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Mathematics professor at Cambridge
Nov 20 |
comment |
Continued fractions using all natural integers
I think you're right -- and indeed I was worried that something like that would happen. So it seems that the quotients have to grow rather fast to make the number transcendental. |
Nov 20 |
answered | Continued fractions using all natural integers |
Nov 20 |
answered | Are there any good nonconstructive “existential metatheorems”? |
Nov 19 |
awarded | Nice Answer |
Nov 18 |
awarded | Nice Answer |
Nov 18 |
answered | Beamer hints and tips |
Nov 18 |
answered | What's so great about blackboards? |
Nov 18 |
comment |
Can infinity shorten proofs a lot?
I'm not sure I count induction either. It seems to me that one is not relying on infinity in a serious way to sum the first million cubes: one is saying, "If it's valid for 1 then it's valid for 2, and if it's valid for 2 then it's valid for 3, and ... and if it's valid for 999999 then it's valid for 1000000." That is a ridiculously long proof, but induction allows one to shorten it by giving a rule for when it's OK to put in the "..." (A more formal way of making the point is that the axiom of infinity isn't part of first-order Peano arithmetic.) |
Nov 17 |
awarded | Critic |
Nov 17 |
awarded | Supporter |
Nov 17 |
comment |
Can infinity shorten proofs a lot?
You can get the inequality quite easily I think. The log of n! is the sum of log m from 1 to n, which is at least the integral from 1 to n of log x, which is nlogn-n. Done. |
Nov 17 |
awarded | Nice Question |
Nov 17 |
awarded | Nice Question |
Nov 17 |
awarded | Student |
Nov 17 |
awarded | Teacher |
Nov 16 |
comment |
Can infinity shorten proofs a lot?
That's a good example, but I think it may already form part of the presentation (when they talk about infinite sets and the work of Cantor). I'll check though. |
Nov 16 |
comment |
Can infinity shorten proofs a lot?
Ultimately what's wanted is a very nice demonstration for the non-mathematician of why infinity is useful even if all you care about is finitary results. |
Nov 16 |
asked | Can infinity shorten proofs a lot? |
Nov 3 |
answered | Memorizing theorems |
Nov 2 |
comment |
Are there any interesting examples of random NP-complete problems?
Hmm, I'm wondering now if there's a boring answer, which is to take a problem that is hard on average and randomly restrict it in some natural way to a small class of instances. If that works, then maybe the question is interesting only in particular cases and not as a general question. |