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As an example, for any one dimensional real valued function $f$, if $q<k$ then $\prod_{i=k}^q f(i)=1$. I thought it was $0$ and that was breaking my intuition about some formulas I'm working with. This makes much more sense. Why is this true? Has the fact that the zeroth power of any real number equals one has anything to do with this?

Garmekain
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    https://en.wikipedia.org/wiki/Empty_product – A. Thomas Yerger May 27 '18 at 15:34
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    $x^n = \prod_{i=1}^nx$, so, yes, this is closely related to the convention that $x^0 = 1$. – Rob Arthan May 27 '18 at 15:38
  • 1, not 0, is the multiplicative identity. If the empty product were $0$ then all products, by induction you have to be $0$ because $0\times\text{the first thing} = 0$. But $1 \times \text{the first thing} = \text{the first thing}$. ... That said I'm not sure your classification of what $\prod_{i=k}^q$ with $q < k$ means is actually correct. – fleablood May 27 '18 at 15:41
  • Ultimately, it is just a convention. It is a convention we chose to follow for the reasons fleablood and Rob Arthan gave, but it is still simply a choice. – Paul Sinclair May 27 '18 at 22:53

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