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?