I want to represent (computable) real numbers in such a way that addition is computable, i.e. there exists a Turing machine $M(x, y, n)$ which halts with the $n$th digit of $x + y$ on its tape.
The most obvious way to encode real numbers is to lead off with a sign bit, i.e. the first digit of a number is 1 if the number is negative or 0 if it is positive. But it seems to me that this will result in addition being non-computable, because $M$ would need to examine an arbitrary number of digits in order to determine if $(x +\epsilon) - x$ is positive.
One thing I could do is to take a continuous bijective function $f:\mathbb R\to[0, 1]$ and encode $x$ as $f(x)$. However, this is very different than how I usually think about the representation of numbers.
Is there a more straightforward way to represent real numbers such that addition is computable? Since all computable functions are continuous, and the sign function is discontinuous, it seems like my intuition about positive versus negative numbers is incompatible with computability.