# Fractional and Integral Parts of a Number

## 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.