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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

$[x]\leq x<[x]+1$

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 $n\leq x$ if and only if $n\leq [x]$ .
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 kx)}{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.