Draft:Gaussian symbol

The Gaussian symbol is a mathematical symbol in the form of square brackets \lbrack x\rbrack, indicating the largest integer not greater than (equal to or less than) the number x, that is, $$x-1< \lbrack x \rbrack \leq x$$.

The Gaussian symbol first appeared in Gaussian mathematics "Arithmetic Research".

Operation example: $$ \lbrack \pi\rbrack =3, \lbrack 2\rbrack =2,\left \lbrack -\frac{5}{2}\right \rbrack =-3$$.

In computer science, the Gaussian symbol is often expressed as the INT function.

Later, in 1962, Kenneth Iverson called the Gaussian symbol in his book A Programming Language ($$\lfloor x\rfloor $$，floor) and introduced it at the same time. Take the top symbol ($$\lceil x\rceil$$，ceil) (used to represent the smallest of the integers not less than x).

Some properties of Gaussian symbols
If and only if x is an integer, the "equals" sign on the left holds.
 * $$ \lbrack x\rbrack  \le x <  \lbrack x\rbrack  + 1$$
 * For all real numbers x, there are:
 * $$ \left \lbrack \frac{x}{2} \right\rbrack  = \frac{1}{4} ((-1)^{ \lbrack x\rbrack } -1 + 2  \lbrack x\rbrack ) $$
 * $$ \left \lbrack \frac{x}{3} \right\rbrack  = \frac{-2}{\sqrt{3}} \sin(\frac{2\pi}{3} \lbrack x\rbrack  +\frac{\pi}{3}) + 1$$


 * When n is a positive integer, there are:
 * $$ \left \lbrack \frac{x}{n} \right\rbrack  = \frac{x-x(\operatorname{rem} n)}{n}$$


 * When x and n are positive numbers, there are:
 * $$ \left \lbrack \frac{n}{x} \right\rbrack  \geq \frac{n}{x} - \frac{x-1}{x} $$


 * For any integer k and any real number x, there are:
 * $$ \lbrack x+k\rbrack   = k +  \lbrack x\rbrack .$$


 * If x is a real number and n is an integer, we have $$n \le x$$ if and only if $$n \le \lbrack  x \rbrack  $$.
 * Using the Gaussian notation, many prime formulas can be generated (but have no practical use).
 * For non-integer real numbers x, the Gaussian function has the following Fourier series expansion:
 * $$ \lbrack x\rbrack = x - \frac{1}{2} + \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}.$$


 * If m and n are positive prime numbers that are relatively prime, then:
 * $$\sum_{i=1}^{n-1} \lbrack  im / n \rbrack  = (m - 1) (n - 1) / 2$$


 * According to Beatty's theorem, every positive irrational number can be divided into a set of integers by Gaussian notation.
 * For each positive integer k, the representation under p carry is $$ \lbrack  \log_p(k) \rbrack  + 1$$  digit.