Euler's number, e, was the first to have been proved to be transcendental (thus, all the more so, irrational).
e^(ln(2)) = 2 by definition.
But is ln(2) irrational?
The equation above could be written:
2^(1/ln(2)) = e
If ln(2) were rational, then so would be 1/ln(2). But then e would be an integral root of an integral power of 2, and thus not transcendental. So ln(2) is irrational also, and e^(ln(2)) represents an irrational number raised to an irrational power that gives a rational result.
Posted by Charlie
on 2004-12-20 17:11:30