> 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).
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!
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.
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!
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.
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?
If 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.
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.
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 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.)
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.
Both Euler's and Catalan's list "(Not proven to be transcendental, but generally believed to be by mathematicians.)". Maybe updated after your comment?
> 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?
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 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.
Indeed. And by similar arguments, there are more uncomputable real numbers than computable real numbers. (And almost all transcendental numbers are uncomputable).
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
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?
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.
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.
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.)
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!
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.
So why bring some numbers here as transcendental if not proven?
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.