Counting sort initialization of the output array, each take time. Therefore, the time for the whole algorithm is the sum of the times for these steps, . Because it uses arrays of length and , the total space usage of the algorithm is also . For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array which uses space . If each item to be sorted is itself an integer, and used as
Non-integer representation field extension of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion must lie in Q(β). On the other hand, the converse need not be true. The converse does hold if β is a Pisot number , although necessary and sufficient conditions are not known. Non-integer representation A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer radix β > 1, the value of is The numbers "d" are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by and