Double Factorials
Hi there. In this short math post, I cover the topic of double factorials.
Reference: https://mathworld.wolfram.com/DoubleFactorial.html
Math text, symbols rendered with LaTeX with Quicklatex.com
Topics
- Factorials
- Double Factorials
- Examples
Factorials
A factorial is a compact way to express a number in the form of multiplying numbers together. The symbol associated with a factorial is the exclamation mark (!). As an example, 3! (three factorial) is 3 x 2 x 1 = 6
. The exclamation mark is a bit unusual. It does not mean that the number is shouting at you.
In general, the definition of a factorial is:
for (n is a positive whole number at least 1).
Zero Factorial Case
When it comes to factorials, zero factorial is not zero. It is actually equal to 1. 0! = 1
.
Expressing A Factorial With The Product Pi Notation
Another compact notation for factorials is the product pi notation. The product pi notation starts at 1 for the index variable i
and increases to all the way to n
in the product.
Factorial Piecewise Function
A piecewise function can be developed around the factorial.
Double Factorials
A double factorial is symbolized by two exclamation marks. As a single factorial has numbers being spaced by 1, a double factorial contains numbers being spaced by two. The double factorial has two versions as the number n
can be either odd or even.
If n
is odd:
For n
being even
With the double factorial, you may have the zero case and the negative one case. In either case they are equal to 1 as defined. That is 0!! = 1
and -1!! = 1
.
Combining all these cases, a piecewise function can be developed with the three cases.
Examples
Example One
Having 3!! would be 3!! = 3 x 1 = 3
.
Example Two
What is the value of 10!!
?
10!! = 10 x 8 x 6 x 4 x 2 = 3840
Example Three
Divide 7!! by 7!.
Example Four
From example three, you can see that the double factorial is less than the single factorial given the same integer number n
. What would be n!
divided by n!!
?
Use the definition of the factorials here, simplify and obtain the answer. This is done for the even and odd cases.
For n > 0
being even:
With the n > 0
being odd case, we have:
In either case for n
, the result would be:
Hello @dkmathstats concretiy your exposition of factorial doubles and the examples developed guide the interested reader. Thank you.