Double Factorials

in StemSocial10 days ago

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


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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.


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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.

 


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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:

 


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Thank you for reading.

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Hello @dkmathstats concretiy your exposition of factorial doubles and the examples developed guide the interested reader. Thank you.