Why is empty product defined to be $1$? [duplicate]

Solution 1:

Note that if $\displaystyle\prod_{i=1}^n a_i=a$ then $\displaystyle\prod_{i=1}^{n-1}a_i = \frac a{a_n}$. Now note that $\displaystyle\prod_{i=1}^1 a=a$, so $\displaystyle\prod_{i=1}^0a=1$.

Solution 2:

Every product should start with $1$. For instance $$\prod_{i=1}^2a_i:=1\cdot a_1\cdot a_2$$ This mode of thinking at least makes the empty product naturally work out to $1$. But it's more than that. If you think of the above in the usual way as $a_1\cdot a_2$,then it's not really symmetric. It reads that you started with the object $a_1$, and then brought in the operation "$\cdot a_2$". If you start with $1$, then you are bringing in both factors as operations: "$\cdot a_1$" and "$\cdot a_2$". This is arguably more symmetric. (Similarly, define $\sum_{i=1}^na_i:=0\;\overbrace{{}+a_1}\;\overbrace{{}+a_2}\;+\cdots$.)