Greatest Integer Function

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

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

In general: If, <= < . Then, 
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: 

👁 GIF2

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:

👁 Image

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 XInteger.
• [-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.

2

Time Complexity: O(1)

Auxiliary Space: O(1)