Greatest Integer and Fractional Part Functions

Greatest Integer and Fractional Part Functions

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Introduction to Greatest Integer Function

What is Greatest Integer Function?

Greatest Integer Function is a function that gives the greatest integer less than or equal to the number.

The greatest integer less than or equal to a number \(x\), or the floor of \(x\), is represented as \(\left \lfloor x \right \rfloor\)

Look at the following examples.

Values of \(x\) \(f(x)=\left \lfloor x \right \rfloor\)
\(3.1\) \(f(3.1)=\left \lfloor 3.1 \right \rfloor = 3\)
\(2.999\) \(f(2.999)=\left \lfloor 2.999 \right \rfloor= 2\)
\(-2.7\) \(f(-2.7)=\left \lfloor -2.7 \right \rfloor =  - 3\)
\(-\sqrt{2}\) \(f(-\sqrt{2})=\left \lfloor -\sqrt{2} \right \rfloor =  - 2\)
\(4\) \(f(4)=\left \lfloor 4 \right \rfloor = 4\)
\(-7\) \(f(-7)= \left \lfloor -7 \right \rfloor =  - 7\)

Using this concept, we can define the greatest integer function, or the floor function, as follows.

\[f\left( x \right) = \left \lfloor x \right \rfloor\]

Clearly, the input variable \(x\) can take on any real value.

However, the output will always be an integer.

Also, all integers will occur in the output set.

Thus, the domain of this function is \(\mathbb{R}\), while its range will be \(\mathbb{Z}\)

Next, let us consider the fractional part of any number \(x\)

This is the difference between \(x\) and its greatest integer part, \(\left \lfloor x \right \rfloor\)

So, \(\left\{ x \right\} = x - \left \lfloor x \right \rfloor\)

Look at the following examples.

Values of \(x\) \(f(x)=\left\{ x \right\}\)
3.1 \(f(3.1)=\left\{ {3.1} \right\} = 3.1 - \left \lfloor 3.1 \right \rfloor = 0.1\)
-2.7 \(f(-2.7)=\left\{ {-2.7} \right\} = - 2.7 -\left \lfloor -2.7 \right \rfloor = 0.3\)

Using the concept of the fractional part, we can define the fractional part function as follows.

\[f\left( x \right) = \left\{ x \right\}\]

For any real number \(x\), its fractional part can be calculated.

Thus, the domain of this function is \(\mathbb{R}\)

Now, let us determine its range.

If \(x\) is an integer, then its fractional part will be 0

If \(x\) is a non-integer, let \(n\) and \(n + 1\) be the integers between which \(x\) lies.

We will have \(\left\{ x \right\} = x - n\), which means that \(\left\{ x \right\}\) must be greater than 0 but smaller than 1

Fractional part functions illustration

We conclude that the range of the fractional part function is \(\left[ {0,1}\right)\)


Graph of Greatest Integer Function

Now, we will plot the graphs of these two functions.

First, consider \(f\left( x\right) = \left \lfloor x \right \rfloor\)

If \(x\) is an integer, then the value of \(f\) will be \(x\) itself. If \(x\) is a non-integer, then the value of \(x\) will be the integer just before \(x\)

For example, for all numbers lying in the interval \(\left[ {0,1} \right)\), the value of \(f\) will be 0

For the entire interval \(\left[ {1,2} \right)\), \(f\) will take the value 1

For the interval \(\left[ { - 1,0}\right)\), \(f\) will take the value \(- 1\) and so on.

These observations lead us to the following graph.

Illustration of Greatest integer functions

Do you observe that the function has a constant value between any two integers?

As soon as the next integer comes, the function value jumps by one unit.

Carefully note the solid and hollow dots.

For example, there is a hollow dot at \(\left( {1,0} \right)\) and a solid dot at \(\left( {1,1} \right)\). This means that the value of \(f\) at \(x =1\) is 1 (and not 0)

Experiment with this simple simulation to observe the effects of parameters \(a\), \(b\), \(h\), and \(k\) on the greatest integer, or step function.


Graph of Fractional Part Function

Now, we will graph the function \(g\left( x \right) = \left\{ x \right\}\)

Let us first start with \(x = 0\)

For this value of \(x\), \(\left\{ x \right\} = 0\)

Now, as \(x\) is increased in the interval \(\left({0,1} \right)\), the fractional part also increases, and has the same value as \(x\)

However, as soon as \(x\) takes the value 1, the fractional part goes back to 0

Let us plot the graph of g in only the interval \(\left[ {0,1} \right]\)

Illustration of Fractional Part Function

Observe how the graph is the same as the line \(y = x\) in the interval \(\left( {0,1} \right)\), and then jumps back to 0 when \(x = 1\)

Now, as \(x\) increases beyond 1,the fractional part will start to rise again (in the same manner as in the interval \(\left( {0,1} \right)\)), and will fall back to 0 when \(x = 2\)

Illustration of Fractional Part Function

Extending this, we obtain the graph of \(g\left( x \right) = \left\{ x \right\}\) over \(\mathbb{R}\) as follows.

Illustration of Fractional Part Function

From the graph, it is evident that the range of the fractional part function is \(\left[{0,1} \right)\)


Greatest Integer Function: Properties

Some useful properties of the greatest integer function are listed below.

1. \(\left \lfloor x+n \right \rfloor=\left \lfloor x \right \rfloor+n\), where \(n \in \mathbb{Z}\)
2. \(\left \lfloor -x \right \rfloor=-\left \lfloor x \right \rfloor\), where \(x \in \mathbb{Z}\)
3. \(\begin{cases}
   & {-\left\lfloor x\right\rfloor}, &  \text{if   } x \in  \mathbb{Z} \\
   &{-\left\lfloor x-1\right\rfloor}, & \text{if   } x \notin  \mathbb{Z} \\
  \end{cases}\)
4. If \(\left \lfloor f(x) \right \rfloor \geq L\), then \(f(x)\geq L\)
 
important notes to remember
Important Notes
  1. If \(x\) is a number between successive integers \(n\) and \(n + 1\), then \(\left \lfloor x \right \rfloor = n\). If x is any integer, then \(\left \lfloor x \right \rfloor = x\)
  2. The domain of greatest integer function is  \(\mathbb{R}\) and its range is  \(\mathbb{Z}\).
  3. The fractional part will always be non-negative, as \(x\) will always be greater than (or equal to) \(\left \lfloor x \right \rfloor\). If \(x\) is an integer, then its fractional part will be 0
  4. The domain of fractional part function is \(\mathbb{R}\) and its range is \(\left[ {0,1}\right)\)

Greatest Integer Function: Examples

Example 1

 

 

What is the domain of:

(a) \(f\left( x \right) =\frac{1}{{\left \lfloor x \right \rfloor}}\)

(b) \(g\left(x \right) = \frac{1}{{\left\{ x \right\}}}\)?

Solution:

(a) The denominator should not be 0, that is, \(\left \lfloor x \right \rfloor \ne 0\)

The greatest integer part of a number is 0 if that number lies in the interval \(\left[ {0,1} \right)\)

Thus, to obtain the domain, this interval must be excluded from the set of real numbers.

This means that the domain of \(f\) is \(\mathbb{R}- \left[ {0,1} \right)\)

(b) Once again, the denominator should not be 0, which means that \(\left\{ x\right\} \ne 0\)

When is the fractional part of a number equal to 0?

Only when that number is an integer!

Thus, the domain of g is  \(\mathbb{R}-\mathbb{Z}\), that is, all integers have been excluded from the set of real numbers.

\(\therefore\) The domain of \(f\) is \(\mathbb{R}- \left[ {0,1} \right)\) and the domain of \(g\) is \(\mathbb{R}-\mathbb{Z}\)
Example 2

 

 

Find the value of \(x\) such that \(\left \lfloor x+3 \right \rfloor=3\)

Solution:

From the definition of the greatest integer function, we have \(3 \leq x+1 <4\)

Subtract 1 in this inequality.

We get, \(2 \leq x <3\)

\(\therefore x\) can take the values greater than or equal to 2 and less than 3
 
Thinking out of the box
Think Tank
  1. Let an interval be of the form \([n,n+1)\), where \(n\) is an integer. Then, what will be the value of the greatest integer function?
  2. When does the fractional part function take the value 0?

Practice Questions

Here are few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is a fractional number?

When one unit is divided in equal parts, then each unit is represented by a fractional number.

When one unit is divided in equal parts, then each unit is represented by a fractional number. A watermelon is divided into two equal halves to show the fraction 1 by 2.

2. What is greatest integer function?

Greatest integer function is a function that gives the largest integer which is less than or equal to \(x\)

This function is denoted by \(f(x)=\left \lfloor x \right \rfloor\)

3. Which is the greatest whole number?

Whole numbers are the collection of natural numbers along with the number 0

Hence, there is no greatest whole number because each number has a successor.

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