Fractional Part Function

Fractional part function is a special type of function in algebra which is defined as the difference of a real number and its integral value. As the name suggests, the fractional part function gives the proper fraction of a number that remains after subtracting its integral value from it, and hence the range of the function is [0, 1). The value of the fractional part function is always a fraction less than 1. Mathematically, it is denoted as f(x) = {x}.

In this article, we will understand the properties of the fractional part function, its domain and range, its graph, and its formula. We will also solve a few examples based on the fractional part function for a better understanding of the concept.

The fractional part function is a function that gives the fractional part of x by subtracting the greatest integer less than x from x, where x is a real number. This function is also known as the decimal part function. In other words, we can also say that fractional part function is defined as the difference between a real number and its greatest integer value (which is determined using the greatest integer function), that is, if f is a fractional part function, then the fractional part of x is written as \(f(x) = \left \{ x \right \} = x - \left \lfloor x \right \rfloor\), where x is a real number. If x is an integer, then the fractional part of x is equal to 0.

Now that we know the meaning of the fractional part function, let us explore the formula of the function and write it mathematically. For a real number x, the fractional part function is written as, \(f(x) = \left \{ x \right \} = x - \left \lfloor x \right \rfloor\). To understand the working of the function, let us consider a few examples consider a non-negative real number and a negative real number.

• For a non-negative real number x = 2.47, the fractional part function works as:
  f(x) = {2.47} = 2.47 - 2 = 0.47 which is the fractional part of 2.47
• For a negative real number x = -3.76, the fractional part functions works as:
  f(x) = {-3.76} = -3.76 - (-4) = -3.76 + 4 = 0.24 which is the fractional part of -3.76
• For integer value of x = 1, the fractional part function works as:
  f(x) = {1} = 1 - 1 = 0 which is the fractional part of 1

We know how the fractional part function works and how to determine the fractional part of x. Next, we will plot the graph of the fractional part function by taking a few points on the graph. The graph of the fractional part of x is similar to the graph of the greatest integer function and does not include any integer value except for 0. Given below is a graph of the fractional part function. As we can see in the graph, whenever the value of x is an integer, the value of the fractional part of x is 0 and is shown by the solid blue dot.

As we can see in the graph of the fractional part function above, the value of the function lies between 0 and 1, and for integer values of x, the value is always zero. It is evident from the graph that the domain of the fractional part function consists of all real numbers as the function is defined for all real numbers. Also, we can see that since the value of the fractional part of x lies between 0 and 1, hence the range of the function is [0,1).

So far, we have understood the concept of fractional part function. Let us now summarize the concept and understand the properties of the fractional part of x. Given below is a list highlighting the important properties of fractional part function:

• The value of the fractional part of x lies between 0 and 1, that is, 0 ≤ {x} < 1.
• For integer values of x, the fractional part function is always equal to 0.
• {x} + {-x} = 0, if x is an integer and {x} + {-x} = 1, otherwise.
• For integers a and b such that b > 0, {a/b} = r/b, where r is the remainder when a is divided by b.

Important Notes on Fractional Part Function

• The range of the fractional part function is [0, 1) and its domain is all real numbers.
• The fractional part of x is 0 if x is an integer.
• The fractional part function of x is defined as the difference between x and the greatest integer less than x.

☛ Related Topics:

• Graphing functions
• Constant function
• Modulus function

What is a Fractional Part Function in Math?

Fractional part function is a special type of function in algebra which is defined as the difference of a real number and its integral value (given by the greatest integer less than the real number).

What is the Range of Fractional Part Function?

The range of the fractional part function is [0, 1) as the value of the fractional part of x lies between 0 and 1, and is equal to 0 if x is an integer.

What is the Derivative of Fractional Part Function?

The derivative of the fractional part function {x} does not exist if x is an integer. However, if x is a real number excluding the integers, then the derivative is equal to 1.

Is Fractional Part Function Continuous?

No, the fractional part function is not continuous as the graph breaks at integral values of x because the fractional part of x is 0 whenever x is an integer.

What is the Domain of Fractional Part of x?

The fractional part of x is defined for all real numbers, hence the domain of the fractional part function includes all real numbers.

Is Fractional Part Function Periodic?

Yes, the fractional part function is periodic and its period is equal to 1 as f(x) = f(x + 1), that is, the value of fractional part of x repeats after every 1 unit.