Why factorial of 0 is 1?

By definition and convention:

• The choice to define 0! as 1 is made for mathematical consistency and simplifies various mathematical formulas and combinatorial calculations.
• It helps in making certain mathematical expressions and series converge properly.
• This definition also aligns with the combinatorial interpretation of permutations and combinations, where there is one way to arrange or choose zero elements from an empty set.

In summary, 0! is defined as 1 by convention in mathematics to maintain consistency and simplify calculations, even though it might seem counterintuitive at first glance.

By mathematical justification:

Mathematically proving that the factorial of 0 is equal to 1 is based on the definition of the factorial function and follows the convention used in mathematics. The factorial function is typically defined as follows:

n! = n × (n – 1) × (n – 2) × … × 2 × 1

For non-negative integers, where n is greater than or equal to 1, this definition works perfectly. However, when n = 0, the definition might seem problematic:

0! = 0 × (0 – 1) × (0 – 2) × … × 2 × 1

The issue arises because the product starts with 0, and it might appear that the result is 0. However, to maintain consistency and mathematical convenience, mathematicians define 0! to be 1.

Now, let’s prove this convention mathematically:

• Start with the definition of n! for non-negative integers:
  • n! = n × (n – 1) × (n – 2) × … × 2 × 1
• When n = 1, we have:
  • 1! = 1 × (1 – 1) = 1 × 0 = 0
• Now, consider the recursive definition of factorial:
  • n! = n × (n – 1)!
• Using this definition, we can calculate 1!:
  • 1! = 1 × (1 – 1)! = 1 × 0! (by substituting n = 1)
• We want to find the value of 0!. Rearrange the equation:
  • 1! = 1 × 0!
• The value of LHS should be equal to RHS as 1! is always equal to 1!
  • For the above condition to be true, The value of 0! must be equal to 1.
  • 0! = 1

So, mathematically, by considering the recursive definition of factorial, we see that 0! must be 1 to maintain consistency with the factorial function’s definition. This convention simplifies various mathematical expressions and combinatorial calculations, making them more elegant and easier to work with.

Have you ever thought about why the factorial of 0 is 1? If yes then here you can find the justification for the same.