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.
Rahul's comment essentially kills off any hope of a natural answer to your question. However, there is a very silly answer.
It's not hard to show that the structure $(\mathbb{Comp}, +)$ (where "$\mathbb{Comp}$" denotes the set of computable reals) is isomorphic to the additive group of the ring $\mathbb{Q}[\pi]$. The latter has a natural computable presentation, hence the former has a (not very natural) computable presentation; that is, there is a structure $(A, \oplus)$, where $A$ is a computable set of natural numbers and $\oplus$ is a total computable binary function, which is isomorphic to $(\mathbb{Comp}, +)$.
The problem, here, is that the unfolding of this interpretation is hard: there is no uniform algorithm to take an element of $A$ and output its decimal expansion (for basically the reason Rahul mentioned in their comment).