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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, .
The Gaussian symbol first appeared in Gaussian mathematics "Arithmetic Research".
Operation example:
。
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" (,floor) and introduced it at the same time. Take the top symbol (,ceil) (used to represent the smallest of the integers not less than x).
Some properties of Gaussian symbols
[edit]- If and only if x is an integer, the equal sign on the left holds.
- For all real numbers x, there are:
- When n is a positive integer, there are:
- When x and n are positive numbers, there are:
- For any integer k and any real number x, there are:
- If x is a real number and n is an integer, we have if and only if .
- 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:
- If m and n are positive prime numbers that are relatively prime, then:
- 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 digit.