Definition 1. **The function [x]** represents the largest integer not exceeding x. In other words, for real x, [x] is the unique integer such that

$x – 1 < [x] le x < [x] + 1 $.

We also define ((x)) to be the fractional part of x. In other words $((x)) =x – [x]$. We now list some properties of [x] that will be used in later or in more advanced courses in number theory.

1. $[x + n] = [x] + n$, if n is an integer.

2. $[x] + [y] le [x + y]$.

3. $[x] + [-x]$ is 0 if x is an integer and $-1$ otherwise.

4. The number of integers m for which $x < m le y$ is $[y] – [x]$.

5. The number of multiples of m which do not exceed x is $[ frac{x}{m} ]$.

Using the definition of [x], it will be easy to see that the above properties are direct consequences of the definition. We now define some symbols that will be used to estimate the growth of number theoretic functions. These symbols will be not be really appreciated in the context of this book but these are often used in many analytic proofs.