Re: Irrational numbers questions

In article <459bd8d2$0$2021$ba620dc5@xxxxxxxxxxxxxxxxxxx> Herman Jurjus <h.jurjus@xxxxxxxxx> writes:
*** T. Winter wrote:
> Assume now that ln(N) is rational = m/n . Then e^m = N^n where the left
> hand side is irrational and the right hand side is rational. Giving a
> contradiction. Thus ln(N) is irrational. Right?

See above. False. You really need that e is transcendental.

Which begs the question: is there an x such that
x, x^2, x^3, ...
are all irrational, but x is not transcendental?

You are right, as each quadratic field shows. Most algebraic numbers
have that property. But to get at a proof that e^m for integer n is
irrational it is (I think) easier to show that e is transcendental than
to show that it is algebraic with certain additional properties.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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