Unit fraction

Combinations

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

Finite sums

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

${\displaystyle \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}=\frac{1}{3}+\frac{1}{5}+\frac{1}{6}+\frac{1}{10}.}$

These sums are called Egyptian fractions, because the ancient Egyptian civilisations used them as notation for more general rational numbers. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham problem and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

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.

Infinite series

Many well-known infinite series have terms that are unit fractions. These include:

• The harmonic series, the sum of all positive unit fractions. This sum diverges, and its partial sums $${\displaystyle \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}}$$ closely approximate the natural logarithm of ${\displaystyle n}$ plus the Euler–Mascheroni constant. Changing every other addition to a subtraction produces the alternating harmonic series, which sums to the natural logarithm of 2: $${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots =\ln 2.}$$
• The Leibniz formula for π is $${\displaystyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots =\frac{\pi }{4}.}$$
• The Basel problem concerns the sum of the square unit fractions: $${\displaystyle 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots =\frac{{\pi }^2}{6}.}$$ Similarly, Apéry's constant is an irrational number, the sum of the cubed unit fractions.
• The binary geometric series is $${\displaystyle 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots =2.}$$

Matrices

A Hilbert matrix is a square matrix in which the elements on the ${\displaystyle i}$th antidiagonal all equal the unit fraction ${\displaystyle 1/i}$. That is, it has elements $${\displaystyle B_{i,j}=\frac{1}{i+j-1}.}$$ For example, the matrix $${\displaystyle \left[\begin{array}{ccc}1& \frac{1}{2}& \frac{1}{3}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}\end{array}\right]}$$ is a Hilbert matrix. It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson (2001) defined a matrix whose elements are unit fractions whose denominators are Fibonacci numbers: $${\displaystyle C_{i,j}=\frac{1}{F_{i+j-1}},}$$ where ${\displaystyle F_i}$ denotes the ${\displaystyle i}$th Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.

Adjacency and Ford circles

Two fractions ${\displaystyle a/b}$ and ${\displaystyle c/d}$ (in lowest terms) are called adjacent if $${\displaystyle ad-bc=\pm 1,}$$ which implies that they differ from each other by a unit fraction: $${\displaystyle \left|\frac{a}{b}-\frac{c}{d}\right|=\frac{|ad-bc|}{bd}=\frac{1}{bd}.}$$ For instance, ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{3}{5}}$ are adjacent: ${\displaystyle 1\cdot 5-2\cdot 3=-1}$ and ${\displaystyle \frac{3}{5}-\frac{1}{2}=\frac{1}{10}}$. However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance, ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{2}{3}}$ differ by a unit fraction, but are not adjacent, because for them ${\displaystyle ad-bc=3}$.

This terminology comes from the study of Ford circles. These are a system of circles that are tangent to the number line at a given fraction and have the squared denominator of the fraction as their diameter. Fractions ${\displaystyle a/b}$ and ${\displaystyle c/d}$ are adjacent if and only if their Ford circles are tangent circles.

Applications

Fair division and mathematics education

In mathematics education, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole. 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.

Probability and statistics

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.

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 ${\displaystyle n}$th item is selected is proportional to the unit fraction ${\displaystyle 1/n}$.

Combinatorial optimization

In the study of combinatorial optimization problems, bin packing problems involve an input sequence of items with fractional sizes, which must be placed into bins whose capacity (the total size of items placed into each bin) is one. Research into these problems has included the study of restricted bin packing problems where the item sizes are unit fractions.

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. An item whose delay is ${\displaystyle k}$ times the length of a message must occupy a fraction of at least ${\displaystyle 1/k}$ 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 ${\displaystyle 1/k}$ as item sizes.

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. In particular, the harmonic bin packing method does exactly this, and then packs each bin using items of only a single rounded unit fraction size.

Physics

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.

Arthur Eddington argued that the fine-structure constant was a unit fraction. He initially thought it to be 1/136 and later changed his theory to 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.