Fractional part

In mathematics, the fractional part of a real number x, denoted \{x\}, is defined as \{x\} = x - \lfloor x \rfloor, where \lfloor x \rfloor is the floor function, representing the greatest integer less than or equal to x.^[1] This function extracts the non-integer portion of x, always yielding a value in the half-open interval [0, 1).^[2] The fractional part function exhibits several fundamental properties that underpin its utility in analysis and number theory. It is periodic with period 1, meaning \{x + n\} = \{x\} for any integer n, which reflects its sawtooth-like graph that repeats every integer interval.^[2] Additionally, for non-integer x, the symmetry property \{x\} + \{-x\} = 1 holds, highlighting the complementary nature of positive and negative fractional parts.^[2] These characteristics make \{x\} a discontinuous yet bounded function, with jumps at integer points where it resets to 0. Beyond its basic definition, the fractional part plays a crucial role in advanced mathematical contexts, particularly in evaluating limits, integrals, and series. For instance, integrals involving \{x\} often connect to special constants, such as \int_0^1 \{1/x\} \, dx = 1 - \gamma, where \gamma is the Euler-Mascheroni constant.^[2] In number theory, it appears in sums like \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \{n/k\}^2 = \ln(2\pi) - 1 - \gamma, linking to harmonic analysis and zeta functions.^[2] Applications extend to double integrals, such as \int_0^1 \int_0^1 \{x/y\}^k \, dx \, dy, and asymptotic behaviors in sequences, demonstrating its relevance in studying uniform distribution and Diophantine approximation.^[2]

Definition and Notation

Formal Definition

The fractional part of a real number x, denoted \{x\}, is defined as \{x\} = x - \lfloor x \rfloor, where \lfloor x \rfloor is the floor function giving the greatest integer less than or equal to x.^[3]^[4] This definition subtracts the integer part of x from x itself, thereby isolating the non-integer remainder that lies in the interval [0, 1).^[3] The floor function serves as the basis for identifying this integer part.^[5] For a positive real number such as x = 3.7, \lfloor 3.7 \rfloor = 3, so \{3.7\} = 3.7 - 3 = 0.7.^[3] This formulation assumes the standard definition of the floor function.^[5]

Common Notations

The standard notation for the fractional part of a real number x is \{x\}. Alternative notations include x - , with $$ representing the floor function (the greatest integer less than or equal to x), a convention appearing in various mathematical contexts. In programming and computational mathematics, \operatorname{frac}(x) is frequently used to express the fractional part.^[3] To distinguish it from set notation in typesetting systems like LaTeX, the symbol is rendered as \{x\} with escaped braces.

Basic Properties

Range and Periodicity

The fractional part function, denoted {x}, always outputs values in the interval [0, 1) for any real number x. This range follows directly from the definition {x} = x - \lfloor x \rfloor, where \lfloor x \rfloor is the greatest integer less than or equal to x. Since \lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 by the properties of the floor function, subtracting \lfloor x \rfloor yields 0 \leq {x} < 1.^[6]^[7] This bounded range ensures that the fractional part captures only the non-integer portion of x, regardless of the magnitude of x itself. For instance, if x = 2.7, then {2.7} = 0.7, which lies within [0, 1). The strict upper bound of 1 (exclusive) arises because equality to 1 would imply x is an integer plus 1, but integers themselves have fractional part 0, resetting the cycle.^[6]^[8] The fractional part function exhibits periodicity with period 1, meaning {x + n} = {x} for any integer n. In particular, {x + 1} = (x + 1) - \lfloor x + 1 \rfloor = (x + 1) - (\lfloor x \rfloor + 1) = x - \lfloor x \rfloor = {x}. This property holds because adding an integer shifts the floor function by the same integer, leaving the fractional remainder unchanged.^[6]^[7] An illustrative example is { \pi + 2 }, where \pi \approx 3.14159, so { \pi } \approx 0.14159 and { \pi + 2 } = { 5.14159 } \approx 0.14159, confirming the invariance under integer addition. This periodicity underscores the function's sawtooth-like behavior, repeating identically every unit interval on the real line.^[6]^[7]

Symmetry and Functional Relations

The fractional part function exhibits a notable symmetry relation involving its behavior under negation. For a positive real number x that is not an integer, the fractional part of -x satisfies \{-x\} = 1 - \{x\}.^[3] When x is a positive integer, \{-x\} = 0, since -x is also an integer. This relation arises from the definition \{y\} = y - \lfloor y \rfloor, where the floor function adjusts the integer part for negative values such that the fractional part remains in [0, 1).^[3]^[7] A key functional identity follows from this symmetry: for non-integer x > 0, \{x\} + \{1 - x\} = 1. This holds because \{1 - x\} = 1 - \{x\}, leveraging the periodicity of the fractional part under integer shifts and the negation property.^[7] At integer points, the identity does not hold, as both \{x\} = 0 and \{1 - x\} = 0, summing to 0 rather than 1. For example, if x = 0.3, then \{x\} = 0.3 and \{1 - 0.3\} = \{0.7\} = 0.7, so $0.3 + 0.7 = 1. The fractional part function is discontinuous at every integer value of x. As x approaches an integer n from the left, \{x\} approaches 1, but at x = n, \{x\} = [0](/page/0), and it remains near 0 just to the right of n. This jump discontinuity reflects the sawtooth nature of the function, with the left-hand limit at integers being 1 and the right-hand limit (and function value) being 0.^[7]

Extension to Negative Numbers

Definition for Negative Reals

The fractional part function extends to negative real numbers via the formula {x} = x - \lfloor x \rfloor, where \lfloor x \rfloor denotes the floor function, the greatest integer less than or equal to x. This application of the floor function for negative x yields a value in the interval [0, 1), consistent with the behavior for positive reals.^[3] To illustrate, for x = -1.3, \lfloor -1.3 \rfloor = -2, so {-1.3} = -1.3 - (-2) = 0.7.^[6] A defining feature of this approach is its uniformity: the fractional part lies in [0, 1) for every real number x, avoiding negative outputs that arise in some other conventions for negative inputs.^[3] This convention for negative reals became standard in mid-20th century texts on mathematical analysis, as seen in Walter Rudin's Principles of Mathematical Analysis (1953), where it supports key properties such as x = \lfloor x \rfloor + {x}.

Behavior and Examples

The fractional part function, when extended to negative real numbers using the floor function, yields a value in the interval [0, 1), even though the input is negative. For instance, consider x = -0.7. The floor of -0.7 is -1, so the fractional part is \{-0.7\} = -0.7 - (-1) = 0.3, demonstrating that the result remains positive despite the negative input.^[3]^[7] This behavior arises because any negative real number x can be expressed as x = -n + f, where n is a positive integer and $0 < f < [1](/page/1). In this form, the fractional part simplifies to \{x\} = f, which mirrors the fractional part obtained for positive numbers with the same f.^[3]^[9] In contrast to positive x, where the fractional part directly corresponds to the decimal expansion beyond the integer part, the computation for negative x effectively "borrows" from the more negative floor value to ensure the result lies in [0, 1). For example, with x = -1.23, the floor is -2, yielding \{-1.23\} = -1.23 - (-2) = 0.77, adjusting the apparent negative decimal to a positive equivalent.^[3]^[9]^[7] At negative integers, the fractional part aligns with the positive case, as \{-3\} = -3 - \lfloor -3 \rfloor = -3 - (-3) = 0.^[3]

Connection to Integer Part Functions

The fractional part function is fundamentally related to the floor function through the decomposition of any real number x as x = \lfloor x \rfloor + \{x\}, where \lfloor x \rfloor denotes the greatest integer less than or equal to x, and \{x\} is the fractional part in the interval [0, 1).^[3] This relation holds for all real x, with the floor function providing the integer component and the fractional part capturing the remainder.^[5] For the ceiling function, which gives the smallest integer greater than or equal to x and is denoted \lceil x \rceil, the fractional part can be expressed for non-integer x as \{x\} = x - (\lceil x \rceil - 1).^[10] This follows from the identity \lceil x \rceil = \lfloor x \rfloor + 1 when x is not an integer, substituting into the floor decomposition to yield x = (\lceil x \rceil - 1) + \{x\}, and rearranging gives the expression.^[5] In contrast to the floor and ceiling, the nearest integer rounding function, often denoted \operatorname{round}(x) or \operatorname{nint}(x), selects the integer closest to x, with ties typically resolved by rounding to the nearest even integer or away from zero depending on convention.^[11] The difference x - \operatorname{round}(x) represents the signed distance to the nearest integer, ranging from -0.5 to $0.5, whereas \{x\} is always nonnegative in [0, 1); for example, with x = 1.6, \operatorname{round}(x) = 2 and x - \operatorname{round}(x) = -0.4, highlighting the adjustment relative to the standard positive fractional part of $0.6.^[11] The floor function is also known as the greatest integer function, historically denoted $$ in some notations, a convention introduced by Carl Friedrich Gauss in 1808 for the largest integer not exceeding x.^[5] This alternative notation persists in certain texts but has largely been superseded by the modern \lfloor x \rfloor symbol.^[5]

Role in Continued Fraction Expansion

The continued fraction expansion of a real number x > 1 begins by setting the first partial quotient a_0 = \lfloor x \rfloor, the greatest integer less than or equal to x. The process then recurses on the reciprocal of the fractional part, $1/\{x\}, where \{x\} = x - \lfloor x \rfloor, to generate subsequent partial quotients.^[12]^[13] In the algorithmic steps, each subsequent partial quotient a_i = \lfloor 1/\{x_{i-1}\} \rfloor for i \geq 1, with x_0 = x and x_i = 1/\{x_{i-1}\} serving as the input for the next iteration. This reliance on the fractional part isolates the non-integer remainder at each stage, transforming the expansion into an infinite sequence for irrational numbers or a finite one for rationals.^[12]^[13] For example, consider x = \pi \approx 3.14159. Here, \lfloor \pi \rfloor = 3, so \{\pi\} \approx 0.14159, and the next term arises from $1/0.14159 \approx 7.0625, yielding a_1 = 7. The process continues similarly, producing the expansion [3; 7, 15, 1, 292, \dots].^[14] This iterative use of the fractional part ensures that the continued fraction captures the "irrational tail" of the number by repeatedly isolating and inverting the remainder, facilitating optimal rational approximations through the convergents.^[12]^[13]