Number factorial – the mathematical concept applicable only for the whole non-negative numbers. This size represents the work by all natural number from 1 to the factorial basis. The concept finds application in combination theory, the theory of numbers and the functional analysis.
1. To find a number factorial, it is necessary to calculate the work of all numbers, in an interval from 1 to the set number. The general formula looks thus: n! = 1*2 * …*n where n is any whole non-negative number. The factorial can be designated by the exclamation mark.
2. Main properties of factorials: • 0! = 1; • n! = (n-1) N*!; • n! ^2 ≥ n^n ≥ n! ≥ n. The second property of a factorial is called a recursion, and a factorial – elementary recursive function. Recursive functions are often applied in the theory of algorithms and in writing of computer programs as many algorithms and functions of programming have recursive structure.
3. It is possible to determine a factorial of a large number by a formula Stirlinga which gives, however, approximate equality, but with a small error. The full formula looks as follows: n! = (n/e) ^n * √ (2*π*n) * (1 + 1 / (12*n) + 1 / (288*n^2) + …) ln (n!) = (n + 1/2) *ln n – n + ln √ (2*π), where e – the basis of a natural logarithm, number of Euler which numerical value is accepted approximately equal 2.71828 …; π – a mathematical constant which value is accepted equal 3.14. Use of a formula Stirlinga in a look is widespread: n! ≈ √ (2*π*n) * (n/e) ^n.
4. There are various generalizations of a concept of a factorial, for example, double, m-fold, decreasing, increasing, praymoriat, a superfactorial. The double factorial is designated!! also it is equal to the work of all natural numbers on an interval from 1 to the number, having the same parity, for example, 6!! = 2*4*6.
5. a m-fold factorial – the general case of a double factorial for any whole non-negative number m: for n = mk – r is fair n!...!! = ∏ (m*I - r) where r – a set of integers from 0 to m-1, I – belongs to a set of numbers from 1 to k.
6. The decreasing factorial registers as follows: (n_k = n! / (n - k)! Increasing: (n) ^k = (n + k-1)! / (n - 1)!
7. Praymorial of number is equal to the work of prime numbers less than the number and is designated #, for example: 12# = 2*3*5*7*11, it is obvious that 13# = 11# = 12#.Суперфакториал it is equal to the work of factorials of numbers on an interval from 1 to initial number, i.e.: sf(n) = 1! *2! *3 * … (n - 1)! *n!, for example, sf(3) = 1! *2! *3! = 1*1*2*1*2*3 = 12.