Hermite's identity

In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds:^[1][2]

$\sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n}\right\rfloor=\lfloor nx\rfloor .$

Proof

Split $x$ into its integer part and fractional part, $x=\lfloor x\rfloor+\{x\}$. There is exactly one $k'\in\{1,\ldots,n\}$ with

$\lfloor x\rfloor=\left\lfloor x+\frac{k'-1}{n}\right\rfloor\le x<\left\lfloor x+\frac{k'}{n}\right\rfloor=\lfloor x\rfloor+1.$

By subtracting the same integer $\lfloor x\rfloor$ from inside the floor operations on the left and right sides of this inequality, it may be rewritten as

$0=\left\lfloor \{x\}+\frac{k'-1}{n}\right\rfloor\le \{x\}<\left\lfloor \{x\}+\frac{k'}{n}\right\rfloor=1.$

Therefore,

$1-\frac{k'}{n}\le \{x\}<1-\frac{k'-1}{n} ,$

and multiplying both sides by $n$ gives

$n-k'\le n\, \{x\}<n-k'+1.$

Now if the summation from Hermite's identity is split into two parts at index $k'$, it becomes

$\begin{align} \sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n}\right\rfloor & =\sum_{k=0}^{k'-1} \lfloor x\rfloor+\sum_{k=k'}^{n-1} (\lfloor x\rfloor+1)=n\, \lfloor x\rfloor+n-k' \\[8pt] & =n\, \lfloor x\rfloor+\lfloor n\,\{x\}\rfloor=\left\lfloor n\, \lfloor x\rfloor+n\, \{x\} \right\rfloor=\lfloor nx\rfloor. \end{align}$

References

1. Savchev, Svetoslav; Andreescu, Titu (2003), "12 Hermite's Identity", Mathematical Miniatures, New Mathematical Library 43, Mathematical Association of America, pp. 41–44, ISBN 9780883856451.
2. Matsuoka, Yoshio (1964), "Classroom Notes: On a Proof of Hermite's Identity", The American Mathematical Monthly 71 (10): 1115, doi:10.2307/2311413, MR 1533020.