What Exactly is a Factorial?

A factorial of a natural number ( n ), denoted as ( n! ), is the multiplication of all positive integers up to ( n ). For instance:

• ( 4! = 4 times 3 times 2 times 1 = 24 )
• ( 5! = 5 times 4 times 3 times 2 times 1 = 120 )

But what about zero?

Here's an interesting fact: ( 0! = 1 ). It may seem odd, but this is a well-established mathematical principle that simplifies various computations and equations.

Calculating Factorials

Determining a factorial can be likened to a countdown of multiplications. Begin with your target number and multiply it by every preceding integer until you reach 1. For example, to find ( 5! ):

1. Start with 5.
2. Multiply by 4 (the next smaller number): ( 5 times 4 = 20 ).
3. Multiply by 3: ( 20 times 3 = 60 ).
4. Multiply by 2: ( 60 times 2 = 120 ).
5. Multiply by 1 (which does not alter the value): ( 120 times 1 = 120 ).

Hence, ( 5! = 120 ).

Applications of Factorials

Factorials play a crucial role in various fields of mathematics, including combinations and permutations, which are methods for arranging items. For example, if you're curious about how many different ways you can organize 4 books on a shelf, simply calculate ( 4! ), resulting in 24.

Tips to Remember

• A factorial of a number is the product of all positive integers up to that number.
• ( 0! = 1 ).
• Factorials are essential for computing combinations and permutations.

So, next time you encounter the term "factorial," think of it as a way to multiply numbers in descending order. Now, how would you calculate the factorial of 6?

The first video titled "The Factorial Function!" provides a quick overview of what factorials are and how they function in mathematics.