Greatest Integer Function

The greatest Integer Function [X] indicates an integral part of the real number [Tex]x    [/Tex]which is the nearest and smaller integer to [Tex]x  [/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, [Tex]n    [/Tex]<= [Tex]X    [/Tex]< [Tex]n+1    [/Tex]. Then, [Tex](n \epsilon Integer)\Longrightarrow [X]=n    [/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.  

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[Tex]\epsilon    [/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.

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)