3.8 Greatest and least integer functions

In this module, we shall study a family of functions which return integers based on certain rule, corresponding to a real number. Greatest integer function (floor), least integer function (ceiling) and nearest integer function form part of this family.

Greatest integer function (floor function)

Greatest integer function returns the greatest integer less than or equal to a real number. In other words, we can say that greatest integer function rounds “down” any number to the nearest integer. This function is also known by the names of “floor” or “step” function. The greatest integer function (GIF) is denoted by the symbol “[x]” .

Interpretation of Greatest integer function is straight forward for positive number. Consider the values “0.23” and “1.7”. The greatest integers for two numbers are “0” and “1”. Now, consider a negative number “-0.54” and “-2.34”. The greatest integers less than these negative numbers are “-1” and “-3” respectively.

We can observe here that greater integer function is actually a function that returns the integral part of a positive real number. This interpretation is clear for positive number. Interpretation for negative numbers needs some explanation. We interpret these values in the context of the fact that every real number can be decomposed to have two parts (i) integral and (ii) fractional part. From this point of view, the negative number can be thought as :

$$\text{-0.54 (real number) = -1 (integral part) + 0.36 (fraction part)}$$

$$\text{-2.34 (real number) = -3 (integral part) + 0.66 (fraction part)}$$

We may be tempted to disagree (why not -2 + -0.34 = -2.34?). But, we should know that this is how greatest integer function (GIF) treats a negative number. It returns "-3" for "-2.34" - not "-2". Subsequently, we shall define a function called fraction part function (FPF) that returns fraction part of real number. We shall find that the function exactly returns the same fraction for negative number as has been worked out. The fraction part function (FPF) returns a fraction, which is always positive. It is denoted as {x}. Because of these aspects of GIF and FPF, we can understand the reason why negative number is treated the way it has been presented above. In terms of integral and fraction parts, we write a real number "x" as :

$$x=\left[x\right]+\left\{x\right\}$$

In the nutshell, we can use any of the following interpretations of greatest integer function :

• [x] = Greatest integer less than equal to “x”
• [x] = Greatest integer not greater than “x”
• [x] = Integral part of “x”

The value of "[x]" is an integer (n) such that :

$$f\left(x\right)=\left[x\right]=n;\text{if}n\le x<n+1n\in Z$$

Working rules for evaluating greatest integer function are two step process :

• If “x” is an integer, then [x] = x.
• If “x” is not an integer, then [x] evaluates to greatest integer less than “x”.

Graph of greatest integer function

Few initial function values are :

$$For-2\le x<-\mathrm{1,}f\left(x\right)=\left[x\right]=-2$$

$$For-1\le x<\mathrm{0,}f\left(x\right)=\left[x\right]=-1$$

$$For0\le x<\mathrm{1,}f\left(x\right)=\left[x\right]=0$$

$$For1\le x<\mathrm{2,}f\left(x\right)=\left[x\right]=1$$

$$For2\le x<\mathrm{3,}f\left(x\right)=\left[x\right]=2$$

The graph of the function is shown here :