#### Greatest Integer And Fractional part function.

### What is Greatest Integer and Fractional Part Function?

The **Greatest integer function is denoted by [f(x)]** where as the **Fractional part function is denoted by {f(x)}.**

Where [f(x)] is read as greatest integer value of f(x) where as {f(x)} is read as fractional part of f(x).

Let Domain of f(x) is $x\in D$ then **Domain of [f(x)]** is $x\in D$, And **Outcome of Greatest Integer function are Integers.** i.e $[f(x)]\in I$

The **Greatest integer of x is denoted by [x]**, it is an integer less than or equal to x. That is if we consider all integers less than or equal to x the greatest of all such integers is **Greatest Integer Value** of x.

The easy way to understand the same is assume that you are at point P (x) on the real axis, as you start moving on real axis towards negative infinity from point P. The first integer you get is **Greatest Integer Value** of x. This also include the point P. If P is an integer point.

**If $n\leq x \lt (n+1)$ then [x] = n and vise versa, If [x]=n the $x \in [n, n+1)$.**

**Example :**

[2.31] = 2

[$\pi$] = 3

[-2.31] = -3

[-$\pi$] = -4

[$\sqrt{2}$ -1] = 0

[1 – $\sqrt{3}$] = -1

[ 5 ] =5 and [- 3 ] = -3

### Fractional part Function.

**What is Fractional part function?**

**Fractional part function** f(x) is denoted by {f(x)}, is read as fractional part of f(x). Domain of Fractional Part function is same as Domain of function f(x) And Outcome always lie in 0 to 1 i.e $0\leq \{f(x)\} \lt 1$

**Fractional part function** is denoted by {x}, is read as fractional part of x. Let [x] be the greatest integer value of a real number x then fractional part of x is **{x}=x-[x]**.

**If $x\in [n, n+1)$ then [x] = n and {x} = x – n.**

As $n\leq x \lt (n+1)$ $\Rightarrow \, 0\leq x – n \lt 1$

$\therefore \, 0\leq \{x\} \lt 1$

**Example :**

{2.31} = 2.31 – 2 = 0.31

{$\pi$} = $\pi$ – 3

{-2.31} = – 2.31 – (-3 ) = 0.69

{-$\pi$} = -$\pi$ – (-4) = 4 – $\pi$

{$\sqrt{2}$ -1} = ($\sqrt{2}$ – 1) – 0 = $\sqrt{2}$ – 1

{1 – $\sqrt{3}$} = 1-$\sqrt{3}$ – (-1) = 2 – $\sqrt{3}$

{ 5 } =5 – 5 = 0 and {- 3 } = -3 – (-3) = 0

### Properties Of Greatest Integer and Fractional Part Function

- [x]=x if $x\in I$
- If $n\leq x\lt (n+1)$ then [x]=n where $n\in I$
- $x-1\lt [x] \leq x$ for all $x\in R$
- If $x\in [n,n+1)$ then Fraction part of x is {x}=x-n
- [x+n] = [x] + n ; For $n\in I$
- {x+n} = {x} For $n\in I$ .Hence Fractional x is a periodic function with fundamental period as 1 and $n\in N$ is one of its period.
- $[x+y] = \begin{cases} [x]+[y] &; \text {if {x}+{y}<1} \\ [x]+[y]+1 &; \text{ if {x}+{y}$\geq$ 1 } \end{cases}$
- [$x_1+x_2+x_3...+x_n$]=[$x_1$]+[$x_2$]+[$x_3$]+...+[$x_n$]+r if $r\leq \{x_1\}$+{$x_2$}+{$x_3$}+...+{$x_n\}\lt (r+1)$ where $r\in I$ and $0\leq r \leq (n-1)$
- $[x] + [-x] = \begin{cases} 0 &; x\in I \\ -1 &; x\notin I \end{cases}$
- $\{x\} + \{-x\} = \begin{cases} 0 &; x\in I \\ 1 &; x\notin I \end{cases}$

### Properties Of Greatest Integer and Fractional Part Function

- [x]=x if $x\in I$
- If $n\leq x\lt (n+1)$ then [x]=n where $n\in I$
- $x-1\lt [x] \leq x$ for all $x\in R$
- [x+n] = [x] + n ; For $n\in I$
- {x+n} = {x} For $n\in I$ .Hence Fractional x is a periodic function with fundamental period as 1 and $n\in N$ is one of its period.
- $[x+y] = \begin{cases} [x]+[y] \\ ;\text {if {x}+{y}<1} \\ [x]+[y]+1 \\; \text{ if {x}+{y}$\geq$ 1 } \end{cases}$
- [$x_1+x_2+x_3...+x_n$]=[$x_1$]+[$x_2$]+[$x_3$]+...+[$x_n$]+r if $r\leq \{x_1\}$+{$x_2$}+{$x_3$}+...+{$x_n\}\lt (r+1)$ where $r\in I$ and $0\leq r \leq (n-1)$
- $[x] + [-x] = \begin{cases} 0 &; x\in I \\ -1 &; x\notin I \end{cases}$
- $\{x\} + \{-x\} = \begin{cases} 0 &; x\in I \\ 1 &; x\notin I \end{cases}$