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Greatest Integer Function

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

  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:  

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 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: 

  • [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\epsilon    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. 

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);
 
?>

                    

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>

                    

Output: 
2

 

Time Complexity: O(1)

Auxiliary Space: O(1)



Last Updated : 27 Jul, 2022
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