Greatest Integer Function and its Applications

The greatest integer function, denoted by [x], gives the greatest integer less than or equal to the given number x.

To put it simply, if the given number is an integer, then the greatest integer gives the number itself, otherwise it gives the first integer towards the left of the number of x on the number line.

For example,

[1.4] = 1

[4]= 4

[3.4] = 3

[ - 2.3] = - 3

[ - 5.6] = - 6, and so on.

NOTE: We can see that [1.4] = 1 + 0.4 or x = [x] + {x}, where {x} is the fractional part of x. For x = - 2.3, [x] = - 3 and {x} = 0.7

In the following figure, we can see that the greatest integer function gives the number itself (when the given number is an integer) or the first integer to the left of the number on the number line.

The graph of greatest integer function is given below. Note that the red dot indicates that integer value on the number line is not included while the green dot indicates that the integer value is included.