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

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Answer 1 · 2016-06-27T06:28:25

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

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 $1$ .