Recently while in a circle of friends and sipping on a glass of tea, the topic of conversation came around to what humanity’s most remarkable discovery was. Someone put forward the notion that it was the internet, and many others readily agreed with this claim. As the conversation on the matter expanded further, I had gotten stuck on the phrase “humanity’s greatest discovery” and wholly disengaged from the ongoing conversation as I started to ponder what humanity’s most remarkable discovery just might be…
As I wandered among terms such as “electricity,” “soccer,” “spacecraft,” and “iPhone” in my head, I suddenly came across what deserved the distinction to be coined as “humanity’s greatest discovery.” In the words of the beloved late artist Bob Ross, “let’s draw a happy tree here” and similarly “let’s draw a circle here and call it zero” to illustrate Al-Khwarizmi’s effort to define nothingness 1200 years ago, which is arguable, in my opinion, humanity’s most remarkable discovery.
That is because the discovery of the concept of zero led to the invention of Algebra. After air, water, and food, humanity’s greatest needs are computers and the internet, both of which are only possible due to the concept of algorithms that forms their basis. That is why the son of man’s greatest discovery must be zero.
In fact, is there even such a thing as meaningful as zero, which defines nothingness? When everything is jolly, swell, and running without a hitch, does anything disrupt that flow as much as the concept of zero does? Why do mathematicians need to come up with unique definitions and rules for math when the topic in question is zero?
For example, while all numbers are divisible by each other, why is dividing by zero an issue? Why does every number multiplied by zero disappear? Whereas each number’s factorial follows the same rule, why does 0! (zero factorial) boggle our minds? I had written an article about dividing by zero earlier; you can read it below.
But today, I wanted to tackle why 0! must be equal to 1.
While researching this topic online, I found the most logical explanation on 0! from my old college professor Ali Nesin’s class. I will try to elaborate on what I learned in his class and explain it in my own words.
So, why is 0! equal to 1?
In mathematics, a factorial is when an exclamation mark is put next to any positive number x, the outcome is the product of multiplying all the numbers from 1 to that positive number x. For example, 7! (7 factorial) is the product of all numbers from 1 to 7. 3! is multiplying the three numbers from 1 to 3. TO put it mathematically:
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
6! = 6 x 5 x 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
4! = 4 x 3 x 2 x 1
3! = 3 x 2 x 1
2! = 2 x 1
1! = 1So what is 0!? To execute the operation of 0! how many numbers do we need to multiply? By definition, we need to multiply the numbers from zero to zero. Well, then how can we multiply zero numbers? Of course, we can’t multiply because there are no numbers to multiply. So in order to get out of this dilemma, we need to come up with a new definition.
And we do this just as we define x⁰ as one and x•0 as 0. If you remember exponential numbers, x⁰ means multiplying zero x’s by themselves. But we can’t multiply zero x’s by each other. Since 1 is the identity number for multiplication and does not impact multiplicative operations, we assume x⁰ = 1 because no numbers multiplied by themselves should equal one by definition. Instead, that’s how we define it. Or we say x times 0 is 0 because we can’t add zero x’s. Because 0 is the identity number for addition and doesn’t impact summative operations, we say x•0 = 0 because adding no numbers equals nothing, or 0.
So mathematicians wanted x⁰ = 1 and x•0 = 0 because it suited them.Let’s keep going with sets. What is the union of zero sets? We can’t complete the operation because there are no sets. That is why we need a new definition. If you remember, when doing set operations, the identity element is the empty set. That’s why the union of zero sets is always the empty set.
Anyways, let’s get back to our original topic. Think of a set with n elements. We know that any set with n number of elements is bound to have subsets. For example, how many subsets of 5 elements are possible from a set of 10 elements? To get the answer, we must perform the combination operation. If you don’t remember, you can find the formula for the combination below.
The total number of combinations of n objects taken k at a time denoted n_C_k, is given by
So, the number of subsets of 5 elements from a set of 10 elements would be
And the number of subsets of 5 elements from a set of n elements would be
What about the number of subsets of n elements from a set of n elements? So, in other words, how many subsets of 10 elements are possible from a set of 10 elements? As you might guess, there is one, which is itself. So the number of the subset of n elements subsets of a set of n elements is 1.If we operate above;
However, we need to simplify it. If the n!’s are simplified, we are left with 1!/0!.Since 1! = 1, we need to find 1/0!1/0! = 1 must be true, so 0! = 1.
As you can see, 0! must be 1. This works out favorably for us. Otherwise, 0! by itself doesn’t mean anything. We define it, and if we wanted to, we could define 0! as 10, but then life would be in a state of absurdity.
Lastly, take 0⁰, for example. There is no clear definition for 0⁰. Some mathematical works define it as 0⁰= 1, whereas others just label it as undefined. That is because 0⁰= 1 is convenient algebraically, and 0⁰=undefined is convenient from an analytical perspective.
So yes, in summary, mathematics is built upon a bunch of definitions. Before the common era, there was no such thing as 0!. Nor was 0! God’s command. We just defined it as such.
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