Representing Computations on Transcendental Numbers

Consider the set of transcendental numbers that are not compressible to a finite base-2 representation. How can I compute multiples (more generally, any algebraic computation) of one of these numbers, while maintaining an error less than $\epsilon$?

I'm curious because I know that I can approximately store a transcendental number in a finite computer, however with each subsequent operation that I perform on the preceding result, starting with the original number, my result becomes less accurate against what it's meant to approximate.

That is, assuming I have stored an approximation to $\pi$, $\pi'$, when I multiply it by two I'm left with $2\pi'$ whose error from $2\pi$ is, presumably, twice as large as the original error between $\pi'$ and $\pi$. (And so on, multiplying by two, my error doubles each time.)

What are the usual methods of reducing this error when using transcendental numbers in algebraic computation?

• Real algebraic numbers can be manipulated exactly (e.g. Tarski arithmetic). If you need transcendental numbers, however, there are two usual techniques, namely continued fractions and linear fractional transformations (a.k.a. Möbius transformations), and a few other less-common techniques (e.g. B-adic numbers). Here's a good start: hal.inria.fr/inria-00075792/document – Pseudonym Dec 14 '15 at 0:24
• Thanks for the correction. I've edited the post to read transcendental instead of irrational and will take a look at the link. A preliminary overview of the problem and solutions is still appreciated. – geofflittle Dec 14 '15 at 0:56
• – user12859 Dec 14 '15 at 2:01