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
Algebraic number may consider numbers "explicitly" defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as or ln 2. An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are , and . Note, therefore, that the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all In this sense, algebraic integers are to algebraic numbers what integers are