The Mystery of 0! and 1! - A Deep Dive into Factorials

Factorials are a fundamental concept in mathematics, often appearing in various areas such as combinatorics and calculus. One of the most intriguing aspects of factorials is the definition of 0!, which is often misunderstood. This article explores why 0! equals 1 and why this does not imply that 0 is equal to 1.

Understanding Factorials

A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! 5 x 4 x 3 x 2 x 1 120. The definition of factorials for positive integers is straightforward, but what about 0!?

Why is 0! Equal to 1?

The equation 1! 0! is true, and we will explore why this is the case. According to the definition of a factorial:

n! 1 x 2 x 3 x ... x n

The Definition of 0!

For n 1, we have:

1! 1

For n 2, we have:

2! 2 x 1 2

Now, for n 0...

According to combinatorics, the number of ways to arrange 0 objects is 1 (the empty arrangement). Therefore, 0! 1. Mathematically, this is consistent with the property that the factorial of any n can be expressed as:

n! n. (n)1. (n)2. ... .1

When n 1, this simplifies to:

0! 0 . (1) . (2) . ... . 1 1

This definition is crucial for maintaining consistency in mathematical formulas and maintaining the integrity of the factorial function.

The Implication of 1! 0!

Although 1! 0! simplifies to 1 1, which is true, this does not imply that 0 is equal to 1. The equation simply reflects a specific property of factorials. In mathematics, 0 and 1 are distinct numbers with different properties:

0 represents the absence of quantity, while 1 represents a single unit or quantity. These definitions are fundamental and do not change even with the definition of 0!.

Extending the Definition of Factorials

The definition of factorials can be extended to other numbers, such as negative integers and fractional numbers, but this is not done through proof but through definition. One common extension is through the gamma function, which generalizes the factorial to non-integer values:

Γ(n) (n - 1)! for positive integers

For n 1 or 0, this extends naturally:

Γ(1) 0! 1

This extension is useful in various areas of mathematics, including calculus and probability.

Conclusion

The equality 1! 0! does not mean that 0 equals 1. It is simply a reflection of a specific property of factorials. The definition of 0! as 1 is crucial for maintaining the consistency of mathematical formulas and does not change the fundamental definitions of 0 and 1.

For a deeper understanding, you can watch Numberphile's video on this topic: