Integral part

entier, integer part of a (real) number $x$

The largest integer not exceeding $x$. It is denoted by $[x]$ or by $E(x)$. It follows from the definition of an integer part that $[x]\leq x<[x]+1$. If $x$ is an integer, $[x]=x$. Examples: $[3.6]=3$; $[1/3]=0$, $[-13/5]=-5$. The integral part is used in the factorization of, for example, the number $n!=1\dots n$, viz.

$$n!=\prod_{p\leq n}p^{\alpha(p)},$$

where the product consists of all primes $p$ not exceeding $n$, and

$$\alpha(p)=\left[\frac np\right]+\left[\frac{n}{p^2}\right]+\mathinner{\ldotp\ldotp\ldotp\ldotp}$$

The function $y=[x]$ of the variable $x$ is piecewise continuous (a step function) with jumps at the integers. Using the integral part one defines the fractional part of a number $x$, denoted by the symbol $\{x\}$ and given by

$$x-[x];\quad0\leq\{x\}<1.$$

The function $y=\{x\}$ is a periodic and piecewise continuous.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)