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.
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.
Even crazier than that: almost all numbers cannot be defined with any finite expression.
i tried Math.random(), but that gave a rational number. i'm very lucky i guess?
When you apply the same test to the output of Math.PI, does it pass?
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.
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)
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...
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.
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".
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?
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.
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.
(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 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.
Seems like Cantor would have been all over this.
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!
So why bring some numbers here as transcendental if not proven?
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".
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.
"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.
"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.
No, my dudes. Just no. If it’s not proven transcendental, it’s not to be considered such.
Indeed. And by similar arguments, there are more uncomputable real numbers than computable real numbers. (And almost all transcendental numbers are uncomputable).
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.
Does a number not matter "in practice" even if it's used to compute a more commonly use constant? Very odd framing.
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.
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.
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
Just escape any asterisks in your post that you want rendered as asterisks: this: \* gives: *.
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.
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
The most famous transcendental numbers
https://sprott.physics.wisc.edu/pickover/trans.html