Fractional Part function

10. Fractional Part f(x) = {x} 

Every number 'x' can be written as the sum of its integer and fractional parts.

x = [x] + {x}

Therefore fractional part,

{x} = x - [x]
For example, 4.7 = 4 + 0.7

or, 0.7 = 4.7 - 4

i.e. fractional part = Number - Integral Part

x = 4.7 , [x] = [4.7] = 4

and {x} = {4.7} = 0.7

Consider, {-4.7}

{-4.7} = -4.7 - [-4.7] = -4.7 - (-5) 

or, {-4.7} = 0.3

if the number is integer, its fractional part is obviously zero.

{6} = 6 - [6] = 6 - 6 = 0

Note- fractional part is always positive and it never becomes 1.

in other words, fractional part of a number is the difference between the number and its integral part.

Remember, 

i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part.

How to draw the graph,

note this,

, if 

because the fractional part is equal to the number if number is between 0 and 1

i.e. {0.6} = 0.6, {0.78} = 0.78

or, {x} = x - [x]

if , [x] = 0

therefore, {x} = x , if 

As soon as x becomes 1, f(x) drops again to 0, and then start increasing as 'x' increases.

i.e. {x} = x + 1 ,  ( since [x] = 1 when )

{x} = x + 2, 

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Properties of {x}:

1. {x+I} = {x} ( where I is an integer)

2. {-x} = 1 - {x}

3. {x} = 0, if 

consider 1st property,

, where I is an integer,

i.e. {9.3} = { 1.3} = {2.3} 
or {3.5 + 7} = {3.5} = 0.5


consider 2nd property,


{-6.9} = 0.1 = 1 - {6.9} = 0.1

Third is obvious any integer has no fraction part i.e. fraction part is zero

if, 
then , 

here y or f(x) = {x} repeats its value after every interval of 1.

i.e. {x} is periodic ,


Graph:

If y = {x} = x - [x]

then, y = x , when 

         y = x - 1, when  ( between 1 and 2 the graph of {x} = x - 1 )

         y = x -2 , when  

and so on...