Zero factorial, written as 0!, is defined as 1. This may seem a bit counterintuitive at first, but it has important mathematical reasons behind it.
Meaning and Definition:
- 0! = 1 by definition. This is a standard convention in mathematics.
Why Is 0! Defined as 1?
The definition of factorials is typically expressed as:
- n! = n × (n-1) × (n-2) × … × 1 for n ≥ 1.
- For 1! = 1 (since 1! = 1).
To maintain consistency and make certain mathematical formulas work smoothly (such as in combinatorics), the value of 0! is set to 1.
One key reason is that it allows the formula for combinations to work universally. In combinatorics, the number of ways to choose k items from n items is given by the formula:
C(n,k)=n!k!(n−k)!
When k = 0, it would be problematic to calculate 0! if it were anything other than 1, as the result would be undefined or inconsistent.
Examples:
- 3! = 3 × 2 × 1 = 6
- 2! = 2 × 1 = 2
- 1! = 1
- 0! = 1
In conclusion, 0! = 1 is a well-defined, convenient convention that helps maintain consistency across mathematical principles.