Fractional Part Function - Formula, Properties, Examples

What is Fractional Part Function?

In mathematics, every real number can be expressed as the sum of its integer part and its fractional part. The fractional part function, denoted by {x}, isolates that non-integer portion. Formally, it represents the difference between a number and its greatest integer value.

This function is also frequently referred to as the decimal part function. It ensures the remainder of any number remains separate from its whole number "anchor." This provides a value that is always non-negative, even if the original number was negative.

Fractional Part Function Formula

The fractional part function formula is straightforward but requires a clear understanding of the floor function. It is expressed as:

{x} = x - [x]

In this equation:

• {x} represents the fractional part.

• x is the original real number.

• [x] is the Greatest Integer Function (floor function), which rounds x down to the nearest integer.

Every real number can be written in the form: x = [x] + {x}. By rearranging the fractional part function formula, you can see that any number is simply its floor plus its fractional remainder.

Enroll Now for Maths Online Classes

Properties of Fractional Part Function

According to the fractional part function properties found in standard calculus, there are specific rules that define how this function behaves across the number line.

Range and Domain

The domain includes all real numbers, as it is defined for every point on the number line. However, the value of the fractional part always stays between 0 and 1. We write this as 0 <= {x} < 1. It can hit exactly 0, but it never reaches 1.

Behaviour with Integers

For any integer value of x, such as 5, 0, or -4, the result is always 0. An integer simply has no decimal remainder to extract.

The Sum Property

The relationship between a number and its negative counterpart follows a specific pattern:

• {x} + {-x} = 0 if x is an integer.

• {x} + {-x} = 1 if x is not an integer.

Read More - Proper Fraction - Definition, Difference, Uses, Examples

Periodicity and Remainder

The function is periodic with a period of 1. This means {x + 1} = {x}. Additionally, for integers a and b, {a/b} = r/b, where r is the remainder of the division.

Fractional Part Function Graph

The visualisation of this function helps in understanding the peculiarity of this function. If we draw this function, we obtain the pattern of the "sawtooth wave." This function has parallel lines with the slope of 1, starting with a point at (n, 0) with a solid dot and rising up to the point (n+1, 1) with an open circle. This represents the function's return to zero every time it crosses an integer value, thus illustrating the discontinuity of the function.

Read More - Mixed Fraction - Definition, Formula and Examples

Fractional Part Function Examples

Let’s check out some fractional part function examples to see how these rules actually work when we use real numbers.

Example 1: Working with Positive and Negative Decimals

The whole process of computing the fractional part is all about computing the decimal "leftover." The above is how you'd do it for three different kinds of numbers:

• If the number is a positive decimal (x = 5.26), we compute the fractional part by subtracting the floor (which is 5). {5.26} = 5.26 - 5 = 0.26

• If the number is a negative decimal (x = -2.91), this is the part of the problem that tends to trip people up. We subtract the floor, which is the next integer down from the original number (-3). {-2.91} = -2.91 - (-3) = -2.91 + 3 = 0.09

• If the number is a whole number (x = 4), the fractional part is zero, as there is no decimal to speak of. {4} = 4 - 4 = 0

Example 2: Finding the Domain of a Reciprocal

Problem: What is the domain of the function f(y)=1{y}f(y) = \frac{1}{\{y\}}f(y)={y}1​?

Solution: In mathematics, division by zero is not allowed. The fractional part of any integer—like 1, 2, or 3—is always 0. So if you try to put an integer in the denominator, the function would break. This means the function works for every number except integers.

Answer: The domain is all real numbers except integers, written as R∖Z\mathbb{R} \setminus \mathbb{Z}R∖Z.

Make Maths Simple and Fun with CuriousJr

CuriousJr helps children build a strong foundation in maths by removing fear and boosting confidence. Our Mental Maths online classes for students from Classes 1 to 8 are designed to improve speed, accuracy, and logical thinking through simple techniques and interactive learning.

With a dual-mentor system, students attend engaging live classes and receive dedicated support for doubt-solving after every session. Animated lessons, fun activities, and exciting challenges make maths easy to understand and enjoyable to learn.

Parents stay informed with regular progress updates and review sessions, ensuring complete transparency in their child’s learning journey.

Book a demo class today and see how CuriousJr makes maths simple, engaging, and confidence-building for your child.