Unit fraction

It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number.

Every rational number can be represented as a sum of distinct unit fractions; these representations are called Egyptian fractions based on their use in ancient Egyptian mathematics.

Many infinite sums of unit fractions are meaningful mathematically.

In geometry, unit fractions can be used to characterize the curvature of triangle groups and the tangencies of Ford circles.

Unit fractions are common in probability theory due to the principle of indifference.

They also have applications in combinatorial optimization and in analyzing the pattern of frequencies in the hydrogen spectral series.

The unit fractions are the rational numbers that can be written in the form

[4] In modular arithmetic, any unit fraction can be converted into an equivalent whole number using the extended Euclidean algorithm.

[5][6] This conversion can be used to perform modular division: dividing by a number

The extended Euclidean algorithm for the greatest common divisor can be used to find integers

[7] Several constructions in mathematics involve combining multiple unit fractions together, often by adding them.

Any positive rational number can be written as the sum of distinct unit fractions, in multiple ways.

[8] The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham problem[9] and the Erdős–Straus conjecture[10] concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

[11] In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.

[12] Many well-known infinite series have terms that are unit fractions.

It has the unusual property that all elements in its inverse matrix are integers.

[19] Similarly, Richardson (2001) defined a matrix whose elements are unit fractions whose denominators are Fibonacci numbers:

[22][23] A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of fair division are a standard classroom example in teaching students to work with unit fractions.

[24] In a uniform distribution on a discrete space, all probabilities are equal unit fractions.

Due to the principle of indifference, probabilities of this form arise frequently in statistical calculations.

[25] Unequal probabilities related to unit fractions arise in Zipf's law.

This states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the

[27][28] One motivation for this is as a test case for more general bin packing methods.

Another involves a form of pinwheel scheduling, in which a collection of messages of equal length must each be repeatedly broadcast on a limited number of communication channels, with each message having a maximum delay between the start times of its repeated broadcasts.

of the time slots on the channel it is assigned to, so a solution to the scheduling problem can only come from a solution to the unit fraction bin packing problem with the channels as bins and the fractions

[27] Even for bin packing problems with arbitrary item sizes, it can be helpful to round each item size up to the next larger unit fraction, and then apply a bin packing algorithm specialized for unit fraction sizes.

[28] The energy levels of photons that can be absorbed or emitted by a hydrogen atom are, according to the Rydberg formula, proportional to the differences of two unit fractions.

An explanation for this phenomenon is provided by the Bohr model, according to which the energy levels of electron orbitals in a hydrogen atom are inversely proportional to square unit fractions, and the energy of a photon is quantized to the difference between two levels.

[29] Arthur Eddington argued that the fine-structure constant was a unit fraction.

This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.