Re: prove it if you can
On 02-11-2005 11:24, vaishakh wrote:
This is an interesting question but not that easy to be proved. The statement
is that an irrational number again with an irrational power can be rational.
It is very easy to point out at the numbers in natural logarithm table
developed to the base 'e' which is irrational and all the values except log1
are irrational. But, can you prove this using simple algebra rather than an
example. What I need is a concrete proof, which one can write even without the
logarithm table. The proof should be also convincing. Statements such as 'the
summation of this series is irrational since each term adds a new digit to the
actual value' will not at all be considered since they have no mathematical
certainty.
Let _s_ be the square root of 2. I don't know whether s^s is rational or
not. If it is, then you're done. Otherwise, consider
(s^s)^s = s^{s^2} = s^2 = 2.
So, if s^s is irrational, you have the example that you want.
Best regards,
Jose Carlos Santos
.
