For any natural number


This means that the factorial of any natural number (also called the counting numbers or positive integers; meaning all integers from 1 to infinity) is equal to the product of all of the integers from 1 up to that number.

Examples                Explanation



Calculating Factorials

The factorial operation is only defined for positive integers and 0. 0! is defined to be equal to 1. This means that 0! never needs to be calculated. The factorial of will never appear when you are simply calculating the factorial of a larger number, but it may appear when working more complicated problems.

It can be easier to calculate factorials of large numbers if you know the factorial of a smaller number.

Examples                Explanation


The example immediately above was solved using the value of 5! which was calculated in an earlier example.

Because any number multiplied by 1 does not change in value, this step can be skipped when calculating factorials.

Examples                Explanation


Try these exercises:


  1. Find , and express the difference as a factorial.
  2. Why doesn’t the factorial of a negative number make sense?

Answers to questions:

  1. The factorial of a number is calculated by multiplying integers from 1 up to the number. A similar process for negative numbers would require the operands to get smaller instead of larger.