! (math)

Selected members of the factorial sequence (sequence A000142 in the OEIS); values specified in scientific notation are rounded to the displayed precision
n n!
0 1
1 1
2 2
3 6
4 24
5 120
6 720
7 5040
8 40320
9 362880
10 3628800
11 39916800
12 479001600
13 6227020800
14 87178291200
15 1307674368000
16 20922789888000
17 355687428096000
18 6402373705728000
19 121645100408832000
20 2432902008176640000
25 1.551121004×1025
50 3.041409320×1064
70 1.197857167×10100
100 9.332621544×10157
450 1.733368733×101000
1000 4.023872601×102567
3249 6.412337688×1010000
10000 2.846259681×1035659
25206 1.205703438×10100000
100000 2.824229408×10456573
205023 2.503898932×101000004
1000000 8.263931688×105565708
10100 1010101.9981097754820

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,

The value of 0! is 1, according to the convention for an empty product.[1]

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence. These arrangements are called the permutations of the set of objects.

The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.

History[]

Factorials were used to count permutations at least as early as the 12th century, by Indian scholars.[2] In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells.[3] After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):

Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body.[4]

The notation n! was introduced by the French mathematician Christian Kramp in 1808.[5]

Definition[]

The factorial function is defined by the product

initially for integer n ≥ 1. This may be written in the Pi product notation as

From these formulas, one may derive the recurrence relation

For example, one has

and so on.

Factorial of zero[]

In order for the recurrence relation to be extended to n = 0, it is necessary to define

so that

There are several independent reasons to view this definition as harmonious. These include the following:

.
More generally, the number of ways to choose all n elements among a set of n is
.

Factorial of a non-integer[]

The factorial function can also be defined for non-integer values using more advanced mathematics (the gamma function n! = Γ(n + 1)), detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica.

Applications[]

Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.

possibilities. This however produces the k-combinations in a particular order that one wishes to ignore; since each k-combination is obtained in k! different ways, the correct number of k-combinations is
This number is known[8] as the binomial coefficient (n
k
)
, because it is also the coefficient of xk in (1 + x)n.
while this is inefficient as a means to compute that number, it may serve to prove a symmetry property[7][8] of binomial coefficients:
where Dn xn is Euler's notation for the nth derivative of xn.[11]

Rate of growth and approximations for large n[]

Plot of the natural logarithm of the factorial

As n grows, the factorial n! increases faster than all polynomials and exponential functions (but slower than double exponential functions) in n.

Most approximations for n! are based on approximating its natural logarithm

The graph of the function f(n) = ln n! is shown in the figure on the right. It looks approximately linear for all reasonable values of n, but this intuition is false. We get one of the simplest approximations for ln n! by bounding the sum with an integral from above and below as follows:

which gives us the estimate

Hence ln n! ∼ n ln n (see Big O notation). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). From the bounds on ln n! deduced above we get that

It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all n we have (n/3)n < n!, and for all n ≥ 6 we have n! < (n/2)n.

Comparison of Stirling's approximation with the factorial

For large n we get a better estimate for the number n! using Stirling's approximation:

This in fact comes from an asymptotic series for the logarithm, and n factorial lies between this and the next approximation:

Another approximation for ln n! is given by Srinivasa Ramanujan (Ramanujan 1988)

Both this and the previous approximation give a relative error on the order of 1/n3, but Ramanujan's is about four times more accurate. However, if we use two correction terms (as in Ramanujan's approximation) the relative error will be of order 1/n5:

Computation[]

If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers up to n (if any) will compute n!, provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.

The main practical difficulty in computing factorials is the size of the result. To assure that the exact result will fit for all legal values of even the smallest commonly used integral type (8-bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers commonly used in personal computers, however many languages support variable length integer types capable of calculating very large values.[12] Floating-point representation of an approximated result allows going a bit further, but this also remains quite limited by possible overflow. Most calculators use scientific notation with 2-digit decimal exponents, and the largest factorial that fits is then 69!, because 69! < 10100 < 70!. Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.

Most software applications will compute small factorials by direct multiplication or table lookup. Larger factorial values can be approximated using Stirling's formula. Wolfram Alpha can calculate exact results for the ceiling function and floor function applied to the binary, natural and common logarithm of n! for values of n up to 249999, and up to 20000000! for the integers.

If the exact values of large factorials are needed, they can be computed using arbitrary-precision arithmetic. Instead of doing the sequential multiplications ((1 × 2) × 3) × 4..., a program can partition the sequence into two parts, whose products are roughly the same size, and multiply them using a divide-and-conquer method. This is often more efficient.[13]

The asymptotically best efficiency is obtained by computing n! from its prime factorization. As documented by Peter Borwein, prime factorization allows n! to be computed in time O(n(log n log log n)2), provided that a fast multiplication algorithm is used (for example, the Schönhage–Strassen algorithm).[14] Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.[15]

Number theory[]

Factorials have many applications in number theory. In particular, n! is necessarily divisible by all prime numbers up to and including n. As a consequence, n > 5 is a composite number if and only if

A stronger result is Wilson's theorem, which states that

if and only if p is prime.[16][17]

Legendre's formula gives the multiplicity of the prime p occurring in the prime factorization of n! as

or, equivalently,

where sp(n) denotes the sum of the standard base-p digits of n.

Adding 1 to a factorial n! yields a number that is divisible by a prime larger than n. This fact can be used to prove Euclid's theorem that the number of primes is infinite.[18] Primes of the form n! ± 1 are called factorial primes.

Series of reciprocals[]

The reciprocals of factorials produce a convergent series whose sum is the exponential base e:

Although the sum of this series is an irrational number, it is possible to multiply the factorials by positive integers to produce a convergent series with a rational sum:

The convergence of this series to 1 can be seen from the fact that its partial sums are less than one by an inverse factorial. Therefore, the factorials do not form an irrationality sequence.[19]

Extension of factorial to non-integer values of argument[]

The gamma and pi functions[]

The gamma function interpolates the factorial function to non-integer values. The main clue is the recurrence relation generalized to a continuous domain.

Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis.

One function that "fills in" the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted Γ(z). It is defined for all complex numbers z except for the non-positive integers, and given when the real part of z is positive by

Its relation to the factorial is that for any natural number n

Euler's original formula for the gamma function was

Another function that also "fills in" the values of the factorial (but with no shift in the argument), originally introduced by Carl Friedrich Gauss, that is sometimes used, is called the pi function, denoted Π(z) for real numbers z ≥ 0. It is defined by

In terms of the gamma function, the pi function is

The factorial function, generalized to all real numbers except negative integers. For example, 0! = 1! = 1, (−1/2)! = π, 1/2! = π/2.

The pi function properly extends the factorial in that

In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value z where it is defined

In fact, this is no longer a recurrence relation but a functional equation. Expressed in terms of the gamma function this functional equation takes the form

Since the factorial is extended by the pi function, for every complex value z where it is defined, we can write:

The values of these functions at half-integer values is therefore determined by a single one of them; one has

from which it follows that for nN,

For example,

It also follows that for nN,

For example,

The pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is analytic wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function. For instance, the Bohr–Mollerup theorem states that the gamma function is the only function that takes the value 1 at 1, satisfies the functional equation Γ(n + 1) = nΓ(n), is meromorphic on the complex numbers, and is log-convex on the positive real axis. A similar statement holds for the pi function as well, using the Π(n) = nΠ(n − 1) functional equation.

However, there exist complex functions that are probably simpler in the sense of analytic function theory and which interpolate the factorial values. For example, Hadamard's 'gamma' function (Hadamard 1894) which, unlike the gamma function, is an entire function.[20]

Euler also developed a convergent product approximation for the non-integer factorials, which can be seen to be equivalent to the formula for the gamma function above:

However, this formula does not provide a practical means of computing the pi function or the gamma function, as its rate of convergence is slow.

Applications of the gamma function[]

The volume of an n-dimensional hypersphere of radius R is

Factorial at the complex plane[]

Amplitude and phase of factorial of complex argument

Representation through the gamma function allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let

Several levels of constant modulus (amplitude) ρ and constant phase φ are shown. The grid covers the range −3 ≤ x ≤ 3, −2 ≤ y ≤ 2, with unit steps. The scratched line shows the level φ = ±π.

Thin lines show intermediate levels of constant modulus and constant phase. At the poles at every negative integer, phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.

For |z| < 1, the Taylor expansions can be used:

The first coefficients of this expansion are

n gn approximation
0 1 1
1 γ −0.5772156649
2 π2/12 + γ2/2 0.9890559955
3 ζ(3)/3π2/12γ3/6 −0.9074790760

where γ is the Euler–Mascheroni constant and ζ(z) is the Riemann zeta function. Computer algebra systems such as SageMath can generate many terms of this expansion.

Approximations of the factorial[]

For the large values of the argument, the factorial can be approximated through the integral of the digamma function, using the continued fraction representation. This approach is due to T. J. Stieltjes (1894).[citation needed] Writing z! = eP(z) where P(z) is

Stieltjes gave a continued fraction for p(z):

The first few coefficients an are[21]

n an
0 1/12
1 1/30
2 53/210
3 195/371
4 22999/22737
5 29944523/19733142
6 109535241009/48264275462

There is a misconception that ln z! = P(z) or ln Γ(z + 1) = P(z) for any complex z ≠ 0.[citation needed] Indeed, the relation through the logarithm is valid only for a specific range of values of z in the vicinity of the real axis, where −π < Im(Γ(z + 1)) < π. The larger the real part of the argument, the smaller the imaginary part should be. However, the inverse relation, z! = eP(z), is valid for the whole complex plane apart from z = 0. The convergence is poor in the vicinity of the negative part of the real axis;[citation needed] it is difficult to have good convergence of any approximation in the vicinity of the singularities. When |Im z| > 2 or Re z > 2, the six coefficients above are sufficient for the evaluation of the factorial with complex double precision. For higher precision more coefficients can be computed by a rational QD scheme (Rutishauser's QD algorithm).[22]

Non-extendability to negative integers[]

The relation n! = n × (n − 1)! allows one to compute the factorial for an integer given the factorial for a smaller integer. The relation can be inverted so that one can compute the factorial for an integer given the factorial for a larger integer:

Note, however, that this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division by zero, and thus blocks us from computing a factorial value for every negative integer. Similarly, the gamma function is not defined for zero or negative integers, though it is defined for all other complex numbers.

Factorial-like products and functions[]

There are several other integer sequences similar to the factorial that are used in mathematics:

Double factorial[]

The product of all the odd integers up to some odd positive integer n is called the double factorial of n, and denoted by n!!.[23] That is,

For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.

The sequence of double factorials for n = 1, 3, 5, 7,... starts as

1, 3, 15, 105, 945, 10395, 135135,... (sequence A001147 in the OEIS)

Double factorial notation may be used to simplify the expression of certain trigonometric integrals,[24] to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres,[25] and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs.[23][26]

Multifactorials[]

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more (see generalizations of the double factorial). The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. One can define the k-tuple factorial, denoted by n!(k), recursively for positive integers as

In addition, similarly to 0! = 1!/1 = 1, one can define:

For sufficiently large n ≥ 1, the ordinary single factorial function is expanded through the multifactorial functions as follows:

In the same way that n! is not defined for negative integers, and n!! is not defined for negative even integers, n!(k) is not defined for negative integers divisible by k.

Primorial[]

The primorial (sequence A002110 in the OEIS) is similar to the factorial, but with the product taken only over the prime numbers.

Superfactorial[]

Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n factorials. So the superfactorial of 4 is

In general

Equivalently, the superfactorial is given by the formula

which is the determinant of a Vandermonde matrix.

The sequence of superfactorials starts (from n = 0) as

1, 1, 2, 12, 288, 34560, 24883200, 125411328000,... (sequence A000178 in the OEIS)

By this definition, we can define the k-superfactorial of n (denoted sfk(n)) as:

The 2-superfactorials of n are

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000,... (sequence A055462 in the OEIS)

The 0-superfactorial of n is n.

Pickover’s superfactorial[]

In his 1995 book Keys to Infinity, Clifford Pickover defined a different function n$ that he called the superfactorial. It is defined by

This sequence of superfactorials starts

(Here, as is usual for compound exponentiation, the grouping is understood to be from right to left: abc = a(bc).)

This operation may also be expressed as the tetration

or using Knuth's up-arrow notation as

Hyperfactorial[]

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

For n = 1, 2, 3, 4,... the values of H(n) are 1, 4, 108, 27648,... (sequence A002109 in the OEIS).

The asymptotic growth rate is

where A = 1.2824... is the Glaisher–Kinkelin constant.[27] H(14) ≈ 1.8474×1099 is already almost equal to a googol, and H(15) ≈ 8.0896×10116 is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.

The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the K-function.

See also[]


References[]

  1. ^ Graham, Knuth & Patashnik 1988, p. 111.
  2. ^ Biggs, Norman L. (May 1979). "The roots of combinatorics". Historia Mathematica. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0. ISSN 0315-0860 – via ScienceDirect.
  3. ^ Stedman 1677, pp. 6–9.
  4. ^ Stedman 1677, p. 8.
  5. ^ Higgins 2008, p. 12
  6. ^ Cheng, Eugenia (2017-03-09). Beyond Infinity: An expion to the outer limits of the mathematical universe. Profile Books. ISBN 9781782830818.
  7. ^ a b Conway, John H.; Guy, Richard (1998-03-16). The Book of Numbers. Springer Science & Business Media. ISBN 9780387979939.
  8. ^ a b Knuth, Donald E. (1997-07-04). The Art of Computer Programming: Volume 1: Fundamental Algorithms. Addison-Wesley Professional. ISBN 9780321635747.
  9. ^ "18.01 Single Variable Calculus, Lecture 37: Taylor Series". MIT OpenCourseWare. Fall 2006. Archived from the original on 2018-04-26. Retrieved 2017-05-03.
  10. ^ Kardar, Mehran (2007-06-25). "Chapter 2: Probability". Statistical Physics of Particles. Cambridge University Press. pp. 35–56. ISBN 9780521873420.
  11. ^ "18.01 Single Variable Calculus, Lecture 4: Chain rule, higher derivatives". MIT OpenCourseWare. Fall 2006. Archived from the original on 2018-04-26. Retrieved 2017-05-03.
  12. ^ "wesselbosman/nFactorial". GitHub. Archived from the original on 26 April 2018. Retrieved 26 April 2018.
  13. ^ "Factorial Algorithm". GNU MP Software Manual. Archived from the original on 2013-03-14. Retrieved 2013-01-22.
  14. ^ Borwein, Peter (1985). "On the Complexity of Calculating Factorials". Journal of Algorithms. 6: 376–380.
  15. ^ Luschny, Peter. "Fast-Factorial-Functions: The Homepage of Factorial Algorithms". Archived from the original on 2005-03-05.
  16. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics archive, University of St Andrews.
  17. ^ Weisstein, Eric W. "Wilson's Theorem". MathWorld. Retrieved 2017-05-17.
  18. ^ Bostock, Chandler & Rourke 2014, pp. 168.
  19. ^ Guy 2004, p. 346.
  20. ^ Luschny, Peter. "Hadamard versus Euler – Who found the better Gamma function?". Archived from the original on 2009-08-18.
  21. ^ "5.10". Digital Library of Mathematical Functions. Archived from the original on 2010-05-29. Retrieved 2010-10-17.
  22. ^ Luschny, Peter. "On Stieltjes' Continued Fraction for the Gamma Function". Archived from the original on 2011-05-14.
  23. ^ a b Callan, David (2009), A combinatorial survey of identities for the double factorial, arXiv:0906.1317, Bibcode:2009arXiv0906.1317C.
  24. ^ Meserve, B. E. (1948), "Classroom Notes: Double Factorials", The American Mathematical Monthly, 55 (7): 425–426, doi:10.2307/2306136, MR 1527019
  25. ^ Mezey, Paul G. (2009), "Some dimension problems in molecular databases", Journal of Mathematical Chemistry, 45 (1): 1–6, doi:10.1007/s10910-008-9365-8.
  26. ^ Dale, M. R. T.; Moon, J. W. (1993), "The permuted analogues of three Catalan sets", Journal of Statistical Planning and Inference, 34 (1): 75–87, doi:10.1016/0378-3758(93)90035-5, MR 1209991.
  27. ^ Weisstein, Eric W. "Glaisher–Kinkelin Constant". MathWorld.

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