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Lawrence
guest

May0509, 08:11 AM (EST) 

"geometric proof"

Dear Alexander Bogomolny, I love your 'cut the knot website', which is indeed both accurate, serious, and authoritative in your mathematical literature which I find very suitable for both the young's and old readers. I have a question and wonder whether you have encountered any simpler proof which comes a geometrical interpretation of the below theorem found in your webpage. gcd(N,M) * lcm(N,M) = N * M Noting the fact that this theorem is only true for set of two numbers ( only true for twodimensional situation)I wonder whether we can use the area of a rectangle with dimension of their length to represent N and M. Thanks very much Gratefully Yours Lawrence 

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lawrwence
guest

Jun0109, 08:31 PM (EST) 

3. "log problem"
In response to message #0

Dear Alexander Bogomolny, I have an interesting problem and hope you can shed some light on it I wonder whether you have encountered any simpler proof which comes with a geometrical interpretation of the below theorem . loga to the base b times logb to the base a equals 1 Noting the fact that this theorem is only true for set of two numbers (only true for twodimensional situation) I wonder whether we can use the area of a rectangle with dimensions of their length and breadth represented by loga to the base a and logb to the base a. I also wonder whether it is possible to do some form of transformation to the area under the log curve bounded by x=a and x=b to a rectangle whose sides is always a reciprocal of each other so that their product equal 1; Thanks very much Gratefully Yours Lawrence


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alexb
Charter Member
2382 posts 
Jun0109, 11:07 PM (EST) 

4. "RE: log problem"
In response to message #3

Off the top of my hat, I do not know of a geometric argument. I'll post it here if anything comes to mind. However, that identity holds for any number of factors, e.g., log_{a}b × log_{b}c × log_{c}a = 1. Also, the identity is an immediate consequence of the equivalence of a^{k} = b and b^{1/k} = a. What can be easier? 

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