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:
∑ k = 0 n − 1 ⌊ x + k n ⌋ = ⌊ n x ⌋ . Split x into its integer part and fractional part, x = ⌊ x ⌋ + { x } . There is exactly one k ′ ∈ { 1 , … , n } with
⌊ x ⌋ = ⌊ x + k ′ − 1 n ⌋ ≤ x < ⌊ x + k ′ n ⌋ = ⌊ x ⌋ + 1. By subtracting the same integer ⌊ x ⌋ from inside the floor operations on the left and right sides of this inequality, it may be rewritten as
0 = ⌊ { x } + k ′ − 1 n ⌋ ≤ { x } < ⌊ { x } + k ′ n ⌋ = 1. Therefore,
1 − k ′ n ≤ { x } < 1 − k ′ − 1 n , and multiplying both sides by n gives
n − k ′ ≤ n { x } < n − k ′ + 1. Now if the summation from Hermite's identity is split into two parts at index k ′ , it becomes
∑ k = 0 n − 1 ⌊ x + k n ⌋ = ∑ k = 0 k ′ − 1 ⌊ x ⌋ + ∑ k = k ′ n − 1 ( ⌊ x ⌋ + 1 ) = n ⌊ x ⌋ + n − k ′ = n ⌊ x ⌋ + ⌊ n { x } ⌋ = ⌊ n ⌊ x ⌋ + n { x } ⌋ = ⌊ n x ⌋ . 