# Why does 0! = 1?

## Main Question or Discussion Point

I understand that a factorial is for example

5! = 1 *2 * 3 * 4 * 5, but how does that tie in with 0! ?

There is nothing to multiply, and I don't understand the statement "because the product of no number at all is 1". How does this work?

CompuChip
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It is defined that way.

I think there are multiple reasons to want to have 0! = 1, but at least the link shows that the definition is not arbitrary.

arildno
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Hmm, I would definitely say that definition is arbitrary, but not reasonless..

A popular explanation is you can only choose zero elements in one way.

D H
Staff Emeritus
Another way to look at it: The standard rule for computing the next factorial is (n+1)! = (n+1)*n! Suppose instead you knew (n+1)!; this expression can also be used to solve for n! Simply invert the expression: n! = (n+1)!/(n+1). This is true for all positive integers n. It also enables an extension to n=0 via 0!=1!/1=1. This downward extension stops at n=0 because going into negative numbers results in division by zero.

nicksauce
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Another reason why it is nice to have 0!=1 is that
$$\Gamma(n+1) = n!$$
For all natural numbers n, and
$$\Gamma(0) = 1$$

So it is nice to extend the first equation by having 0! = 1.

sylas
There is nothing to multiply, and I don't understand the statement "because the product of no number at all is 1". How does this work?
Another way to look at it. When you have "nothing to multiply", that IS "1". When you are doing multiplications, "1" is the null case representing nothing to do. Multiplying by one does nothing; it is the same as doing no multiplication at all.

Hmm, I would definitely say that definition is arbitrary, but not reasonless..
I totally disagree. It is reasonless, but not arbitrary.

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arildno
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I totally disagree. It is reasonless, but not arbitrary.
Why?

There are good, heuristic reasons why we ought adopt that definition, but we are not necessitated by anything to adopt it.
Thus, it is arbitrary.

sylas
Why?

There are good, heuristic reasons why we ought adopt that definition, but we are not necessitated by anything to adopt it.
Thus, it is arbitrary.
Only if you think 1 + 1 = 2 is arbitrary. We could define a different operation to conventional addition, and define it so that 1 + 1 = 3, and everywhere else is the same as currently defined. You'll recognize this as a silly example; but it's actually the same thing.

The factorial function, like the additional function, is a mathematical object with a whole pile of properties. Those properties imply that 0! = 1.

If you defined it any other way, you'd be defining a different function, that was the same as conventional factorial in most cases, but not at zero. The factorial function is a specific function with certain properties. It is only become of those properties that we can do useful thing like define

Comb(n,r) = n!/r!/(n-r)!

This is the number of ways of selecting r objects from n different possibilities. That definition refers to the factorial function; but it won't hold for some arbitrary function with an arbitrarily chosen value at zero.

The reasons that 0! = 1 are not heuristic; they are integral to what the factorial function is. I'm not sure; but I think I am here expressing a particular philosophy of mathematics -- specifically, that mathematics is discovered rather than invented.

Cheers -- sylas

arildno
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You are quite right concerning differing philosophies, sylas.

To me, maths is better likened as a game (or a set of games), where we choose which rules are to govern the particular game we want to play.

WHY we would want to play any particular game at all is not answered within the field of maths, but by pointing to "useful purposes" outside maths.

As for the distinction between "invented/discovered", I don't regard it as of much interest.

Chess was definitely invented, but are strategies leading to victory invented, too?

I'd say they are discovered.

That is to say: Even if maths is to be regarded as arbitrarily invented, it does not follow that we won't do any discoveries within the field invented by our own imagination.

Maths is richer than its inventor(s) ever imagined, to do maths is to discover just how rich it is.

The factorial function is just a shorthand way of writing some particular thing. It is convenient because we often need to make products like 5*4*3*2*1 and so on. And apparently it is also convenient to have 0!=1, because it avoids all kinds of special cases in the formulas.

sylas
Chess was definitely invented, but are strategies leading to victory invented, too?

I'd say they are discovered.
Great example! This shows well that it's a bit of both; not a hard and sharp distinction.

That is to say: Even if maths is to be regarded as arbitrarily invented, it does not follow that we won't do any discoveries within the field invented by our own imagination.
That makes sense to me.

IMHO, 0! = 1 is best seen as a discovery; because the factorial function was not invented arbitrarily. It came up as a strategy in many of the games we play... Taylor expansion, counting combinations, etc, etc.

Maths is richer than its inventor(s) ever imagined, to do maths is to discover just how rich it is.
Yes! It can really suck you in as you find unexpected associations and deep connections between the various abstractions we play with in maths.

Cheers -- sylas

The factorial function is just a shorthand way of writing some particular thing. It is convenient because we often need to make products like 5*4*3*2*1 and so on. And apparently it is also convenient to have 0!=1, because it avoids all kinds of special cases in the formulas.
More importantly than just being a short-hand, the factorial is a tool to add to a mathematicians toolbox (for lack of a better word) that can allow us to write more things in a closed form.

I'd also subscribe to the philosophy that maths is both invented and discovered. We state our axioms (the rules of play) and then try and find out just what can be derived from that (a process of discovery not invention). Complex numbers are a great example of this... we define i as the solution to sqrt(-1) and then discover that gives us a huge range of other things, such as:
more sensible numbers of roots for every power equation (3 for a third power, 4 for a fourth power, etc)
roots of unity
holomorphic functions
a relationship between the 'fundamental' maths constants e, i, pi, 1 and 0 in the form e^(i*pi) + 1 = 0
etc etc
So i is an invention, but the things that derive from it are discoveries (and complex numbers are a particularly rich set of discoveries from a single invention

You are quite right concerning differing philosophies, sylas.

To me, maths is better likened as a game (or a set of games), where we choose which rules are to govern the particular game we want to play.

WHY we would want to play any particular game at all is not answered within the field of maths, but by pointing to "useful purposes" outside maths.

As for the distinction between "invented/discovered", I don't regard it as of much interest.

Chess was definitely invented, but are strategies leading to victory invented, too?

I'd say they are discovered.

That is to say: Even if maths is to be regarded as arbitrarily invented, it does not follow that we won't do any discoveries within the field invented by our own imagination.

Maths is richer than its inventor(s) ever imagined, to do maths is to discover just how rich it is.
What about the natural numbers, arithmetic and constants like pi and e? These come before any strategies, but are certainly discovered like we discover facts of nature. Godel proved you can't fully axiomize arithmetic.