zero factorial, why 0! should be 1, 4 reasons

why 0! should be 1,

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In this video, I give 4 of my reasons on why we should have 0! to be 1. For the first reason, be sure you know the difference that I am not using the usual definition of n! to "prove" that 0! =1. I am saying that 0! should be 1 so that the pattern will still work.

math for fun,
zero factorial,
factoreo,

[calyodelphi]

Honestly my favorite out of these is reason #2 because of how philosophical it is. How many ways can you arrange nothing? Exactly one way: no arrangement at all. The complete absence of an arrangement is itself a valid arrangement when you have nothing to arrange. :D

[MathManMcGreal]

Scream ZERO loud enough and it'll turn into a one...

[nunogirao]

In reason 4, what I really like is that, if you use one of the other reasons to accept 0!=1, than, you can turn the other way around to prove the «convention» 0^0=1.

[MuffinsAPlenty]

I'm a big fan of the empty product. And you can use the empty product to explain 0! = 1.

For nonnegative integers n, you can say that n! is the product of all positive integers less than or equal to n.

For 0!, you are then taking the product of all positive integers less than or equal to 0. But there are no such numbers. Therefore, under this definition, 0! is a product with no factors, i.e., the empty product, which is 1. :)

[Whateverbro]

I really like the reason where you used the pi function, keep up the good work, your videos really help

[Differentox]

The fourth reason doesn't really count. The reason 0! works in the power series is BECAUSE it is defined to be 1. So that’s circular reasoning. If it wasnt 1, the sum would be expressed as
1 + Sum(n = 1)(infinity) x^n/n!

[avelkm]

As an IT guy I see custom thumbnail 0!=1 that reads "zero is not equal to one"

[wlan]

Since exponentials and factorials are constructed by multiplication, it makes sense that their foundation would be the multiplicative identity (1).

[AgentM]

So -1! Must be 1/0 which is undefined?

[gtweak]

man, your channel is gold - you explain all the stuff I had been wondering about, but never had competence to obtain a valuable answer - thanks!

[ARVash]

I really like number 2 because it helps illuminate a purpose driven use case of factorial. I think ultimately what is useful for factorial depends on what you hope to happen when you hit that "hole" in the function. The use case of what you're actually trying to describe matters.

[danerman]

Love this channel. I think the 2nd explanation makes the most sense to me, shows 0!=1 by the meaning of factorial.

[stephenbeck]

Another reason, closely related to #2, is a general combination/permutation problem. If the number of ways to choose k elements out of a set of size n is equal to n!/(k!(n-k)!), then when n=k (that is, you choose all the elements of the set, of which there is only one way to do so), it makes sense to say (n-k)!=0!=1.

[runlinshu]

I like the #4 method the most, because in addition to defining 0! = 1, 0^0 must also be 1 to satisfy that e^0 equals 1

[kachraseth]

2nd one is the most obvious way to make understand a total beginner.

[mr.soundguy]

There is a 5th reason. If we define n! for n >= 1 to be the product of the integers 1 ... n, then 0! is the empty product which by default in mathematics is always 1, just like that the empty sum is always 0.

[semiawesomatic]

Right now Matt Parker is trying to solve a problem really hard. I don't wanna explain it but it's a probability problem about coins with a mild twist. I'd love to see your take on the problem. I know it's really different from what you normally do, but it'd still be cool to see a video on it.

[urusledge]

Thank you for saying "should be." This is an axiom, which I am fine with. It grinds my gears when people say this is true in the sense that it can be proven.

[dfcastro]

Before anything I would like to say that I love all your videos and the way you explain. Although it is clear that the video is not a proof of why zero factorial is 1 but why we can define it as 1 there are some observations.

The reason #1 uses the definition of factorial and this definition finishes at 1. So using the logic of (n-1)! = n! / n loses meaning if n = 1 because the the factorial is defined until 1. This is a logical argument that MAKES SENSE but it not something that is supposed to find shelter on the factorial definition.

The reason #2 is almost philosophical because there are those who could say that if you have no objects to be arranged than there is no way to arrange what does not exists while there are those who could state that if there is nothing to be arranged than this nothingness is arranged itself in only one possible way.

The reason #3 is elegant, truly beautiful, but it is an extension of factorial and in order to be accepted as extension of factorial it must matches 2 conditions (and it does).
- Working with Gamma Function we need to make sure Gamma (1) matches the 1! and it does AND n! = n x (n-1)!. There is a video of yours (beautiful) profing it. So it matches the conditions.
- The PI function fits also but with a parameter shifted by 1.
Thus it is an extension of factorial and when we set n equals to zero it returns 1. So it is fine but what it says is that it matches a convention established when the factorial function was defined. It is not a proof of zero factorial equals 1.

The reason #4 is also elegant but it is based on o power to 0 which is in some situations an indetermination while in other situations it is defined as 1. Once we overcome this it is very beautiful approach.

However there is one approach that is a reasoning alternative for approach #2 that it is not explored. The combination of a set of n elements where we pick all the n elements. It means (n k) as a column matrix like notation. It is solved as (n! / [k! (n-k)!]). If n and k are equal we get
(n! / [n! (n-n)!]) = (n! / [n! (0)!])

We know that there is only one possible way to arrange a set of elements if we pick all the elements of that set.
Therefore (n! / [n! (0)!]) = 1. We cancel the n! and we get 1 / 0! = 1, so we finally get the definition but it is logical however I think is less philosophical and closer to a math proof but I am not confident that we can call it that way.

[copperfield]

Of those I like 2 the most.

another one I like is by defining the factOREO as a productoria (or however is called), defined from 1 to n, therefore when n=0 we get an empty range which give us an empty product which give us the identity of the operator which just 1, much like with summations with empty sum which give us 0