The Greatest Integer Function, also called the floor function, rounds any number down to the nearest whole number. It gives the largest integer that is less than or equal to the given number. This function is often shown as ⌊x⌋ or [x]. For example, ⌊5.6⌋ = 5 and ⌊–2.3⌋ = –3. The graph of this function looks like steps, so it’s also called a step function. The domain (input values) of this function is all real numbers (R), and the range (output values) is all integers (Z). It's useful in rounding off numbers downward in math and programming.

What is Greatest Integer Function?

The Greatest Integer Function, also known as the floor function, gives the largest integer that is less than or equal to a given number. It is written using this symbol: ⌊x⌋ (read as "floor of x").

In simple words, if you give any number to this function, it will round the number down to the nearest whole number. For example:

• ⌊4.8⌋ = 4
• ⌊3⌋ = 3
• ⌊–2.5⌋ = –3

Even if the number is negative or a decimal, the function still picks the closest whole number less than or equal to that number.

Let’s say a number x lies between n and n+1, such that n ≤ x < n+1, where n is an integer. Then, the value of the greatest integer function is n, written as ⌊x⌋ = n.

For example, if x = 6.9, it lies between 6 and 7, so ⌊6.9⌋ = 6.
If x = –1.2, it lies between –2 and –1, so ⌊–1.2⌋ = –2.

This function is widely used in mathematics, programming, and real-life situations where values need to be rounded down.

The range of the greatest integer function is an integer that is (Z) and the domain of the greatest integer function is R i.e. any real number. This implies that for any graph the inputs of the function can take any real number but the output will constantly be an integer.

That is a function represented by [x] recited as step ′x′

It is specified for all x where the domain = (−∞, ∞) and the range is all integers.

In the greatest integer function method, we will simply round off the assigned number to the most adjacent integer that is smaller than or equal to the number itself. Obviously, the input variable x can have any real value. But the output will always be an integer. Some of the examples of the greatest integer function are given in the tabular format:

Greatest Integer Function Graph

The greatest integer function graph is also identified as the step curve because of the step formation of the curve. Let’s understand the graph of the greatest integer function through a plot. Suppose f(x) = ⌊x⌋, if x is an integer, then the value of f will be x itself and if x is not an integer, then the value of x will be the integer just smaller than x.

Example: For all numbers resting in the interval [0,1), the output of f will be 0. That is:

For all numbers resting in the interval [−1,0), f will use the value −1 and so on for the next set of numbers.

Similarly, for all numbers resting in the entire interval [1,2), f will take the value 1.

• So for an integer m, [m, m+1) will hold the value of the greatest integer function as m. From the graph, we can say that the function has a fixed value within any two integers.
• As soon as the subsequent integer appears, the function value shifts by one unit. This indicates that the value of f at x = 1 is 1.
• Therefore there will be a hollow dot at the location (1,0) and a solid dot at the location (1,1).
• Wherein the hollow dot means not involve the value and the solid dot signifies including the value. These observations direct us to the above graph.

Learn about Limit and Continuity

Properties of Greatest Integer Functions

There are different properties of the greatest integer function some of them are as follows:

• \(⌊x+n⌋=⌊x⌋+n,\text{ where },\ n\ ∈Z\)
• \(⌊x⌋=x\ \text{ exists if x is an integer }\ .\)
• \(⌊−x⌋=−⌊x⌋,\ if\ x\ ∈Z\text{ (If x is an Integer)}\)
• \(⌊−x⌋=−⌊x⌋−1,\ ifx\ ∉\ Z\text{ (If x is not an Integer)}\)
• \(\text{ If }⌊\ f(x)⌋\ge L,\text{ then }f(x)\ge L\)

Learn about Logarithmic functions

Important Points on Greatest Integer Functions

The important points on the greatest integer functions are given below:

• If x is a number that lies between successive integers m and m+1 then ⌊x⌋=m.
•  If and only if x is an integer, then the value of ⌊x⌋=x.
• The domain of the greatest integer function is R(all real values) and its range is Z(set of integers).
• The fractional part of a given number will constantly be non-negative, as x will always be higher than or equal to ⌊x⌋. If x is an integer, then its fractional component will be zero.
• The domain of the fractional part function exists in R and its range is [0,1).

Applications of the Greatest Integer Function

The greatest integer function is very useful in real-life situations, especially when dealing with costs, prices, and measurements that need to be rounded down to the nearest whole number.

1. Billing and Invoicing

In shops or industries, when goods are sold by weight or volume, billing is often done using the floor value.
Example: If rice bags weigh 49.9 kg, they might be billed as 49 kg.

2. Programming and Algorithms

Many programming tasks use ⌊x⌋ to ensure integers are used where decimals aren’t valid.
Example: Dividing items among people evenly without fractions.

3. Ticketing and Pricing

When calculating ticket prices based on distance or weight, partial values are ignored using this function.
Example: 5.9 km of travel might be charged for 5 km.

4. Construction and Packaging

Used to calculate how many full items or boxes can fit in a space.
Example: If a shelf can hold 2.9 boxes per row, it will only fit 2 full boxes.

5. Mathematics and Graphing

The function is used in step graphs, and is helpful in solving piecewise functions, inequalities, and real-world math problems.

Greatest Integer Function and Smallest Integer Function

Solved Examples of Greatest Integer Function

Some solved examples of the greatest integer function are given below:

Example 1: ⌊2.4⌋

Remember that number we are looking for must satisfy two conditions.

• The number should be an integer one.
• The number should be lesser than or equal to 2.4.

So a number that is smaller than 2.4 and is an integer is 2.

Therefore ⌊2.4⌋ = 2

Example 2: ⌊-2.66⌋

Again, the number we are viewing for must satisfy the following two conditions.

• The number should be an integer one.
• The number should be lesser than or equal to -2.66.
• A common mistake that students mostly commit is to assume that [-2.66] = -2.
• For ⌊2.66⌋ = 2, it resembles that we just lifted the .5. But that is not the case with negative numbers and ⌊-2.66⌋ = -2 is the wrong answer.

⌊-2.66⌋ is not equal to -2.

Recollect that we are searching for a number smaller than -2.66. Hence the number that is smaller than -2.66 is -3.

So ⌊-2.66⌋= -3

Example 3: ⌊5⌋

Here we are viewing for a number lesser than or equal to 5. As 5 is equivalent to 5.

Therefore ⌊5⌋= 5

Example 4: ⌊0. 56⌋

Till now it would be certain that we would focus on the number that is less or equal and try to neglect as much as possible the word greatest. So the integer that is less than 0.56 is 0.

Since it is 0, ⌊0. 56⌋ = 0

We hope that the above article on Greatest Integer Functions is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.