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

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

Compare how we handle empty sums and empty products:

##color(white)()##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##.

##color(white)()##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##.