BrightChamps Logo
Login

Summarize this article:

Live Math Learners Count Icon247 Learners

Last updated on October 18, 2025

Greatest Integer Function

Professor Greenline Explaining Math Concepts

In mathematics, the greatest integer function rounds a given number down to the nearest integer less than or equal to it. In this article, we will learn about the greatest integer function, its properties, and its graph.

Greatest Integer Function for US Students
Professor Greenline from BrightChamps

What is the 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, since 5 ≤ 5.09 < 6.

Professor Greenline from BrightChamps

Difference Between Greatest Integer Function and Smallest Integer Function

In mathematics, real numbers can be rounded using the greatest and smallest integer functions. In this section, we will learn the differences between the greatest and smallest integer functions.

 

Greatest Integer Function Smallest Integer Function

It is represented by ⌊x⌋

It is represented by ⌈x⌉

For a number x, the greatest integer function is the integer that is less than or equal to x.

For the number x, the smallest integer function is the number that is greater than or equal to x.

The greatest integer function is also known as the floor function

The smallest integer function is also known as the ceiling function

For example, ⌊3.2⌋ = 3, as 3 ≤ 3.2 < 4

For example, ⌈3.2⌉= 4, as 4 ≥ 3.2 > 3

Professor Greenline from BrightChamps

What are the Properties of Greatest Integer Function?

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 


 

  •  If I is an integer, ⌊x + I⌋ = ⌊x⌋ + I. That is, the greatest integer of (x + I) equals the sum 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
Professor Greenline from BrightChamps

How to Represent on Number Line?

A number line can help visually represent the greatest integer function. 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 of a number is the largest integer less than or equal to it, located 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

Professor Greenline from BrightChamps

What are the Domain and Range of Greatest Integer Function?

The greatest integer function is defined for all real numbers and outputs integers. 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.

 

Value of x f(x) = ⌊X⌋ Explanation
5.8 ⌊5.8⌋ = 5 The greatest integer less than or equal to 5.8 is 5
-3.2 ⌊-3.2⌋ = -4 The greatest integer less than or equal to -3.2 is -4
7 ⌊7⌋ = 7 As 7 is an integer
-6.999 ⌊-6.999⌋ = -7 The greatest integer less than or equal to -6.999 is -7

 

Professor Greenline from BrightChamps

How to Represent Greatest Integer Function on Graph?

The greatest integer function can be represented by a graph with a step structure. Hence, the graph is also called a 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, for any integer n, all numbers in [n, n+1) have f(x) = n. When x reaches n + 1, the function value becomes n + 1. Thus, 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)

Professor Greenline from BrightChamps

Tips and Tricks to Master Greatest Integer Function

Learn to understand, visualize, and apply the greatest integer function in calculations and real-life problems.

 

  • Know that the greatest integer function gives the largest integer less than or equal to a number.
     
  • Practice with both positive and negative numbers.
     
  • Plot stepwise graphs to visualize the function.
     
  • Apply it in expressions with addition, multiplication, or fractions.
     
  • Solve real-life problems like rounding, time conversion, and inventory management.
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Greatest Integer Function

Here are some common mistakes that students often make when working with the greatest integer function. By understanding these common mistakes, students can master the greatest integer function.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Mistaking the greatest integer function for rounding

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may think ⌊3.9⌋ = 4, but the correct value is 3. The process of rounding down the number to the integer that is less than or equal to the number is the greatest integer function.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Misapplying properties of addition

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often misapply the addition property of the greatest integer function. As they assume that ⌊X + Y⌋ = ⌊X⌋ + ⌊Y⌋ instead of ⌊X + Y⌋ ≥ ⌊X⌋ + ⌊Y⌋. So, always first find the value of ⌊X⌋ and ⌊Y⌋, then add them.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Errors when finding the greatest integer function of a negative number

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

When calculating ⌊-x⌋, students incorrectly round toward zero. For example, ⌊-2.3⌋ ≠ -2; the correct value is -3. As round down means towards left, as we round the number to the integer which is less than or equal to the number, so, ⌊-2.3⌋ = -3.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing the greatest integer function with the smallest integer function

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often confuse the greatest and smallest integer functions.

 

For example, ⌊1.34⌋ = 2 instead of 1. So always remember that in the greatest integer function, the given number is rounded down to the nearest integer ≤ x. The smallest integer function is rounding up to the nearest integer ≥ x.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Error assumption of the domain and range 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Some students believe the range of ⌊x⌋ includes non-integers, e.g., ⌊x⌋ = 3.5. This is incorrect, as ⌊x⌋ always outputs an integer. So always remember that the range of ⌊x⌋ is always the set of integers, for example, ⌊4.5⌋ = 4. 

arrow-right
arrow-right
Professor Greenline from BrightChamps

Real-World Applications of Greatest Integer Function

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

 

 

  • Rounding down prices or payments: In financial calculations, amounts are often rounded down to the nearest whole number using the greatest integer function.
     
  • Time calculation: Converting minutes to hours or seconds to minutes often involves taking the greatest integer to get complete units.
     
  • Seating arrangements: Determining the number of complete rows or groups that can be formed from a total number of people.
     
  • Inventory management: Calculating the maximum number of full boxes or packs that can be made from a given quantity of items.
  • Computer science and programming: Used in algorithms for indexing, discretization, and mapping continuous values to discrete intervals.
Max from BrightChamps Saying "Hey"
Hey!

Solved Examples of Greatest Integer Function

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Find ⌊3.5⌋?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

⌊3.5⌋ is 3.

Explanation

The greatest integer less than or equal to 3.5 is 3

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Find the value of x in ⌊x + 2⌋ = 4

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The value of x in ⌊x + 2⌋ = 4 is any real number in [2, 3).

Explanation

Given, 

⌊x + 2⌋ = 4 ⇒ 4 ≤  x + 2 < 5

Subtracting each part by 2, 

4 - 2 ≤ x + 2 - 2 < 5 - 2

2 ≤ x <3

So, x ∊ [2, 3)

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Find ⌊-9.23⌋?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

⌊-9.23⌋ = -10

Explanation

For a negative number, the greatest integer is the largest integer less than or equal to it. The nearest integer less than or equal to -9.23 is -10

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Find ⌊12.0001⌋?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

⌊12.0001⌋ is 12

Explanation

The greatest integer less than or equal to 12.0001 is 12, so ⌊12.0001⌋ = 12.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

Find the value of x in ⌊x + 4⌋ = 18

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The value of x in ⌊x + 4⌋ = 18 is x ∈ [14, 15)

Explanation

Given, ⌊x + 4⌋ = 18

⌊x + 4⌋ = 18 ⇒ 18 ≤ x + 4 < 19

Subtracting 4, 

18 - 4 ≤ x + 4 - 4 < 19 - 4 

14 ≤ x < 15
Hence, x can be any real number between 14 and 15.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on Greatest Integer Function

1.What is the greatest integer function?

The greatest integer function of a number is the nearest integer that is less than or equal to the number.

Math FAQ Answers Dropdown Arrow

2.What is ⌊8.59⌋?

The ⌊8.59⌋ is 8

Math FAQ Answers Dropdown Arrow

3.Is ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋?

No, ⌊x + y⌋ ≠ ⌊x⌋ + ⌊y⌋. ⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ ≤ ⌊x⌋ + ⌊y⌋ + 1.

Math FAQ Answers Dropdown Arrow

4.What is the smallest integer function?

The smallest integer function of a number is the nearest integer that is greater than or equal to the number. 

Math FAQ Answers Dropdown Arrow

5.What is ⌊-6.24⌋?

⌊-6.24⌋ = -7

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom