mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Hermite's identity, named after Charles Hermite, gives the value of a

summation In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...

involving the

floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

. It states that for every

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

''x'' and for every positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

''n'' the following identity holds:. :

\sum_^\left\lfloor x+\frac\right\rfloor=\lfloor nx\rfloor .

Proofs

Proof by algebraic manipulation

Split

x

into its

integer part In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

and

fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...

x=\lfloor x\rfloor+\

. There is exactly one

k'\in\

with :

\lfloor x\rfloor=\left\lfloor x+\frac\right\rfloor\le x<\left\lfloor x+\frac\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 \+\frac\right\rfloor\le \<\left\lfloor \+\frac\right\rfloor=1.

Therefore, :

1-\frac\le \<1-\frac ,

and multiplying both sides by

n

gives :

n-k'\le n\, \ Now if the summation from Hermite's identity is split into two parts at index k', it becomes
: \begin

\sum_^\left\lfloor x+\frac\right\rfloor

& =\sum_^ \lfloor x\rfloor+\sum_^ (\lfloor x\rfloor+1)=n\, \lfloor x\rfloor+n-k' \\ pt & =n\, \lfloor x\rfloor+\lfloor n\,\\rfloor=\left\lfloor n\, \lfloor x\rfloor+n\, \ \right\rfloor=\lfloor nx\rfloor.

\end

Proof using functions

Consider the function :

f(x) = \lfloor x \rfloor + \left\lfloor x + \frac \right\rfloor + \ldots + \left\lfloor x + \frac \right\rfloor - \lfloor nx \rfloor

Then the identity is clearly equivalent to the statement

f(x) = 0

for all real

x

. But then we find, :

f\left(x + \frac \right) = \left\lfloor x + \frac \right\rfloor + \left\lfloor x + \frac \right\rfloor + \ldots + \left\lfloor x + 1 \right\rfloor - \lfloor nx + 1 \rfloor = f(x)

Where in the last equality we use the fact that

\lfloor x + p \rfloor = \lfloor x \rfloor + p

for all integers

p

. But then

f

has period

1/n

. It then suffices to prove that

f(x) = 0

for all

x \in [0, 1/n)

. But in this case, the integral part of each summand in

f

is equal to 0. We deduce that the function is indeed 0 for all real inputs

x

References

{{DEFAULTSORT:Hermite's Identity Mathematical identities Articles containing proofs