Why Does Zero Factorial Equal One? (2024)

A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it. The answer is one.

In general, the factorialof a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. For example, 4! = 24 is the same as writing 4 x 3 x 2 x 1 = 24, but one uses an exclamation mark to the right of the factorial number (four) to express the same equation.

It is pretty clear from these examples how to calculate the factorial of any whole number greater than or equal to one, but why is the value of zero factorial one despite the mathematical rule that anything multiplied by zero is equal to zero?

This typically confuses people the first time that they see this equation, but we will see in the below examples why this makes sense when you look at the definition, permutations of, and formulas for the zero factorial.

The Definition of a Zero Factorial

The first reason why zero factorial is equal to one is that the definition of the factorial states that 0! = 1. A definition is a mathematically correct explanation (even if a somewhat unsatisfying one). Still, one must remember that a factorial is the product of all integers equal to or less than the original number—in other words, a factorial is the number of combinations possible with numbers less than or equal to that number.

Read MoreDemystifying Factorials in Math and StatsBy Courtney Taylor

Because zero has no numbers less than it but is still in and of itself a number, there is but one possible combination of how that data set can be arranged: It cannot. This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! is equal to one because there is only a single possible arrangement of this data set.

For a better understanding of how this makes sense mathematically, it's important to note that factorials like these are used to determine possible orders of information in a sequence, also known as permutations, which can be useful in understanding that even though there are no values in an empty or zero set, there is still one way that set is arranged.

Permutations and Factorials

A permutation is a specific, unique order of elements in a set. For example, there are six permutations of the set {1, 2, 3}, which contains three elements, since we may write these elements in the following six ways:

  • 1, 2, 3
  • 1, 3, 2
  • 2, 3, 1
  • 2, 1, 3
  • 3, 2, 1
  • 3, 1, 2

We could also state this fact through the equation 3! = 6, which is a factorial representation of the full set of permutations. Similarly, there are 4! = 24 permutations of a set with four elements and 5! = 120 permutations of a set with five elements. So an alternate way to think about the factorial is to let n be a natural number and say that n! is the number of permutations for a set with n elements.

With this way of thinking about the factorial, let’s look at a couple more examples. A set with two elements has two permutations: {a, b} can be arranged as a, b or b, a. This corresponds to 2! = 2. A set with one element has a single permutation, as the element 1 in the set {1} can only be ordered in one way.

This brings us to zero factorial. The set with zero elements is called the empty set. To find the value of zero factorial, we ask, “How many ways can we order a set with no elements?” Here we need to stretch our thinking a little bit. Even though there is nothing to put in an order, there is one way to do this. Thus we have 0! = 1.

Formulas and Other Validations

Another reason for the definition of 0! = 1 has to do with the formulas that we use for permutations and combinations. This does not explain why zero factorial is one, but it does show why setting 0! = 1 is a good idea.

A combination is a grouping of elements of a set without regard for order. For example, consider the set {1, 2, 3}, wherein there is one combination consisting of all three elements. No matter how we arrange these elements, we end up with the same combination.

We use the formula for combinations, n!/[r! x (n-r)!], with the combination of three elements, n, taken three at a time, r, and see that: 1 = C (3, 3) = 3!/(3! 0!). If we treat 0! as an unknown quantity and solve algebraically, we see that 3! x 0! = 3! and so 0! = 1.

There are other reasons why the definition of 0! = 1 is correct, but the reasons above are the most straightforward. The overall idea in mathematics is that when new ideas and definitions are constructed, they remain consistent with other mathematics, and this is exactly what we see in the definition of zero factorial is equal to one.

Key Takeaways

  1. A factorial is the product of all integers less than or equal to the original number.
  2. As such, zero factorial equals one because it represents the one possible arrangement of an empty set: none at all.
  3. Factorials are used to determine permutations, which represent unique orders of elements in a set.
Why Does Zero Factorial Equal One? (2024)


Why Does Zero Factorial Equal One? ›

This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! is equal to one because there is only a single possible arrangement of this data set.

Why is zero factorial equal to one? ›

Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. But there are no positive values less than zero so the data set cannot be arranged which counts as the possible combination of how data can be arranged (it cannot). Thus, 0! = 1.

Why should 0 equal 1? ›

In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one ibecause any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.

Can you show that 0 != 1 though it has no meaning from the definition? ›

Explanation 1: We define n! as the product of all integers k with 1≤k≤n. When n=0 this product is empty so it should be 1. Explanation 2: If n is a nonnegative integer, we define n! to be the number of orderings on a set with n distinct objects. If n=0, this set is empty.

Why does 0 0 not equal 1? ›

The short answer is that 0 has no multiplicative inverse, and any attempt to define a real number as the multiplicative inverse of 0 would result in the contradiction 0 = 1. Some people find these points to be confusing. These notes may be useful for anyone with questions about dividing by 0.

Is factorial function one to one? ›

In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6.

How do you prove the power of zero equals 1? ›

Let's prove this in steps. Let us consider any number a raised to the power b in the exponent as ab. Since we know a number divided by itself always results in 1, therefore, ab ÷ ab = 1. Therefore, any number 'a' raised to the power zero is always equal to one as it's numerical value is 1.

Why is exponent 0 equal to 1 on Reddit? ›

Exponents are repeated multiplication so raising something to the power of 0 also removes everything. But now we are not doing addition, we are doing multiplication. So what number can you multiply and divide by without changing the answer? The answer is 1.

Is it true that any number raised to zero is equal to one? ›

Now let's focus on the zeroth power. When a number is raised to the power of zero, it always equals one. This might seem counterintuitive at first because a positive power results in multiplying the base by itself multiple times. However, if we consider negative powers as well, we can see a pattern emerge.

How to prove A0 1? ›

A number to the power of 0 is equal to 1 because of the division rule of exponents. a^n/a^n=1 because any value divided by itself is 1.

Who invented zero? ›

Following this in the 7th century a man known as Brahmagupta, developed the earliest known methods for using zero within calculations, treating it as a number for the first time. The use of zero was inscribed on the walls of the Chaturbhuj temple in Gwalior, India.

How much is 1 divided by infinity? ›

Essentially, 1 divoded by a very big number gets very close to zero, so… 1 divided by infinity, if you could actually reach infinity, is equal to 0.

Why do factorials end in 0? ›

Think of all the numbers like 10, 20, 30, etc. that you multiplied in to the factorial of 1000 to arrive at the answer. Each one of the adds a 0 (two zeroes for 100, 200, etc.), since any number multiplied by 10 will end in a 0. They're also composed of a lot of numbers that are multiples of 2 or 5.

What does i1 equal? ›

i0 = 1, this is because for any non-zero number, any number with exponent 0 is equal to 1. i1 = i, this is because any number with exponent 1 is equal to itself, in this case, i itself. We see that i4 = i0, so this is a cycle, so by prediction i5 equals to i, i6 equals -1, and so on…. and in fact that is true.

Is n factorial 1 always prime? ›

n! + 1 is prime for (sequence A002981 in the OEIS): n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, ... (resulting in 24 factorial primes - the prime 2 is repeated)

Why doesn t 1 0 equal 0? ›

As much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be ​true, because anything times 0 is 0.

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