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)
Seems like Cantor would have been all over this.
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".
Indeed. And by similar arguments, there are more uncomputable real numbers than computable real numbers. (And almost all transcendental numbers are uncomputable).
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.
Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...
(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.)
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.
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.
No, my dudes. Just no. If it’s not proven transcendental, it’s not to be considered such.
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.
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.
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.