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Last updated on September 18, 2025
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.
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.
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 |
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The properties of the greatest integer function are used to simplify and solve problems involving rounding. Some properties of the greatest integer function are:
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:
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 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 |
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)
The greatest integer function is used to convert continuous or fractional values into whole numbers. Here are some applications of the 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.
Find ⌊3.5⌋?
⌊3.5⌋ is 3.
The greatest integer less than or equal to 3.5 is 3
Find the value of x in ⌊x + 2⌋ = 4
The value of x in ⌊x + 2⌋ = 4 is any real number in [2, 3).
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)
Find ⌊-9.23⌋?
⌊-9.23⌋ = -10
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
Find ⌊12.0001⌋?
⌊12.0001⌋ is 12
The greatest integer less than or equal to 12.0001 is 12, so ⌊12.0001⌋ = 12.
Find the value of x in ⌊x + 4⌋ = 18
The value of x in ⌊x + 4⌋ = 18 is x ∈ [14, 15)
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.
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.
: He loves to play the quiz with kids through algebra to make kids love it.