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wu :: forums    riddles    medium (Moderators: Eigenray, towr, SMQ, Grimbal, Icarus, ThudnBlunder, william wu)    LCM « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: LCM  (Read 655 times) Christine Full Member     Posts: 159 LCM   « on: Jun 7th, 2015, 2:51pm » Quote Modify What is the LCM of  sqrt(2) and sqrt(3) ?   I was told to use continued fractions, but I don't know how to do it. Help! « Last Edit: Jun 7th, 2015, 2:52pm by Christine » IP Logged pex Uberpuzzler     Gender: Posts: 880 Re: LCM   « Reply #1 on: Jun 7th, 2015, 5:21pm » Quote Modify on Jun 7th, 2015, 2:51pm, Christine wrote: What is the LCM of  sqrt(2) and sqrt(3) ? Is this even well-defined? Assuming LCM is least common multiple, which is commonly defined for integers only. I can see how to extend the concept to rationals, but for irrational numbers...   I suppose a common multiple would be a positive number m such that m = a*sqrt(2) = b*sqrt(3). What are a and b allowed to be? Clearly they can't both be integers, since sqrt(2/3) is irrational. But if they're not integers, nothing prevents us from replacing them by m/2 = (a/2)sqrt(2) = (b/2)sqrt(3) and no least common multiple exists. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: LCM   « Reply #2 on: Jun 7th, 2015, 10:28pm » Quote Modify Perhaps we need to find some approximate LCM, within a certain tolerance. Like: m = a*2 = b*3 with |a*2 - [a*2| < 0.01 and |b*3 - [b*3]| < 0.01, and a,b,m integers   So, for example   9513*3 = 16476.9993 11651*2 = 16477.0022 So, 16477 might be considered the LCM for 2 and 3 with a given tolerance.     [edit] Or perhaps it's simpler to consider: |m - a*2| < t and |m - b*3| < t, and a, b, m integers and some tolerance t e.g m=134421, a=95050, b=77608, t~=0.0021 [/edit] « Last Edit: Jun 7th, 2015, 11:07pm by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB pex Uberpuzzler     Gender: Posts: 880 Re: LCM   « Reply #3 on: Jun 7th, 2015, 11:00pm » Quote Modify ... and good rational approximations of irrational numbers are commonly found using continued fractions, so using towr's interpretation, the hint actually makes sense. Good mind reading! IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: LCM   « Reply #4 on: Jun 7th, 2015, 11:17pm » Quote Modify Quote: Good mind reading! It helped that I googled a bit, I found this, for example. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB 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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