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: