What is Greatest integer function and frictional part.

Greatest integer function:

                                                  The greatest integer function is a real number  x, denoted by [x] is the largest integer less than or equal to x.

The equation of the greatest integer function is:

                                        [x] ≦ x < [x] + 1

For example:

x= 5.6

Then the greater integer function of x is.

[x]=[5.6]=5

Lets do a proof.

Show that if n is an integer than [x+n]=n+[x],where x is a real number.

solution:

proof

     Let [x]=m, so m is an integer 

this implies that,

     m ≦  x < m + 1

now adding n to the inequality we get.

    m + n ≦ x +n < m + n +1

This shows that m + n is the largest integer less than or equal to x + n,

   [x + n]=m + n               as , m=[x]

so,

 [x + n]= [x] + n

Fractional part:

            The fractional part of a real number x,denoted by {x}, is the difference  of  x - [x], Where [x] is the greater integer function.

For example:

the frictional part of  5/4 will be,

 {5/4}= 5/4 -[5/4]

          =5/4 -1

          =1/4

so 1/4 is the frictional part of 5/4