[Tutor] Floating Confusion

[Kent Johnson]

Luke Paireepinart wrote:

> I've always wondered:
> There are numbers in Decimal that can't be represented accurately in 
> Binary without infinite precision.
> But it doesn't go the other way.  Rational base-2 numbers are rational 
> base-10 numbers.  I'm supposing this is because base-10 is a higher base 
> than base-2.

It is because 10 is an exact multiple of 2, not just because it is 
larger. The place values of a decimal fraction are (1/10)**n. The place 
values of a binary fraction are (1/2)**n or (5/10)**n. So any binary 
fraction can be rewritten as a decimal fraction.

> So I've wondered that, even though Pi is an irrational number in 
> base-10, is it possible that it's a simple (rational) number in a higher 
> base?

No. First, rational vs irrational has nothing to do with whether it can 
be represented in a particular base. 1/7 is rational but has no finite 
representation as a decimal fraction.

pi is a transcendental number which means it is not the solution of any 
polynomial equation with rational coefficients. In particular, it is not 
rational. But any number that can be represented as a 'decimal' fraction 
in any (rational) base is rational so this is not possible with pi.

> I mean, I suppose a base of Pi would result in Pi being 1, but what 
> about integer bases?
> Is there some kind of theory that relates to this and why there's not a 
> higher base with an easy representation of Pi?

http://en.wikipedia.org/wiki/Transcendental_numbers

Kent