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.

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.