>
> Here's a question that popped into my head a few days ago..
>
> Are all quantities represented by base 10 integers irrational
> in base pi or base e?
> --
> #ken P-)}
By "irrational" you mean "have no representation other
than the trivial one which looks like the representation of a
rational number in base 10"?
For 3 is trivially 3*pi^0, or 3.0 in base pi. What you
mean is that it can't be written in any other way. Correct.
For example, if 3 were also 2.3011111... in base pi, then we
would have:
3 = 2 + 3*pi^-1 + (1*pi^-3 + 1*pi^-4 + ...) or, using the
standard trick for a geometric series on the repeating part,
3 = 2 + 3*pi^-1 + (1/(pi^3 - pi^2)) but then clearing fractions
we get,
3*pi*(pi^3 - pi^2) = 2*pi*(pi^3 - pi^2) + 3(pi^3 - pi^2) + pi
and pi is a root of a nontrivial polynomial in the rationals.
Since pi is transcendental, this is impossible. The
argument generalizes...
---David
[EMAIL PROTECTED]
