# Greatest Integer Function

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 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

 

Output:

2


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