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Greatest Integer Function
• Last Updated : 26 Nov, 2018

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

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

In general: If, <= < . Then,

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.

1. 0<=x<1 will always lie in the interval [0, 0.9) so here the Greatest Integer Function of X will 0.
2. 1<=x<2 will always lie in the interval [1, 1.9) so here the Greatest Integer Function of X will 1.
3. 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 XInteger.
• [-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 using namespace std;  // Function to calculate the// GIF value of a numberint GIF(float n){    // GIF is the floor of a number    return floor(n);}  // Driver codeint 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 numberstatic int GIF(double n){    // GIF is the floor of a number    return (int)Math.floor(n);}  // Driver codepublic 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

 
Output:
2


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