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.
Input: X = 2.3 Output: [2.3] = 2 Input: X = -8.0725 Output: [-8.0725] = -9 Input: X = 2 Output:  = 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():
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