User:Black cat 99168/Gaussian symbol

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

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

Operation example:

$$[\pi]=3,[2]=2,\left [-\frac{5}{2}\right ]=-3$$.

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

 * $$ [x] \le x < [x] + 1$$


 * If and only if x is an integer, the equal sign on the left holds.


 * For all real numbers x, there are:


 * $$ \left[ \frac{x}{2} \right] = \frac{1}{4} ((-1)^{[x]} -1 + 2 [x]) $$
 * $$ \left[ \frac{x}{3} \right] = \frac{-2}{\sqrt{3}} \sin(\frac{2\pi}{3}[x] +\frac{\pi}{3}) + 1$$


 * When n is a positive integer, there are:


 * $$ \left[ \frac{x}{n} \right] = \frac{x-x(rem ~ n)}{n}$$


 * When x and n are positive numbers, there are:


 * $$ \left[ \frac{n}{x} \right] \geq \frac{n}{x} - \frac{x-1}{x} $$


 * For any integer k and any real number x, there are:


 * $$ [x+k] = k + [x].$$


 * If x is a real number and n is an integer, we have $$n \le x$$ if and only if $$n \le [ x ] $$.
 * 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:


 * $$[x] = x - \frac{1}{2} + \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}.$$


 * If m and n are positive prime numbers that are relatively prime, then:


 * $$\sum_{i=1}^{n-1} [ im / n ] = (m - 1) (n - 1) / 2$$


 * 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  $$[ \log_{p}(k) ] + 1$$  digit.