7 ms·
Khinchin's Constant
- yannis 10y agoInteresting, but where can I use it?
- Retra 10y agoI know some people are down-voting you based on the idea that math "doesn't have to be useful," (which I strongly disagree with), but I've down-voted you for a different reason and I'd like to explain. Mathematics is abstract. That means something. It means when I say "I have $50", you can think of those $50 as peices of paper, portions of goats, amounts of hot dogs, or investment potential. $50 is not a thing, it's a model for things that have arithmetical properties and value. That's how numbers work: they have properties, and you look at the world and map those properties to things that are relevant to you. So you read something about Khinchin's constant, you understand what the terms are, understand the structural implications of it's properties, and then you go and try to find things that behave similarly in the world. Sometimes it fits well, sometimes you have to force it, sometimes it's just overly complex. The point is that the behavior of something like this is a model, which means it will never map perfectly to reality, and also that it may apply with varying degrees of success to any problem. So where can you use it? Everywhere. You just have to think in terms of its properties. Do that enough and maybe you'll find out why it's stupid to use for some problems and smart for others. But there's no point in asking someone else to do all the thinking for you.
- noonespecial 10y agoI don't know because I don't know you. For you maybe nothing? Maybe something? Simple Example: Did you know that the ratio of a circle's circumference to its diameter is the same regardless of the size of the circle? I suppose you could use that to figure out how much stuff you need to make a wheel. I don't. Never have. Would you believe I use it to figure out what to add to an electric circuit to counteract the power factor of the inductive load of an electric motor the size of a pickup truck? Math is funny like that. Once you understand a concept, you might see it in nature in a place you didn't expect it, and it may be suddenly quite useful.
- kilovoltaire 10y agoMy favorite thing about this constant is the counter-intuitive fact that if you uniformly randomly choose a real number, there's a 100% chance it will have the Khinchin behavior, and yet there are almost no known numbers which actually do. Or as Wikipedia puts it: > Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose.
- benkuykendall 10y ago>> if you uniformly randomly choose a real number There is no uniform probability distribution over all real numbers.
- fdej 10y ago> However, the numerical value of K is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower. It can be calculated to n digits in O~(n^2) time, which isn't really that bad. Something like Brun's constant or the area of the Mandelbrot set is probably a better example of a real number that is "notoriously difficult" to compute.
- russdill 10y agoSo then my question would be, are physics constants likely to be Khinchin, computable, or just non-computable?
- 323454 10y agoThe behaviour of Khinchin(e) is seriously crazy! Not only does it not converge to K, it looks O(log(n))...