The signed binary and dyadic digit streams described in sections 3.3.1 and 3.3.2 represent reals in the range [-1,1]. In fact the choice of [-1,1] as endpoints is arbitrary, as we could easily interpret a given stream as a real in any closed interval.

To illustrate, suppose we have a numeral *x* which we would
interpreted as the number . We could just as
easily interpret *x* as a number in the range [*a*,*b*] as

When it is necessary to represent reals on the whole real line, however, the stream representation alone is not sufficient. Perhaps the simplest solution to this problem is to introduce a (mantissa, exponent) representation similar to a floating point number. We do this using a mantissa which is a stream of either signed binary or dyadic digits, and and exponent is an integer of unrestricted size. For simplicity, the exponent represents a power of two, although powers of other numbers could also be used.

Hence, given a numeral *x* with mantissa *m* and exponent *e*, we
have:

Most algorithms for operations on reals represented using a mantissa and exponent in this way reduce to two simple and independent operations on the mantissa and exponent.