Consider a floating point set F defined by (1.1.1) and (1.1.2) with d 4. (a) How many elements of F are there in the real interval [1/2,4], including the endpoints? b) What is the element of F closest to the real number 1/10? (c) What is the smallest positive integer not in F? The real number set R is infinite in two ways: it is unbounded and continuous. In most practical computing, the second kind of infiniteness is much more consequential than the first kind, so we turn our attention there first. We replace R with the set F of floating point numbers, whose members are zero and all numbers of the form where e is an integer clle the epnent, nd 1+ f isthe mantiss, in which b E 0,1, for a fixed integer d. Equation (1.1.2) represents the mantissa as a number in [1,2) in base-2 form. Equivalently, i-1 for an integer z in the set (0, 1,...2d- Consequently, starting at 2 and ending just before 25-+1 there are exactly 2d evenly spaced numbers belonging to F