Implementing exact real numbers in Maple

In Maple, the system variable Digits controls the relative precision of a computation. In most of the applications, however, the measure of precision is the accuracy:

Definition 1   Let $r$ be a real number, and $\varepsilon>0$. A rational number $r_0$ is an approximation of accuracy $\varepsilon$ for $r$ (sometimes called $\varepsilon$-approximation) iff

$$\vert r-r_0\vert\le \varepsilon.$$

Our representation of real numbers is based on the concept of accuracy. From the software engineering point of view, an exact real number $r$ is an object that fulfills the following:

• for every $\varepsilon>0$, it should be possible to compute an $\varepsilon$-approximation of $r$;
• if an approximation of accuracy $\varepsilon$ is known, an approximation of accuracy $\varepsilon/c$ should be easily obtainable, for rational $c>1$;
• the most precise approximation already computed should be instantly available, together with its accuracy;
• the user should be able to handle real numbers without necessarily computing more or less accurate approximations of them.

These specifications can be extended to complex numbers, or to vectors/matrices with real entries (the absolute value is replaced now by a suitable norm). In general, we will call such an object erna object (erna real, erna complex, etc.). Erna stands for exact real number arithmetic, a common appellative for techniques for manipulating computable reals.