Why is $\prod_{i=1}^{0} a_i = 1$?

Question

I just solved an equation and came accross this little guy here $\prod_{i=1}^{0} a_i$. I only get the desired result if I set $\prod_{i=1}^{0} a_i = 1$. Why is this true? Is this somehow defined? What is the intuition behind it?

Answer 1

Note $${\prod_1^ra_i\over\prod_s^ra_i}=\prod_1^{s-1}a_i$$ because you just cancel $a_sa_{s+1}\cdots a_r$ top and bottom, and what's left is $a_1a_2\cdots a_{s-1}$. Now take $s=1$ to get $${\prod_1^ra_i\over\prod_1^ra_i}=\prod_1^{0}a_i$$ But the left side is clearly $1$.