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.<p>It is e^(pi^2/(12 log 2))<p>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.
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:<p>1: Almost all numbers are transcendental.<p>2: If you could pick a real number at random, the probability of it being transcendental is 1.<p>3: Finding new transcendental numbers is trivial. Just add 1 to any other transcendental number and you have a new transcendental number.<p>Most of our lives we deal with non-transcendental numbers, even though those are infinitely rare.
> 1: Almost all numbers are transcendental.<p>Even crazier than that: almost all numbers cannot be defined with any finite expression.
how can i pick a real number at random though?<p>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.
Pick a digit, repeat, don't stop.
How did you test the output of Math.random() for transcendence?<p>When you apply the same test to the output of Math.PI, does it pass?
Use an analog computer. Sample a voltage. Congrats.
I can't believe Champerowne's constant was only analyzed as of 1933.<p>Seems like Cantor would have been all over this.<p><a href="https://en.wikipedia.org/wiki/Champernowne_constant" rel="nofollow">https://en.wikipedia.org/wiki/Champernowne_constant</a>
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.
It's fame comes from the simplicity of its construction rather than its utility elsewhere in mathematics.<p>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.
All the transcendental numbers are "manufactured in a mathematical laboratory somewhere".<p>In fact we can tighten that to all <i>irrational</i> numbers are manufactured in a mathematical laboratory somewhere. You'll never come across a number in reality that you can prove is irrational.<p>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".
Yes, it occurs in the nature of the mathematician's mind.
This guy's books sounds fascinating, <i>Keys to Infinity</i> and <i>Wonder of Numbers</i>. Definitely going to add to Kindle. <i>pi transcends the power of algebra to display it in its totality</i> what an entrace<p>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.
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.
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).
Base pi: <a href="https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_%CF%80" rel="nofollow">https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...</a><p>Base e: <a href="https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#Base_e" rel="nofollow">https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...</a>
I think your parent comment was speaking of a "base-$\alpha$ representation", where $\alpha$ is a <i>single</i> transcendental number—no concerns about countability, though one must be quite careful about the "digits" in this base.<p>(I'm not sure what "the elements of the base need to be enumerable" means—usually, as above, one speaks of a <i>single</i> 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.)
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Don't want to be "that guy," but Euler's constant and Catalan's constant aren't proven to be transcendental yet.<p>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.<p>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.<p>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.
> Did you know that there are "more" transcendental numbers than the more familiar algebraic ones?<p>Indeed. And by similar arguments, there are more uncomputable real numbers than computable real numbers. (And almost all transcendental numbers are uncomputable).
I would have expected more numbers originating from physics, like Reynolds number (bad example since it is not really constant though).<p>The human-invented ones seem to be just a grasp of dozens man can come up with.<p>i to the power of i is one I never heard of but is fascinating though!
> 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.)<p>So why bring some numbers here as transcendental if not proven?
As far I know, Euler's constant hasn't even been proven to be irrational.
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."<p>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"?<p>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).<p>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.<p>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.<p>"Fifteen Most Famous Transcendental Numbers" is indeed not the same as "Fifteen Most Famous Numbers that <i>are known to be</i> transcendental". It is also not the same as "Fifteen Most Famous Numbers that <i>have been proven to be</i> transcendental". Instead, it is the same as "Fifteen Most Famous Numbers that <i>are</i> transcendental".<p>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.<p>"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?<p>No, my dudes. Just no. If it’s not proven transcendental, it’s not to be considered such.
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.<p>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).<p>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.
It took me quite a bit to figure out what you're trying to say here.<p>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).<p>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.<p>Nevertheless, I would say that despite the fact that you would <i>directly</i> use e only relatively rarely, it is still the more important constant.
This comment is quite strange to me. e <i>is</i> 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.<p>Does a number not matter "in practice" even if it's used to compute a more commonly use constant? Very odd framing.
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?<p>e^(i<i>x) = cos(x) + i</i>sin(x). In particular e^(i<i>pi) = -1<p>(1 + 1/n)^n = e. This is part of what makes e such a uniquely useful exponent base.<p>Not applied enough? What about:<p>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...<p>The Fourier transform is defined as integral e^(i</i>omega*t) f(t) dt.<p>And you can't just get rid of e by changing base, because you would have to use log base e to do so.<p>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.<p>Cauchy path integration feels like a cheat code once you fully imbibe it.<p>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.<p>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.<p>In calculations like compound financial interest, radioactive decay and population growth (and many others), e is either applied directly or derived implicitly.<p>> ... 2*pi is the most important transcendental number, not pi.<p>Gotta agree with this one.
>but there is no need to use it in any application.<p>Applications such as planes flying, sending data through wires, medical imaging (or any of a million different direct applications) do not count, I assume?<p>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 <a href="https://en.wikipedia.org/wiki/Fast_Fourier_transform" rel="nofollow">https://en.wikipedia.org/wiki/Fast_Fourier_transform</a>