HN

Comments (54)

• tzs
  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.
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  brianberns
  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!
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  mg
  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.
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  drob518
  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?
• why-o-why
  I can't believe Champerowne's constant was only analyzed as of 1933.Seems like Cantor would have been all over this.https://en.wikipedia.org/wiki/Champernowne_constant
• keepamovin
  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 entraceI 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.
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  zkmon
  If a number system has a transcendental number as its base, would these numbers still be called transcendental in that number system?
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  barishnamazov
  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.
• tshaddox
  > 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).
• nuancebydefault
  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!
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  senfiaj
  > 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?
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  adrian_b
  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.