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        riddles >> medium >> LCM
        
(Message started by: Christine on Jun 7^th, 2015, 2:51pm)

Title: LCM
Post by Christine on Jun 7^th, 2015, 2:51pm

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!

Title: Re: LCM
Post by pex on Jun 7^th, 2015, 5:21pm

on 06/07/15 at 14:51:42, 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.

Title: Re: LCM
Post by towr on Jun 7^th, 2015, 10:28pm

Perhaps we need to find some approximate LCM, within a certain tolerance. Like: m = a*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 = b*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 with |a*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 - [a*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2| < 0.01 and |b*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 - [b*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3]| < 0.01, and a,b,m integers

So, for example
9513*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 = 16476.9993
11651*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 = 16477.0022
So, 16477 might be considered the LCM for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 with a given tolerance.

[edit]
Or perhaps it's simpler to consider: |m - a*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2| < t and |m - b*http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3| < t, and a, b, m integers and some tolerance t
e.g m=134421, a=95050, b=77608, t~=0.0021
[/edit]

Title: Re: LCM
Post by pex on Jun 7^th, 2015, 11:00pm

... 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! :)

Title: Re: LCM
Post by towr on Jun 7^th, 2015, 11:17pm

Quote:

Good mind reading!

It helped that I googled a bit, I found this (http://mathforum.org/kb/message.jspa?messageID=452408), for example.