Draft:Gaussian symbol

• Comment: Fails WP:GNG, requires significant coverage in multiple independent secondary sources. Dan arndt (talk) 09:46, 18 March 2021 (UTC)

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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

• $\lbrack x\rbrack \leq x<\lbrack x\rbrack +1$

If and only if x is an integer, the "equals" sign on the left holds.

• 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 $n\leq x$ if and only if $n\leq \lbrack x\rbrack$.
• 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:

$\lbrack x\rbrack =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.$

• 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.

References

1. Carl Friedrich Gauss (1798). Disquisitiones Arithmeticae.