We introduce a representation of the computable real numbers by continued fractions. This deals with the subtle points of undecidable comparison an integer division, as well as representing the infinite 1/0 and undefined 0/0 numbers. Two general algorithms for performing arithmetic operations are introduced. The algebraic algorithm, which computes sums and products of continued fractions as a special case, basically operates in a positional manner, producing one term of output for each term of input. The transcendental algorithm uses a general formula of Gauss to compute the continued fractions of exponentials, logarithms, trigonometric functions, as well as a wide class of special functions. A prototype system has been implemented in LeLisp, and the performance of these algorithms is promising.