why 0! = 1. Idk it seems counterintuitive.

Question

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In: Mathematics

Answer 1

The best reason is that factorial is a definition that depends on itself. For example, 5! Is the same as 5 * 4! In fact, for any number n, n! is exactly equal to n * (n-1)!

The only way for that to work is to somewhat arbitrarily define 0! as being precisely equal to 1. I say arbitrarily not knowing what order events took place in, but it helps me to understand to consider it such.

For me, I imagine that it’s this way because it makes the math work out and has the side effect of continuing to be useful describing things about our world. It’s a concrete answer that has helped me accept it.

Answer 2

5! = 120

(now divide by five)

4! = 24

(now divide by four)

3! = 6

(now divide by three)

2! = 2

(now divide by two)

1! = 1

(now divide by one)

0! = 1

This is by no means rigorous, but it helps with intuition.

Answer 3

Work through it backwards…

5! / 5 = 4!

4! / 4 = 3!

3! / 3 = 2!

2! / 2 = 1!

1! / 1 = 0!

0! / 0 … whoops there’s no -1 factorial.

Answer 4

There’s an easy proof for it that might help some people understand.

n! = (n+1)! / n+1

For example, if n = 3:

3! = 4! / 4 = 24 / 4 = 6

So 0! = 1! / 1 = 1 / 1 = 1