I 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

I 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.