Factorials

Factorials are products of consecutive positive integers.

The factorial of a positive integer (n) is written as:

and defined by:

For example:

and:

Factorials Using Pi Notation

Factorials can be written compactly using pi notation:

For example:

Important Factorial Values

Proof on Why (0! = 1)

By definition:

$$0! = 1$$

This may seem strange at first, but it keeps factorial patterns consistent.

For example:

$$5! = 5 \cdot 4!$$

so:

$$4! = \frac{5!}{5}$$

Similarly:

$$1! = 1 \cdot 0!$$

Since:

$$1! = 1$$

we must have:

$$0! = 1$$

Factorial Growth

Factorials grow extremely quickly.

For example:

and:

Common Factorial Identities

Recursive Formula

For example:

Cancellation

Example:

Factorial Ratios

For example:

Signum Function

The signum function tells the sign of a number.

It is written as:

and defined by:

For example:

Floor Function / Integer Part Function / Greatest Integer function

The Greatest Integer function, also called the floor function, gives the greatest integer less than or equal to a number. (Hence the name Greatest Integer function)

It is written as:

For example:

and:

because (-3) is the greatest integer less than or equal to (-2.3).

Ceiling Function

The ceiling function gives the smallest integer greater than or equal to a number.

It is written as:

For example:

and:

Fractional Part Function

The fractional part of a number is the part remaining after removing the integer part.

It is written as:

and defined by:

For example:

and:

Relationships Between These Functions

The floor function and fractional part function satisfy:

For example:

Practice Problems