How 0 Factorial is 1? | Why It Makes Sense

Understanding why 0! equals 1 often presents a moment of curiosity for many learners encountering factorials for the first time. This specific mathematical convention underpins significant areas of combinatorics and calculus, making its rationale fundamental to a deeper comprehension of these fields.

Understanding the Factorial Operation

The factorial operation, symbolized by an exclamation mark (!), represents the product of all positive integers less than or equal to a given non-negative integer. For any positive integer ‘n’, n! is calculated as n × (n-1) × (n-2) × … × 3 × 2 × 1.

• For example, 4! = 4 × 3 × 2 × 1 = 24.
• Similarly, 3! = 3 × 2 × 1 = 6.

Factorials find extensive application in counting arrangements, known as permutations, where the order of items matters. For instance, the number of ways to arrange four distinct books on a shelf is 4!, which is 24.

The Recursive Definition of Factorials

A fundamental property of factorials is their recursive definition. For any integer n greater than 1, n! can be expressed as n multiplied by the factorial of (n-1). This relationship is written as:

n! = n × (n-1)!

This recursive rule establishes a clear pattern for calculating factorials. We can observe this pattern by looking at a sequence of factorials:

1. 5! = 5 × 4!
2. 4! = 4 × 3!
3. 3! = 3 × 2!
4. 2! = 2 × 1!

This definition is standard for positive integers and forms the basis for understanding 0! = 1. It provides a consistent mathematical structure that allows us to extend the concept of factorials beyond just positive integers.

Extending the Factorial Pattern Downward

To determine the value of 0!, we apply the recursive definition in reverse. Starting from a known factorial, we can divide by ‘n’ to find the factorial of ‘n-1’.

Consider the known values:

• 4! = 24
• 3! = 6 (which is 24 ÷ 4)
• 2! = 2 (which is 6 ÷ 3)
• 1! = 1 (which is 2 ÷ 2)

Following this consistent pattern, to find 0!, we would divide 1! by 1:

0! = 1! ÷ 1

Since 1! is equal to 1, substituting this value yields:

0! = 1 ÷ 1

This calculation results in 0! = 1. This downward extension of the recursive definition ensures that the factorial function maintains its mathematical coherence and pattern as we approach zero.

Combinatorics and the “Empty Set”

The definition of 0! = 1 gains significant justification from the field of combinatorics, which deals with counting arrangements and selections. Factorials are fundamental in counting permutations and combinations.

Permutations

The number of permutations of ‘n’ items taken ‘k’ at a time is given by the formula:

P(n, k) = n! / (n-k)!

Consider the case where we arrange all ‘n’ items, meaning k = n. The formula becomes:

P(n, n) = n! / (n-n)! = n! / 0!

We know that P(n, n) represents the number of ways to arrange ‘n’ distinct items, which is simply n!. For the formula to hold true, n! / 0! must equal n!. This implies that 0! must be 1.

Consider also arranging 0 items from a set of ‘n’ items, P(n, 0). There is only one way to arrange zero items: by doing nothing. Applying the formula:

P(n, 0) = n! / (n-0)! = n! / n! = 1

This result directly supports the idea that 0! must be 1 for combinatorial formulas to remain consistent.

Combinations

The number of combinations of ‘n’ items taken ‘k’ at a time (choosing ‘k’ items without regard to order) is given by:

C(n, k) = n! / (k! × (n-k)!)

Consider choosing 0 items from a set of ‘n’ items, C(n, 0). There is only one way to do this: choose nothing. Applying the formula:

C(n, 0) = n! / (0! × (n-0)!) = n! / (0! × n!) = 1 / 0!

Since C(n, 0) must equal 1, it follows that 1 / 0! must equal 1. This equation is only satisfied if 0! = 1. This combinatorial argument provides a robust justification for the definition of 0! as 1, ensuring that these fundamental counting principles remain valid and consistent.

The Gamma Function: A Generalization

A more advanced mathematical perspective on factorials involves the Gamma function, denoted as Γ(z). The Gamma function extends the concept of factorials to complex numbers, not just non-negative integers. For any positive integer ‘n’, the Gamma function relates to the factorial by the identity:

Γ(n+1) = n!

The Gamma function is defined by the integral:

Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt

To determine the value corresponding to 0!, we need to calculate Γ(1), because Γ(0+1) = 0!. Substituting z = 1 into the integral definition:

Γ(1) = ∫₀^∞ t^(1-1)e^(-t) dt = ∫₀^∞ t^0 e^(-t) dt = ∫₀^∞ e^(-t) dt

Evaluating this improper integral:

Γ(1) = [-e^(-t)] from 0 to ∞ = (-e^(-∞)) - (-e^(-0)) = (0) - (-1) = 1

This result, Γ(1) = 1, directly implies that 0! = 1, as the Gamma function consistently extends the factorial for non-integer values and provides a natural value at zero. This powerful generalization reinforces the convention of 0! = 1 from a higher mathematical standpoint. You can learn more about the Gamma function and its properties on Khan Academy.

Consistency in Series Expansions

The definition of 0! = 1 is also essential for maintaining the consistency of many important mathematical series, particularly Taylor and Maclaurin series. These series represent functions as infinite sums of terms calculated from the function’s derivatives at a single point.

Consider the Maclaurin series for the exponential function, e^x:

e^x = Σ (x^n / n!) from n=0 to ∞

Expanding the first few terms of this series:

e^x = (x^0 / 0!) + (x^1 / 1!) + (x^2 / 2!) + (x^3 / 3!) + ...

For this series to hold true and produce the correct value for e^x, especially when x=0, the first term (when n=0) must be correctly defined. If x=0, then e^0 = 1. Substituting x=0 into the series:

e^0 = (0^0 / 0!) + (0^1 / 1!) + (0^2 / 2!) + ...

All terms after the first become 0. The first term simplifies to 1 / 0! (as 0^0 is conventionally defined as 1 in this context to maintain series consistency). For e^0 to equal 1, it is necessary that 1 / 0! also equals 1. This directly implies that 0! must be 1.

Without 0! = 1, these foundational series would either have an undefined first term or would not accurately represent the functions they are designed to approximate. This demonstrates that the definition is not arbitrary but a necessary convention for the broad applicability of mathematical tools.

Historical Context and Acceptance

The convention of 0! = 1 has been widely accepted in mathematics for centuries. Early mathematicians, including Christian Kramp who popularized the factorial notation in the early 19th century, recognized the need for this definition to ensure the consistency of formulas. It is not a value derived from directly multiplying numbers down to zero, which would lead to zero, but rather a definition established to preserve logical patterns across mathematical domains. This convention allows for elegant and unified mathematical expressions in fields ranging from probability to advanced analysis.