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.
- 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
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 ); ?> |
2
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