I have been giving the challenge to proof the natural logarithm function is irrational. I'm trying to proof it by contradcition but I seem to be getting nowhere. Can someone hint me how to?
Thank you.
What do you mean by saying that the whole function is irrational?
This short PDF document was the first in Google results, and it seems pretty good.
Thanks, this slipped my mind somehow.A function f(x) is said to be 'rational' if it is computable with a finite number of 'elementary operations' +,-,* and \... all other functions are said to be 'irrational'...
Then are we allowed to use irrational constants to build rational functions? I.e., if a function is a ratio of two polynomials, can the polynomials' coefficients be irrational? If no, then the document provided above is still sufficient because it shows that $\displaystyle \ln n\notin\mathbb{Q}$ for $\displaystyle n\in\mathbb{Z}$.
From the 'theoretical' point of view, if You have a computer capable to menage irrational numbers [i.e. it has a memory of 'infinite dimension' and an infinite 'speed of computation'...], the ratio of two polynomials can be computed with a finite sequence of elementary operations...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
I am asking whether rational functions can use irrational constants with respect to the OP's problem. If irrational numbers are not allowed, then the problem has been solved. Otherwise, more thought is needed, but I don't want to go there if it's not necessary.
Here is discussion of rational functional.
Use the discussion tab at the top of that page.
Assume $\displaystyle \ln x = \frac{a_{n}x^{n} + ... + a_{1}x + a_{0}}{b_{m}x^{m} + ... + b_{1}x + b_{0}}$ then $\displaystyle x = e^{\frac{a_{n}x^{n} + ... + a_{0}}{b_{m}x^{m}+...+b_{0}}$.
See how this yields a contradiction?