Fractional and Integral Parts of a Number

The largest integer that does not exceed x is called the integral part of the number x (designated by [x]). Thus, [5.6] = 5, [-3.2] = -4. The function [x] is called the largest integer function. The difference {x} = x - [x] is called the fractional part of x (designated by {x}). The following always holds: 0 ^ {x} < 1. The function {x} is a periodic function with a period one. Closely connected with the fractional part of a number is the concept of the distance to the nearest integer x [designated by (x)], which is defined as follows:

(x)=min [x—k] k=0, ±1, ±2, …

All these concepts are extensively used in number theory and other branches of mathematics.