Fractional Part Function, often denoted as {x}, represents the decimal part of a real number x after removing its integer part. In simpler terms, it captures the fractional portion of a number, excluding the whole number component. The fractional part function is particularly useful in various mathematical contexts, such as number theory, analysis, and computer science, where understanding the non-integer portion of a number is essential.

In this article, we will learn about the various concepts related to the fractional part function, like the meaning and definition of the fractional part function, properties of the fractional part function, its formula, application, graph, and solved examples for better understanding.

What is the Fractional Part Function?

Fractional Part Function, denoted as {x} or frac(x), is a mathematical operation that extracts the decimal part of a real number x. It represents the portion of x that comes after the decimal point. Formally, for any real number x, the fractional part is given by the equation:

{x} = x- ⌊x⌋

Here, ⌊x⌋ denotes the greatest integer less than or equal to x. The fractional part function always yields a value between 0 (inclusive) and 1 (exclusive). It exhibits periodicity with a period of 1, meaning that {x + 1} = {x} for any real number x. This function is commonly used in various mathematical applications, including number theory and signal processing.

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Fractional Part Function Definition

Fractional part function, often denoted as {x} or frac(x), gives the fractional part of a real number x, which is the decimal part after the decimal point.

Mathematically, for any real number x: {x} = x- ⌊x⌋ is the Fractional Part Function

where ⌊x⌋ represents the greatest integer less than or equal to x (the floor function). The fractional part function always returns a value between 0 (inclusive) and 1 (exclusive). For example, if x = 3.75, then {x} = 0.75.

Properties of Fractional Part Function

The fractional part function, denoted as {x} or frac(x), has several properties:

1. Range: The fractional part function always returns a value between 0 (inclusive) and 1 (exclusive). Formally, ( 0 ≤ {x} < 1) for any real number x.

2. Periodicity: The fractional part function is periodic with a period of 1. This means that {x + 1} = {x} for any real number x.

3. Integer Part Relation: The fractional part and integer part of a number are related by the equation x = ⌊x⌋ + {x}, where ⌊x⌋ is the greatest integer less than or equal to x.

4. Symmetry: The fractional part function is symmetric around the integers. This means that {x} = {-x} for any real number x.

5. Additivity: The fractional part function is additive, meaning that {x + y} = {x} + {y} for any real numbers x and y.

6. Continuity: The fractional part function is discontinuous at integers but continuous elsewhere.

Fractional Part Function Formula

The formula for the fractional part, denoted as {x} or frac(x), is given by:

{x} = x – ⌊x⌋

In this formula, ⌊x⌋ represents the greatest integer less than or equal to x (the floor function). The fractional part function subtracts the integer part from the original number x, leaving only the decimal part or the fractional component. The result of the fractional part function is always a value between 0 (inclusive) and 1 (exclusive).

Example: Find the Fractional Part Function for x = 4.75

Solution:

Using the fractional part function formula:

{4.75} = 4.75 – ⌊4.75⌋

Now, find the greatest integer less than or equal to 4.75:

⌊4.75⌋ = 4

Substitute this back into the formula:

{4.75} = 4.75 – 4

{4.75} = 0.75

So, the fractional part of 4.75 is 0.75.

Fractional Part Function Graph

The graph of the Fractional Part Function, often denoted as {x} or frac(x), displays distinct characteristics. It represents the decimal part of a real number x and exhibits some notable features:

• Step Function: The graph consists of horizontal steps at integer values of x, indicating the abrupt change in the fractional part when x crosses an integer. At each integer, the fractional part resets to 0.
• Periodicity: The graph is periodic with a period of 1. This means that the pattern repeats every interval of 1 along the x-axis. The fractional part function is the same for x and x + 1, x + 2, and so on.
• Values between 0 and 1: The function’s range is between 0 (inclusive) and 1 (exclusive), reflecting the fact that the fractional part is always a decimal value between 0 and 1.
• Discontinuity at Integers: The graph has discontinuities at integer values of x, where there are vertical jumps. These jumps occur because the fractional part experiences an abrupt change at integer values.
• Symmetry: The graph is symmetric around the integers. This means that the portion of the graph to the right of an integer mirrors the portion to the left.

Fractional Part Function Domain and Range

The domain and range of the Fractional Part Function, denoted as {x} or frac(x).

Domain

The Fractional Part Function is defined for all real numbers. In mathematical notation, the domain is represented as \([Tex]\mathbb{R} [/Tex]) or (-∞, ∞), indicating that the function is applicable to any real value of x.

Range

The range of the Fractional Part Function is limited to the interval [0, 1), which means it includes 0 but excludes 1. The fractional part is always a decimal between 0 (inclusive) and 1 (exclusive).

Fractional Part Function – Conclusion

We can conclude about Fractional Part Function in following points

• Fractional Part Function represent the decimal or the fractional value of a function excluding the integral value
• Fractional Part Function Formula is given as {x} = x – ⌊x⌋ where, ⌊x⌋ is greatest integer function value of x
• Domain of Fractional Part Function is (-∞, ∞)
• Range of Fractional Part Function is [0, 1)
• The Graph of Fractional Part Function is periodic, has value lying between 0 and 1, discontinued at integers and symmetric in nature

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• Ceiling Function
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Fractional Part Function Solved Examples

Example 1: Calculate the fractional part of 5.63 using the Fractional Part Function.

Solution:

Fractional Part Function is given by:

{x} = x – ⌊x⌋

For x = 5.63:

{5.63} = 5.63 – ⌊5.63⌋

Now, find the greatest integer less than or equal to 5.63:

⌊5.63⌋= 5

Substitute this back into the formula:

{5.63} = 5.63 – 5

{5.63} = 0.63

So, the fractional part of 5.63 is 0.63.

Example 2: If y = { -3.8 }, determine the value of y.

Solution:

Fractional Part Function is given by:

{x} = x – ⌊x⌋

For (x = -3.8):

{-3.8} = -3.8 – ⌊-3.8⌋

Find the greatest integer less than or equal to -3.8:

⌊-3.8⌋=-4

Substitute this back into the formula:

{-3.8} = -3.8 – (-4)

{-3.8} = -3.8 + 4

{-3.8} = 0.2

So, if y = { -3.8 }, then y = 0.2

Fractional Part Function – Practice Questions

Q1. Evaluate { 9.25 } – { 4.6 } using the Fractional Part Function.

Q2. Find the sum { 2.4} + { 1.7 } using the Fractional Part Function.

Q3. If ( x = 7.89 ), calculate { x }

Q4. Solve for (a) if { a + 1.5 } = 0.3

Q5. Determine the range of the Fractional Part Function and express it in interval notation.

Fractional Part Function – FAQs

What is a Fractional Part Function in Math?

The Fractional Part Function, denoted as {x} or frac(x), in mathematics, extracts the decimal part of a real number x. It represents the fractional component of x after the decimal point, excluding the integer part.

What is Range of Fractional Part Function?

The range of the Fractional Part Function is the interval [0, 1), meaning it includes 0 (inclusive) but excludes 1. The function produces decimal values between 0 and 1, showcasing its periodic nature.

What is Derivative of Fractional Part Function?

The Fractional Part Function is not differentiable at integer points due to its step-like behavior. Therefore, the derivative is undefined at these points. However, the derivative exists elsewhere and equals 0 for non-integer values.

Is Fractional Part Function Continuous?

The Fractional Part Function is discontinuous at integer values, where abrupt changes occur in its value. While exhibiting discontinuities, it is continuous in the intervals between integers.

What is Domain of Fractional Part of x?

The domain of the Fractional Part Function encompasses all real numbers. It is applicable to any real value of x, denoted as R or (-∞, ∞).

Is Fractional Part Function Periodic?

Yes, the Fractional Part Function is periodic with a period of 1. The pattern of the function repeats every interval of 1 along the x-axis. This periodicity is evident in its step-like graph and the fact that {x+1}={x} for any real number x.