or may not be transcendental: - Percolation Thresholds: https://en.wikipedia.org/wiki/Percolation_threshold - Critical scalings in 3d: https://en.wikipedia.org/wiki/Universality_class#Ising_model
> 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?
Because it still might be transcendental. Just because you don't know if the list is correct, doesn't mean it isn't.
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.
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".
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.
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.
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.
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.
I think the Oscars should go to the algebraic numbers - think about it - they are far less common ...
Yes, but the most famous ones are boring, we already know these! Let's get a list of the least famous transcendental numbers.
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.
I can't believe Champerowne's constant was only analyzed as of 1933.
Seems like Cantor would have been all over this.
> 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).
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?
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.
Okay, fair. It just seemed to me to have pretty limited utility.
Hm who cares about utility in this case?
Well, 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.
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".
> 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?
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.
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?
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.
They're accurately modeled. Just as Newtownian phenomena are accurately modeled, until they aren't. Reality is not necessarily reflective of any model.
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.
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.
> 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...
Related Numberphile video: https://www.youtube.com/watch?v=5TkIe60y2GI One of my favourites which I happened to look up just yesterday.
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.
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.
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 π or 2π is that the natural periodicity of trigonometric functions is 2π or π (for tan/cot). If you're using a different period, you consequently need to multiply or divide by 2π, 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.
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).
Uuuuuum no?
e^(ix) = cos(x) + isin(x). In particular e^(ipi) = -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^(iomega*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.
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
> 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: *.
> 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.
>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