Factorial

Figure 1-5

n{displaystyle n}	n!{displaystyle n}
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40.320
9	362.880
10	3,628,800
15	1.307.674.368,000
20	2.432 e.008.176.640.000
25	15.511.210.043.330.985.984,000.000
50	30.414.093.201.713.378.043 × 10^45
70	1,19785717... × 10^100
450	1,73336873... × 10^1000
3.249	6,41233768... × 10^10 000
25.206	1,205703438... × 10^100 000
100,000	2,8242294079... × 10^456 573

The factorial of a positive integer n, the factorial of n or n factorial is defined in principle as the product of all positive integers from 1 (ie the natural numbers) to n. For example:

5!=1× × 2× × 3× × 4× × 5=120.{displaystyle 5!=1times 2times 3times 4times 5=120. }

The factorial operation appears in many areas of mathematics, particularly combinatorics and mathematical analysis. Basically, the factorial of n represents the number of different ways of ordering n different objects (elements without repetition). This fact has been known for several centuries, in the 12th century by Hindu scholars.

The definition of the factorial function can also be extended to non-natural numbers while maintaining their fundamental properties, but advanced mathematics is required, particularly mathematical analysis. French mathematician Christian Kramp (1760-1826) was the first person to use the current mathematical notation n!, in 1808.

Definition by product and induction

We can define the factorial of a positive integer n, expressed n!, as the product of all positive integers less than or equal to n.

n!=1× × 2× × 3× × 4× × ...× × (n− − 1)× × n{displaystyle n!=1times 2times 3times 4times...times (n-1)times n}.

The above multiplication can also be represented using the producer operator:

n!= k=1nk{displaystyle n!=prod _{k=1}^{n}k}.

It is also possible to define it by the recurrence relation

<math alttext="{displaystyle n!={begin{cases}1&{text{si, }}n1end{cases}}}" xmlns="http://www.w3.org/1998/Math/MathML">n!={1Yeah,n.2(n− − 1)!× × nYeah,n▪1{displaystyle n!={begin{cases}1 fake{text{si, }}n vis2(n-1)!times n pretend{text{si, }n rigid1end{cases}}}}}}}<img alt="{displaystyle n!={begin{cases}1&{text{si, }}n1end{cases}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df01318a5eb46249679bca02374522a590ea4aa" style="vertical-align: -2.505ex; width:30.492ex; height:6.176ex;"/>

In this second definition the domain of the function is the set of non-negative integers ℤ_≥0 and the codomain is the set of positive integers ℤ₊. In this case there is a recurring sequence, the successive calculation of its elements is called recursive process and the equality n! = (n - 1)!n is called a recursive equation.

The second definition incorporates the premise that

0!=1{displaystyle 0!=1}

Zero factorial

A common extension, however, is the definition of factorial of zero. According to the empty product mathematical convention, the value of 0! should be defined as:

0!=1{displaystyle 0!=1}

It is possible, however, to give an intuitive argument to justify the choice, as follows:

• For each positive integer n greater or equal to 1, it is possible to determine the value of the previous factorial by using the following identity:

[chuckles](n− − 1)!=n!n]=[chuckles]n⋅ ⋅ (n− − 1)!=n!]{displaystyle {bigg}(n-1)!={frac {n}{n}{bigg}}=[ncdot (n-1)!

valid for any number greater than or equal to 1.

Thus, if it is known that 5! is 120, so 4! it's 24 because

5!5=1205=24{displaystyle {frac {5}{5}}}{frac {120}{5}=24}

and therefore 3! must necessarily be 6 since

4!4=244=6{displaystyle {frac {4}{4}}}{frac {24}{4}=6}

The same process justifies the value of 2! = 2 and 1! = 1 since:

2!=3!3=63=2,1!=2!2=22=1{displaystyle 2!={frac {3}}}={frac {6}}}=2,qquad 1!={frac {2}{2}}}={frac {2}{2}}{2}}}=1}

If we apply the same rule to the case where n = 1 we would have 0! corresponds to:

0!=1!1=11=1{displaystyle}={frac {1}{1}}={frac {1}{1}}=1}

Although the argument may be somewhat convincing, it's important to note that it's just an informal argument, and the actual reason for taking the 0! = 1 is because it is a special case of the empty product convention used in many other branches of mathematics.

Properties

• Yeah. m and n are positive integers and m. n implies that m! n!
• Yeah. m. n Turns out mIt's factor or divider which... nAnd you have: n! = n(n-1)...(m+1).m(1)
• Number n(n-1)...(m+1) is the product of the n-m higher exposed factors n!
• n-m is less than n and replacing in (1) you get n! = n(n-1)...(n-m+1).(n-m!
• <math alttext="{displaystyle n!n!.(n+12)n{displaystyle n}{left({frac {n+1}{2}}}right)^{n}}}<img alt="{displaystyle n!, for n. It applies property that the geometric mean of the first positive integers does not exceed the arithmetic mean of them.

Applications

Factorials are widely used in the branch of mathematics called combinatorics, through the Newton binomial, which gives the coefficients of the expanded form of (a + b)ⁿ:

(a+b)n=(n0)an+(n1)an− − 1b+(n2)an− − 2b2+ +(nn− − 1)abn− − 1+(nn)bn=␡ ␡ k=0n(nk)an− − kbk{displaystyle(a+b)^{n}={n choose 0}a^{n}{n +{n choose 1}a^{n-1}b+{n choose 2}a^{n}{n}{nx}{n}{nx}{n}{nx}{n}{nx}{nx}{n}{n}{n}{n

where (nk){displaystyle {n choose k} represents a binomial coefficient:

(nk)=n!(n− − k)!⋅ ⋅ k!{displaystyle {n choose k}={frac {n!}{(n-k)!cdot k}}}}}}

Similarly, it can be found in the derivation by the product rule for derivatives of higher order in a similar way to Newton's binomial:

dnxdxn(f(x)g(x))=(fg)(n)=(n0)fg(n)+(n1)f♫g(n− − 1)+(n2)f♫g(n− − 2)+ +(nn− − 1)f(n− − 1)g♫+(nn)f(n)g=␡ ␡ k=0n(nk)f(k)g(n− − k)♪♪ !

Where f⁽ⁿ⁾ is the nth derivative of the function f.

By means of combinatorics, the factorials intervene in the calculation of probabilities. They also intervene in the field of analysis, in particular through the polynomial development of functions (Taylor's formula). They are generalized to the reals with the gamma function, of great importance in number theory.

For large values of n, there is an approximate expression for the factorial of n, given by Stirling's formula:

n!≈ ≈ 2π π n(ne)n(1+112n+1288n2+ ){displaystyle n!approx {sqrt {2pi n}}left({frac {n}{e}{e}right)^{n}left(1+{frac {1}{12n}}}}+{frac {1}{288n^{2}}}+cdots right)}}}}

The advantage of this formula is that it does not require induction, and therefore allows n! to be evaluated faster when n is larger.

Extension

The given definition of factorial is valid for nonnegative numbers. It is possible to extend the definition to other contexts by introducing more sophisticated concepts, especially it is possible to define it for any real number except for negative integers and for any complex number except again for negative integers. The factorial of n is generalized to any real number n by the gamma function such that

Interpreter Interpreter (n)=(n− − 1)!=∫ ∫ 0∞ ∞ tn− − 1e− − tdt{displaystyle Gamma (n)=(n-1)!=int _{0}^{infty };t^{n-1}e^{-t};dt,}

for n > 0. It can be further generalized, for every complex number z that is not equal to a non-positive integer, by the following definition:

Interpreter Interpreter (z)=(z− − 1)!=limn→ → ∞ ∞ n!nzz(z+1) (z+n){displaystyle Gamma (z)=(z-1)!=lim _{nto infty }{frac {n!;n^{z}}{z;(z+1)cdots (z+n)}}{,}

Similar products

Primary

The primorial (sequence A002110 in OEIS) is defined in a similar way to the factorial, but only the product of prime numbers less than or equal to n is taken:

n# # = p≤ ≤ npCousinp{displaystyle n#=prod _{pleq n atop p{text{ prime}}p}.

Double factorial

The double factorial of n is defined by the recurrence relation:

n!!={1Yeah.n=0(n− − 2)!!⋅ ⋅ nYeah.nI was. I was. 0{textstyle n!}=left{begin{array}{lcl}1 fake{mbox{si}}}{bn=0(n-2)!cdot n stranger{mbox{si}{si}{neq 0end{array}}}right. !

For example:

8!!=2⋅ ⋅ 4⋅ ⋅ 6⋅ ⋅ 8=384{displaystyle 8!=2cdot 4cdot 6cdot 8=384}

9!!=1⋅ ⋅ 3⋅ ⋅ 5⋅ ⋅ 7⋅ ⋅ 9=945{displaystyle 9!=1cdot 3cdot 5cdot 7cdot 9=945}

The sequence of double factorials (sequence A006882 in OEIS) for:

n=0,1,2,...... {displaystyle n=0,1,2,dots }

Start like this:

1,1,2,3,8,15,48,105,384,945,3840,...... {displaystyle 1,1,2,3,8,15,48,105,384,945,3840,dots }

The above definition can be extended to define the double factorial of negative numbers:

(n− − 2)!!=n!!n{displaystyle(n-2)}!

And this is the sequence of double factorials for:

n=− − 1,− − 3,− − 5,− − 7,...... {displaystyle n=-1,-3,-5,-7,dots }

1,− − 1,13,− − 115,...... {displaystyle 1,-1,{frac {1}{3}},-{frac {1{15}}},dots }

The double factorial of an even negative number is undefined.

Some identities of the double factorials:

1. n!=n!!(n− − 1)!!{displaystyle n!=n!(n-1)!,}
2. (2n)!!=2nn!{displaystyle (2n)!=2^{n}n!
3. (2n+1)!!=(2n+1)!(2n)!!=(2n+1)!2nn!{displaystyle (2n+1)!={(2n+1)! over (2n)!}={(2n+1)! over 2^{n}n!}
4. (2n− − 1)!!=(2n− − 1)!(2n− − 2)!!=(2n)!2nn!{displaystyle (2n-1)!={(2n-1)! over (2n-2)!}={(2n)! over 2^{n}n!}
5. Interpreter Interpreter (n+12)=π π (2n− − 1)!!2n{displaystyle Gamma left(n+{1 over 2}right)={sqrt {pi}},,{(2n-1)! over 2^{n}}}}}}}
6. Interpreter Interpreter (n2+1)=π π n!!2(n+1)/2{displaystyle Gamma left({n over 2}+1right)={sqrt {pi}},,{n!! over 2^{(n+1)/2}}}}}}}}

References and citations