What is 3 2 faculty

What is a faculty?

......Every natural number n has a factorial. It is the product of the natural numbers that are less than or equal to the number n.
You write it as n! = 1 * 2 * 3 * ... * (n-1) * n and read it n faculty
It is useful to define 1! = 1 and also 0! = 1.

One can n! as a function with the domain D = | N, i.e. as a sequence.
Then an= n! or written recursively ai= i * ai-1. (i = 1, 2, 3, ..., n).


These are the first 15 numbers, such as those provided by the TI-30 pocket calculator.
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!=8.717829120*1010
15!=1.307674368*1012
The last two numbers are rounded and given in scientific notation.
All numbers are composite numbers that have ever increasing prime numbers as divisors. Among them are always pairs of 2 and 5, which lead to a growing number of ending zeros.

On this page you will find mostly entertaining faculties.

Great faculties Top
calculator
Faculties are growing rapidly. Up is 15! already 13 digits.
The largest number the TI-30 calculator can display is 69! = 1.711224524 * 1098
In between there is a number that is easy to remember, namely 25! is about 1025, more precisely 1.6 * 1025.


Google calculator
Larger faculties up to 170! can be found by typing them into the Google search box and pressing Enter. For example, the Google calculator finds 100! = 9.33262154 × 10157.

Tables
......Since stochastics found its way into schools, faculties have gained in importance there.

Tables with faculties and binomial coefficients are available today.


Numerical representations
Even larger faculties can be found online at Wolfram Demonstration Project or at Nitrxgene (URLs below) and then also in the exact number spelling.
100!=9332621544394415268169923885626670049071596826438162146859296389521759999322
9915608941463976156518286253697920827223758251185210916864000000000000000000000000 
The number has 158 digits, including 24 ending zeros.

Stirling formula
Large faculties can also be determined with the help of the Stirling formula (and other approximation formulas).
The symbol ~ is similar to an equal sign and is intended to express that the terms approach each other more and more as the number n increases. In other words: The quotient of both terms tends towards 1 for n towards infinity.
Obviously the quotient does not converge strongly, as the following numerical example shows.
50! = 50 ^ 50 * e ^ (- 50) sqrt (2 * pi * 50) = 4.2 * 1064. - It should be 50! = 3.0 * 1064

It is noteworthy that the constants e and pi appear. In the background is the gamma function, which can be used to prove the Stirling formula. See also (2), page 97f.

Number of ending zeros Top
The top turns 100! mentioned with 158 digits and 24 ending zeros.
The question arises as to how one can determine the number of ending zeros.
The following rule applies.
It is enough to examine the last number, here 100. In a first step, you divide 100 by 5, which is 20. Then you divide 20 again by 5 and get 4. The sum of the quotients is 20 + 4 = 24, and that is the number of final zeros.
This example is not typical because there are no remainders when dividing.


A second example is 81! .
In a first step 81 is divided by 5, which is 16. The rest is of no interest.
Then divide 16 by 5 again and get 3. Again, leave out the remainder.
The sum of the quotients is 16 + 3 = 19, and that is the number of ending zeros.

Number of digits Top
A game with faculties is to determine the number of their digits. (1)
It then allows the numbers to be written as figures.

1081396758240
2909005041013
0580032964972
0646107774902
5791441766365
7322653190990
5153326984536
5268082403397
7639893487202
9657993872907
8134368160972
8000000000000
0000000000000 
1
081
39675
8240290
900504101
30580032964
9720646107774
902579144176636
57322653190990515
3326984536526808240
339776398934872029657
99387290781343681609728
000000 0000000000000000000
The number 105! has 169 digits.

So the digits form a 13 * 13 square.

Because of 169 = 1 + 3 + 5 + ... + 25 = 13²
also creates a triangle.


5
7 9
7 1 2
6 0 2 0
7 4 7 3 6
7 9 8 5 8 7
9 7 3 4 2 3 1
5 7 8 1 0 9 1 0
5 4 1 2 3 5 7 2 4
4 7 3 1 6 2 5 9 5 8
7 4 5 8 6 5 0 4 9 7 1
6 3 9 0 1 7 9 6 9 3
8 9 2 0 5 6 2 5 6
1 8 4 5 3 4 2 4
9 7 4 5 9 4 0
4 8 0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 0
0 0
0

The number 81! has 121 digits.

This number is the sum of the triangular numbers d10+ d11=55+66.

Therefore you can build a figure from two triangles.


8 2 4 7 6 5
0 5 9 2 0 8 2
4 7 0 6 6 6 7 2
3 1 7 0 3 0 6 7 8
5 4 9 6 2 5 2 1 8 6
2 5 8 5 5 1 3 4 5 4 3
7 4 9 2 9 2 2 1 2 3
1 3 4 3 8 8 9 5 5
7 7 4 9 7 6 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 
The hexagon is given by the number 91 of the faculty 65! certainly.

It applies (i.e.10+ d11) - 2 * d5=55+66-2*15=91


Prime numbers Top
The powers of two 2n with many factors lead to prime numbers if they are decreased by 1 or increased by 1. These are then the well-known Mersenne numbers 2n-1 or Fermat numbers 2n+1. The question is whether one also gets prime numbers if one forms n! -1 and n! +1 with the help of the faculties.
An answer can be found at N. J. A. Sloane.
n! -1 is a prime number for 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, ...
n! +1 is a prime number for 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, ...


Wilson theorem
For every prime number, the following applies: The term (n-1)! + 1 is divisible by n if and only if n is a prime number.
To illustrate the sentence, a table with examples follows. The red numbers are prime numbers. Then the division opens.
n
(n-1)! + 1
[(n-1)! + 1] : n
2
2
1
3
3
3
4
7
1,8
5
25
5
6
121
20,2
7
721
103
11
3628801
329891
13
479001601
36846277 

It also applies: If the term t (p) = (p-1)! + 1 is divisible by p², then p is a Wilson prime number.
With N. J. A. Sloane one finds the following Wilson primes 5, 13 and 563 known up to now.

Derivatives Top
The nth derivative of the power function f (x) = xn is f (n)(x) = n!


Faculties appear in the Taylor series.
f (x) = f (a) + [f '(a) / 1!] (x-a) + [f' '(a) / 2!] (x-a) ² + ... + [f(n)(a) / n!] (x-a)n
The meaning of this formula lies in the fact that one can use it to lead to series expansions of i.a.x, ln (1 + x), sin (x), cos (x), tan (x) and arc tan (x). You can find it at Wikipedia Taylor series.

Permutations Top
If one gives n different elements of a set, then there are n! different arrangements of the elements.
This is how you encounter the faculties at school.


example
Let the set {a, b, c} be given.
The order of the three elements can be changed in six ways, namely to abc, acb, bac, bca, cab, cba.
That's 3! = 6 arrangements or, as they say, 3! Permutations

If one wants to prove in general that n elements n! Have permutations, one chooses the method of complete induction.
The initial set is {a1, a2, a3, ..., an-1, an} or for simplicity {1, 2, 3, ..., n-1, n}.
proof
Prerequisite: The number of permutations of n elements is n !.
The cases n = 1 and n = 2 are trivial.
For n = 3 the following applies (see above): There are 3! = 6 permutations of three elements.
It has to be shown that n + 1 elements (n + 1)! Permutations have, provided that n elements n! Have permutations.
If you add to the n! Permutations add the element n + 1, so you can put it in n + 1 places of each of the n! Set permutations. So there are a total of n! * (N + 1) permutations, and that is (n + 1) !, wzbw.

Binomial coefficient Top
Pascal triangle

Binomial coefficients are the numbers of the Pascal triangle.
In the 2nd line are the numbers 1, 2, 1. These are the prefixes of (a + b) ² = a² + 2ab + b².
The 3rd line contains the numbers 1, 3, 3, 1. These are the prefixes of (a + b) ³ = a³ + 3a²b + 3ab² + b³.
In the 4th line are the numbers 1, 4, 6, 4, 1 from (a + b)4= a4+ 4a³b + 6a²b² + 4ab³ + b4.
...
The nth line contains the prefixes of the calculated term to (a + b)n
The peculiarity of the triangle is the law of formation, namely that their sum is below every two numbers.

The number that is in the kth position in the nth line can be calculated explicitly as "n over k", as an expression that is formed from factorials.

They are used in several places on my homepage.

Triangular numbers

Numbers lottery
If one sets k = 6 and n = 49, then the following applies:
n! / [k! (n-k)!]
=49!/(6!43!)
=(49*48*47*46*45*44)/(2*3*4*5*6)...
=13 983 816. 
In the lottery 6 out of 49, the number of possibilities to tick six numbers is calculated as a binomial coefficient.

More on my page 13 983 816. There n! / [K! (N-k)!] Is derived.


Taxi route
......In taxi geometry, the problem arises, among other things, of determining the number of routes with a distance of 8 from A to B.

Soma body

Curiosities Top

4!+1=5²5!+1=11²7!+1=71²145=1!+4!+5!40585=4!+0!+5!+8!+5!10!=6!*7!=3!*5!*7!
inter alia (1)

Faculties on the Internet Top

German

Martin Steen
Faculty (Numbers between 1 and 2500 are allowed as input)

Thomas Peters
The faculties

Wikipedia
Faculty (mathematics), faculty prime, Wilson prime, Stirling formula, gamma function, binomial coefficient, Taylor series



English

Eric W. Weisstein (MathWorld)
Factorial, Factorial Prime

Jiel Beaumadier, Matej Hausenblas
All about factorial notation

Nitrxgene
Factorial Calculator (Allowed range: 0 to 200,000)

N. J. A. Sloane (On-Line Encyclopedia of Integer Sequences)
Factorial numbers: A000142,
Pascal's triangle read by rows: A007318
Numbers n such that n! - 1 is prime. A002982
Numbers n such that n! +1 is prime. A002981
Wilson primes: primes p such that (p-1)! == -1 mod p ^ 2. A007540

The Wolfram Demonstrations Project
Factorial

Wikipedia
Factorial, Factorial prime, Wilson prime, Stirling's approximation, Gamma function, Binomial coefficient, Taylor series


credentials Top
(1) Martin Gardner: Geometry with Taxis, the heads of the Hydra and other mathematical gimmicks, Basel 1997
[ISBN 3-7643-5702-9]
(2) Jean-Paul Delahaye: Pi - The Story, Basel, Boston, Berlin 1999 [ISBN 3-7643-6056-9]


Feedback: Email address on my main page

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2009 Jürgen Köller

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