Factorial factor is an operation symbol invented by Keyston Kramp (1760 ~1826) in 1808, and it is a mathematical term.

The factorial of a positive integer is the product of all positive integers less than or equal to this number, and the factorial of 0 is 1. The factorial writing of natural number n! . In 1808, Keyston Kaman introduced this symbol.

N factorial representation of any natural number greater than or equal to 1;

? Still n! = 1×2×3×...×n？ Or?

Factorial of 0? 0! = 1。

Necessity of definition

Because the factorial of a positive integer is a continuous multiplication operation, the result of multiplying 0 by any real number is 0. Therefore, we can't generalize or deduce 0 with the definition of positive integer factorial! = 1. That is, "0! = 1"。 Give "0!" The definition is only to facilitate the expression and operation of related formulas. In the definition of combination number in discrete mathematics, it is meaningful for any positive integer that meets the conditions, especially in time.

. However, for the combination number formula, it is a great embarrassment because there is no factorial of 0 defined in time. According to the conclusion and formula, we define "0! = 1 "is very necessary. In this way, the combination number formula will be unimpeded in time and there will be no embarrassment.

The universality of use (1) explicitly uses "0! = 1 ",without which it can only be expressed as.

(2) As a continuation of factorial, γ function is a meromorphic function defined in the range of complex numbers, and there are also β functions closely related to it (called Euler second integral and Euler first integral respectively).

It is just a factorial symbol on the special "form" defined. It can't be demonstrated by deduction.

"Why 0! = 1 "is a pseudo question that beginners always ask. This shows that we haven't put "about' 0! = 1' is just a concept of' definition'.

Of course, it is a big mistake to regard the above necessity and rationality as the derivation process of some teaching AIDS. Necessity and rationality are just a few limited examples, "0! The definition of = 1 "cannot be proved by a few examples.

But the use of this definition has been tested and convenient so far, and there has never been any logically unreasonable phenomenon.

defining range

Usually, the factorial we are talking about is defined in the range of natural numbers (most scientific calculators can only calculate the factorial of 0 ~ 69), and decimal scientific calculators have no factorial function, such as 0.5! ,0.65! ,0.777! It's all wrong But sometimes we define the Gamma function as the factorial of non-integers, because when x is a positive integer n, the value of the Gamma function is the factorial of n- 1