Greatest Integer Function

The function used to find the greatest integer less than or equal to the given number is the greatest integer function. It is represented as ⌊x⌋ for the number x. Mathematically, the greatest integer function ⌊x⌋ is represented as: 
⌊x⌋ = n, where n ≤ x < n + 1, where n is an integer. 

For example, ⌊5.09⌋ is 5, as 5 ≤ 5.09 < 6.

To round a real number to an integer in math, we use the greatest and smallest integer functions. In this section, we will learn the differences between the greatest and smallest integer functions.

The properties of the greatest integer function are used to simplify and solve problems involving rounding. Some properties of the greatest integer function are:

• ⌊x⌋ = x, if x is an integer 
   
•  ⌊x + I⌋ = ⌊x⌋ + I, where I is an integer. The greatest integer of x + I is equal to the sum of the greatest integer of x and I
   
• ⌊x + y⌋ ≥ ⌊x⌋ + ⌊y⌋, for any real numbers x and y, the greatest integer of their sum is greater than or equal to the sum of their individual greatest integers 
   
• If ⌊f(x)⌋ ≥ I, then f(x) ≥ I
   
• If ⌊f(x)⌋ ≤ I, then f(x) < I + 1
   
• ⌊-x⌋ = -⌊x⌋, if X is an integer 
   
• ⌊-x⌋ = -⌊x⌋-1, if x is not an integer

For a better understanding of the concept of the greatest integer function, we can use a number line for visual representation. In this section, we will learn how to represent the greatest integer function on a number line by following these steps: 

• First, we draw a number line and mark the numbers in equal intervals

• Then we mark the real number

• On the number line, the greatest integer function of a number is the next integer less than or equal to that number, that is, the number to its left. 

For example, ⌊4.9⌋

To represent ⌊4.9⌋ on a number line, start drawing the number line and mark the point 4.9. The greatest integer less than or equal to 4.9 is the number left to it, which is 4, the number immediately to the left. So, the ⌊4.9⌋ = 4

The greatest integer function has both a domain and a range. The set of all real numbers (R) is the domain of the greatest integer function. The set of all possible output values is the range, and it is an integer.

The greatest integer function can be represented using a graph, which is a curve with a step structure. So the graph is also known as the step function. For understanding how to plot the function, consider f(x) = ⌊x⌋. Where, if x is an integer, then f(x) = x, and if x is not an integer, then f(x) ≤ x, the integer left to x. 

For instance, for any x in the interval [2, 3), f(x) = 2
For any x the interval [-2, -1), F(x) = -2

In other words, that is for any integer n, all the numbers in the interval [n, n+1) will have f(x) = n. When x reaches n + 1, the function value will be n + 1, so the graph has a step structure:
A solid dot at the point (n, n) indicates the value included, and the values excluded are indicated using a hollow dot at (n + 1, n)

The greatest integer function is used to convert continuous or fractional values into whole numbers. Here are some applications of the greatest integer function:

• In shipping and packing, the greatest integer function is used to calculate how many full boxes are needed to pack the items. For example, if we need to pack 47 items, and we pack 10 items per box, ⌊47 ÷ 10⌋ = 4, which means 4 boxes are required to pack. 

• In event planning, the greatest integer function helps with seating arrangements, planning food, scheduling and timing, and resource allocation. 

• In logistics, we use the greatest integer function for inventory management.