Question

Why is 0! (zero factorial) defined to be 1?

Answer 1

`0! = 1` because it is an empty product.

Explanation:

Compare how we handle empty sums and empty products:

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Sums

The number `0` is the identity under addition, i.e.:

`0+a = a+0 = a`

for any number `a`.

Consider (the formula for triangular numbers):

`T_n = sum_(k=0)^n k`

If `n=0`, then the sum is empty and equal to the additive identity `0`.

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Products

The number `1` is the identity under multiplication, i.e.:

`1*a = a*1 = a`

for any number `a`.

Consider (the formula for factorials):

`n! = prod_(k=0)^n k`

If `n=0`, then the product is empty and equal to the multiplicative identity `1`.