> From: David Hobby <[EMAIL PROTECTED]> > > > > > 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...
The real question is should you use Rational Integer exponents in an Irrational number system. Somehow that seems...wrong. Lets use a number that is bigger than pi this time, say 33.
