Number to Power of Zero Falling is One

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $x \in \R$ be a real number.

$x^{\underline 0} = 1$

where $x^{\underline 0}$ denotes the falling factorial.


Proof

\(\displaystyle x^{\underline 0}\) \(=\) \(\displaystyle \prod_{j \mathop = 0}^{-1} \left({x - j}\right)\) Definition of Falling Factorial
\(\displaystyle \) \(=\) \(\displaystyle 1\) Product is Vacuous

$\blacksquare$