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March 9th, 2015, 06:27 PM   #1
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From: Lake Placid

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proof of Irrationality of e

I'm an AP Calc BC student. We just learned Euler's way to prove the irrationality of e. I just suddenly came up with this weird proof in class. But I couldn't find anything wrong with it. Can you guys check this proof and point out any mistake hopefully? Thanks a lot!

if e is rational, then e=m/n(simplest form), where m,n are unequal integers(n is not 0)

then ln(e)=ln(m/n)

then 1=ln(m/n)=logn(m)

that turns out m=n, which is not true and also contradicted to "m,n are unequal integers"

then e is irrational.
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March 9th, 2015, 06:47 PM   #2
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Are you saying $\displaystyle \ln\left(\frac{m}{n}\right)=\log_n{m}$ ?

If so, that isn't true ... $\displaystyle \ln\left(\frac{m}{n}\right) \ne \frac{\ln{m}}{\ln{n}}=\log_n{m}$

$\displaystyle \ln\left(\frac{m}{n}\right) = \ln{m}-\ln{n}$, remember?
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March 10th, 2015, 03:47 AM   #3
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Thanks a lot. I misused the change base formula
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