Hermite's identity

In mathematics, the Hermite's identity states that for every real number x and positive integer n the following holds:

\sum_{k=0}^{n-1}\left\lfloor x%2B\frac{k}{n}\right\rfloor=\lfloor nx\rfloor .

Proof

Write x=\lfloor x\rfloor%2B\{x\}. There is exactly one k'\in\{1,...,n\} with

\lfloor x\rfloor=\left\lfloor x%2B\frac{k'-1}{n}\right\rfloor\le x<\left\lfloor x%2B\frac{k'}{n}\right\rfloor=\lfloor x\rfloor%2B1
\Rightarrow 0=\left\lfloor \{x\}%2B\frac{k'-1}{n}\right\rfloor\le \{x\}<\left\lfloor \{x\}%2B\frac{k'}{n}\right\rfloor=1 \, \Rightarrow \, 1-\frac{k'}{n}\le \{x\}<1-\frac{k'-1}{n} \, \Rightarrow \, n-k'\le n\, \{x\}<n-k'%2B1.

Now

\sum_{k=0}^{n-1}\left\lfloor x%2B\frac{k}{n}\right\rfloor =\sum_{k=0}^{k'-1} \lfloor x\rfloor%2B\sum_{k=k'}^{n-1} (\lfloor x\rfloor%2B1)=n\, \lfloor x\rfloor%2Bn-k' =n\, \lfloor x\rfloor%2B\lfloor n\,\{x\}\rfloor=\left\lfloor n\, \lfloor x\rfloor%2Bn\, \{x\} \right\rfloor=\lfloor nx\rfloor.

See also