# Greatest Integer Function

Last Updated : 05 Apr, 2024

The greatest Integer Function [X] indicates an integral part of the real numberÂ [Tex]xÂ Â Â Â [/Tex]which is the nearest and smaller integer toÂ [Tex]xÂ Â [/Tex]. It is also known as the floor of X.

[x]=the largest integer that is less than or equal to x.

In general: If,Â [Tex]nÂ Â Â Â [/Tex]<=Â [Tex]XÂ Â Â Â [/Tex]<Â [Tex]n+1Â Â Â Â [/Tex]. Then,Â [Tex](n \epsilon Integer)\Longrightarrow [X]=nÂ Â Â Â [/Tex]
This means if X lies in [n, n+1), then the Greatest Integer Function of X will be n.

In the above figure, we are taking the floor of the values each time. When the intervals are in the form of [n, n+1), the value of the greatest integer function is n, where n is an integer.Â Â

1. 0<=x<1 will always lie in the interval [0, 0.9), so here the Greatest Integer Function of X will be 0.
2. 1<=x<2 will always lie in the interval [1, 1.9), so here the Greatest Integer Function of X will be 1.
3. 2<=x<3 will always lie in the interval [2, 2.9), so here the Greatest Integer Function of X will be 2.

Examples:Â Â

Input: X = 2.3
Output: [2.3] = 2

Input: X = -8.0725
Output: [-8.0725] = -9

Input: X = 2
Output: [2] = 2

Number Line Representation

• If we examine a number line with the integers and plot 2.7 on it, we see:Â
• The largest integer that is less than 2.7 is 2. So [2.7] = 2.Â
• If we examine a number line with the integers and plot -1.3 on it, we see:Â

Since the largest integer that is less than -1.3 is -2, so [-1.3] = 2.
Here, f(x)=[X] could be expressed graphically as:

Note: In the above graph, the left endpoint at every step is blocked(dark dot) to show that the point is a member of the graph, and the other right endpoint (open circle) indicates the points that are not part of the graph.

Properties of Greatest Integer Function:Â

• [X]=X holds if X is an integer.
• [X+I]=[X]+I, if I is an integer, then we can I separately in the Greatest Integer Function.
• [X+Y]>=[X]+[Y], means the greatest integer of the sum of X and Y is the equal sum of the GIF of X and the GIF of Y.
• If [f(X)]>=I, then f(X) >= I.
• If [f(X)]<=I, then f(X) < I+1.
• [-X]= -[X], If X[Tex]\epsilonÂ Â Â Â [/Tex]Integer.
• [-X]=-[X]-1, If X is not an Integer.

It is also known as the stepwise function or floor of X.

The below program shows the implementation of the Greatest Integer Function using floor() method.Â

C++ // CPP program to illustrate // greatest integer Function #include <bits/stdc++.h> using namespace std; // Function to calculate the // GIF value of a number int GIF(float n) { // GIF is the floor of a number return floor(n); } // Driver code int main() { int n = 2.3; cout << GIF(n); return 0; }  Java // Java program to illustrate // greatest integer Function class GFG{ // Function to calculate the // GIF value of a number static int GIF(double n) { // GIF is the floor of a number return (int)Math.floor(n); } // Driver code public static void main(String[] args) { double n = 2.3; System.out.println(GIF(n)); } } // This code is contributed by mits  Python3 # Python3 program to illustrate # greatest integer Function import math # Function to calculate the # GIF value of a number def GIF(n): # GIF is the floor of a number return int(math.floor(n)); # Driver code n = 2.3; print(GIF(n)); # This code is contributed by mits  C# // C# program to illustrate // greatest integer Function using System; class GFG{ // Function to calculate the // GIF value of a number static int GIF(double n) { // GIF is the floor of a number return (int)Math.Floor(n); } // Driver code static void Main() { double n = 2.3; Console.WriteLine(GIF(n)); } } // This code is contributed by mits  Javascript <script> // Javascript program to illustrate // greatest integer Function // Function to calculate the // GIF value of a number function GIF(n) { // GIF is the floor of a number return Math.floor(n); } // Driver code var n = 2.3; document.write(GIF(n)); // This code is contributed by Ankita saini </script>  PHP <?php // PHP program to illustrate // greatest integer Function // Function to calculate the // GIF value of a number function GIF($n) { // GIF is the floor of a number return floor($n); } // Driver code $n = 2.3; echo GIF($n); ?> 

Output

2

Time Complexity: O(1)

Auxiliary Space: O(1)