Factorials

Introduction

Factorials in math arose out of the study of the number of ways a set of things can be arranged. We can get an idea by doing this simple problem.

You are given 3 numerals 5, 6 & 8. How many different 3-digit numbers can be written using each of these numerals only once in a number. With a little effort we can see that there are 6 numbers – 568, 586, 658, 685, 856 & 865.

Mathematicians arrived at a formula for this which can be written as 3*2*1=6.

Writing as a formula makes it easy for us to apply the idea to any set of numbers. For example, you can check for yourselves that if we start with 4 different numerals, we will get 4*3*2*1= 24 numbers! That is the power of mathematics.

Relations between Factorials

Since 4! = 1*2*3*4, we can write 4! = 4X 3!

5! / 4! = 5

3! + 4! = 3! (1 + 4) = 5 X 3!

History

Almost all ancient cultures including India were interested in this idea of “arranging a set of things in all possible ways”.

Mathematicians like abstractions and using codes to denote elaborate computations. So in the 19thcentury this idea of 4*3*2*1 was written as 4! by a French mathematician Christian Kramp, in a book published in 1808. Before this notation, the computation (4!) was called a “factorial”. 4! was called “4 factorial”.

In the 19thcentury, factorials also started appearing in many areas of advanced math. Hence the notation (4!) was very useful in writing out such expressions.

The idea of factorials can easily be understood by primary school children. It is also a powerful idea which has applications in many areas of advanced math like “Permutations & Combinations” and “Combinatorics”

0!

6! = 5! X 6 & 5! = 4! X 5. Extending this idea to smaller numbers we get 1! = 0! X 1. From this we get the value of 0! as 1.

We can also interpret this result as, if there are no crayons in a box, there is only one way of arranging them.

There is a famous math puzzle which asks you write 24 with just four 0s and any valid arithmetic symbols. Hint – it uses the above idea. Try it for some time before looking at the answer on the next page.

Solution

24 = 4! = (1 + 1 + 1 + 1)! = (0! + 0! + 0! + 0!)!

Factorials in Advanced Math

The idea of factorial started with the physical reality of rearranging objects. Then as we saw above, several relations between factorials of numbers were discovered.

However, mathematicians extended the application of factorial, to what they called Gamma function.

This enabled the idea of factorials to be extended to integers and even complex numbers (with the exception of negative integers). This application is often utilized in mathematical analysis and complex analysis.