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wu :: forums    riddles    medium (Moderators: william wu, towr, Grimbal, SMQ, ThudnBlunder, Eigenray, Icarus)    Irrationality of e « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Irrationality of e  (Read 613 times) william wu wu::riddles Administrator     Gender: Posts: 1291 Irrationality of e   « on: Dec 12th, 2007, 3:28pm » Quote Modify Prove that the constant e is irrational.   (Apologies if this has been posted already; moderators, feel free to delete this thread if so.)   IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: Irrationality of e   « Reply #1 on: Dec 12th, 2007, 5:16pm » Quote Modify I don't remember seeing a thread on this (nor did a search seem to reveal any threads...)     Here is a proof which involves the infinite series representation of e     We use the following e = Sum 1/n!   Suppose e = p/q where q > 1 (p, q are integers)   Then consider q!e = Sum_{n=1 to q} q!/n! + Sum_{n=q+1 to oo} q!/n!   The n=1 to q part is an integer.   Consider the nth tern of remaining: q!/n! = 1/(q+1)(q+2)...(q+n-q) < 1/qn-q     Thus Sum_{n=q+1 to oo} q!/n! < Sum_{n=q+1 to oo} 1/qn-q = 1/(q-1) <= 1   But q!e must be an integer.   Hence e is irrational. « Last Edit: Dec 12th, 2007, 5:30pm by Aryabhatta » IP Logged JiNbOtAk Uberpuzzler Hana Hana No Mi     Gender: Posts: 1187 Re: Irrationality of e   « Reply #2 on: Dec 12th, 2007, 5:20pm » Quote Modify What do you know, Wikipedia has a topic on this.     « Last Edit: Dec 12th, 2007, 5:21pm by JiNbOtAk » IP Logged Quis custodiet ipsos custodes? cool_joh Guest Re: Irrationality of e   « Reply #3 on: Dec 14th, 2007, 2:43am » Quote Modify Remove on Dec 12th, 2007, 5:16pm, Aryabhatta wrote: We use the following e = Sum 1/n! How can we prove that? IP Logged pex Uberpuzzler     Gender: Posts: 880 Re: Irrationality of e   « Reply #4 on: Dec 14th, 2007, 3:13am » Quote Modify on Dec 14th, 2007, 2:43am, cool_joh wrote: How can we prove that? As far as I know, we don't have to; it's the definition of e. IP Logged Hippo Uberpuzzler     Gender: Posts: 919 Re: Irrationality of e   « Reply #5 on: Dec 17th, 2007, 12:52pm » Quote Modify on Dec 14th, 2007, 2:43am, cool_joh wrote: How can we prove that?   Suppose e is such a constant that (ex)'=ex. Than we can approximate ex in 0 by the "infinite polynomial" such that it agrees in each derivative in 0. The polynomial is \sumk=0\inftyxk/k!. We can show the error of approximation goes to 0 for an arbitrary large x as more and more sum elements are added together.   Therefore ex=\sumk=0\inftyxk/k! Special case e=e1=\sumk=0\infty1k/k!. « Last Edit: Dec 17th, 2007, 12:57pm by Hippo » IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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