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 $0\leq r \leq (n-1)$
- $[x] + [-x] = \begin{cases} 0 &; x\in I \\ -1 &; x\notin I \end{cases}$
- $\{x\} + \{-x\} = \begin{cases} 0 &; x\in I \\ 1 &; x\notin I \end{cases}$