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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
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Nov
17
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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.