Greatest Integer Function

Greatest Integer Function [X] indicates an integral part of the real number $x$ which is nearest and smaller integer to $x$ . It is also known as floor of X .

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

In general: If, $n$ <= $X$ < $n+1$ . Then, $(n \epsilon Integer)\Longrightarrow [X]=n$

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 greatest integer function is n, where n is an integer.

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

Examples:

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:

The largest integer that is less than 2.7 is 2. So [2.7] = 2.

If we examine a number line with the integers and plot -1.3 on it, we see:

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 in 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 the part of the graph.

Properties of Greatest Integer Function:

[X]=X holds if X is 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 sum of X and Y is equal sum of GIF of X and GIF of Y.
If [f(X)]>=I, then f(X) >= I.
If [f(X)]<=I, then f(X) < I+1.
[-X]= -[X], If X $\epsilon$ Integer.
[-X]=-[X]-1, If X is not an Integer.

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

Below program shows the implementation of Greatest Integer Function using floor():

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  

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:

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