Clay Kitchings ::
EMAT 6600 :: Rational or Irrational?
This
problem was also inspired from MATH 6000 at UGA in the same semester I took
EMAT 6600. Most of us (mathematics
educators and mathematicians) have proved that is irrational.
This problem is on the EMAT 6600
website of problems.
However, I
wondered whether the same pattern (or shell of a proof) could be followed for
numbers other than in order to
verify that they are irrational. After seeing a proof for , I conjectured that all primes could be proven irrational
similarly.
Problem:
Show that is irrational
for any prime number p.
We shall
prove it is irrational by contradiction.
So, letŐs suppose that is rational and
that it can be written as (We insert
the relatively prime requirement to basically say we can write the rational
number in lowest terms.)
This
implies:
It is worth
noting that other irrational numbers may be proven irrational by somewhat
similar methods. It turns out that
proof by contradiction can be a powerful tool in this endeavor, even for
logarithms that are irrational.