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The factorial of a positive integer is the product of all integers less than or equal to it, and is denoted by the symbol !. For example, the factorial of the number 4 is expressed as 4! and is equal to 24:
4! = 1 x 2 x 3 x 4 = 24
In particular, the factorial of the number 0, (that is, 0!), is defined equal to 1, although this value does not arise from the definition of factorial, which is valid only for any integer greater than or equal to 1. Why Why is the factorial of the number 0 defined as 1 if there is a mathematical rule that says that any number multiplied by zero is equal to zero?
Beyond the confusion that this situation may give rise to, it should be noted that the value of the factorial of the number 0 is a definition ; that is, mathematically it is defined that 0! = 1. Let’s see below the foundations of this definition.
The definition of the factorial of the number 0
As we already mentioned, the first thing to note is that the assignment of the value 1 to the factorial of the number 0 (0! = 1) is a definition, although in principle this does not lead to a satisfactory explanation if we only look at the definition of factorial.
Recall that the definition of a factorial of a positive integer is the product of all integers equal to or less than it. Note that this definition also implies that the factorial is associated with all possible combinations of numbers less than or equal to the number we are considering.
The number 0 has no positive integers less than it but it is still a number and there is only one possible combination of this particular set of numbers made up of only the number 0. That combination is one, just as in the case of the number 1.
To better understand the mathematical meaning of this definition, it must be taken into account that the factorial concept also involves other information contained in a number, specifically the possible permutations of its factors. Even in the empty set represented by the number 0 it can be thought that there is a way to order this set.
Permutations and factorials
The concept of factorial is used in the branch of mathematics called combinatorics, a discipline in which the concept of permutation of elements is defined. A permutation is a specific and unique order of the elements that make up a certain set. For example, there are six possible permutations of the set {1, 2, 3}, containing three elements, since we can 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 express this concept through the factorial expression of three, 3! = 6, which allows us to calculate the complete set of permutations of a group of 3 elements. Similarly, there are 24 permutations (4!=24) of a set with four elements and 120 possible permutations (5!=120) of a set with five elements. So, an alternative way of thinking about the concept of factorial is to put aside the idea that it is associated with a natural number n and think that n ! is the number of permutations of a set consisting of n elements.
Let’s see some examples considering now this new conception of the factorial of a number. A set consisting of two elements has two possible permutations: {a, b} can be ordered as (a, b) or as (b, a). This is associated with the definition of factorial of the number 2; 2! = 2. A set made up of a single element, {a}, has only one possible permutation, and is associated with the definition of factorial of the number 1; 1! = 1.
Let us return now to the case of the factorial of 0. The set integrated by zero elements is called the empty set. To find the value of the factorial of 0 we can ask ourselves, in how many ways can we order a set without elements? And while one answer may be that there is nothing to order in an empty set, we also have the alternative that even empty is a set, so the answer could be 1, and so 0! = 1.
Other applications of the factorial
As we already said, the factorial concept is used in combinatorics and this mathematical tool is used to perform calculations in formulas that express permutations and combinations of groups of elements. Although these applications do not provide a direct justification for the assignment of 1 to the factorial of the number 0, it can be understood why it is defined in this way.
The concept of combination of a group of elements refers to the number of subgroups that can be obtained with them, regardless of the order in which they are considered. For example, the set {1, 2, 3} has only one join if three elements are taken, regardless of order. But if we took them by two elements we would have three possible combinations, {1, 3}, {2, 3} and {1, 2}, just as if we took them by one element, {1}, {2} and {3}. The general formula to calculate the number of combinations without repetition of a certain set of n elements taken in subgroups of p elements is C ( n , p ) = n !/ p !( n–p ) !.
If we use this formula to determine the combination number of three elements taken three, we see that the result must be 1, expressed by C (3, 3) = 3! / 3! (3-3)! = 3! / 3! 0!, so it is necessary to define the 0! = 1 for the mathematical expression to make sense.
In the same way, there are other situations that make it necessary to define the factorial of the number 0 as 1, 0! = 1, as part of the general conception in the development of mathematics that indicates that when new ideas are built and new definitions are incorporated there must be compatibility with pre-existing structures.
Bibliography
Zero factorial or 0!. Khan Academy .
Is there a factorial of 0? YouTube channel Drifting .
