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Unit fraction

## Summary

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc.

## Elementary arithmetic

Multiplying any two unit fractions results in a product that is another unit fraction:

${\displaystyle {\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.}$

However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:

${\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}}$
${\displaystyle {\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}}$
${\displaystyle {\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}.}$

## Modular arithmetic

Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that

${\displaystyle \displaystyle ax+by=1,}$

from which it follows that

${\displaystyle \displaystyle ax\equiv 1{\pmod {y}},}$

or equivalently

${\displaystyle a\equiv {\frac {1}{x}}{\pmod {y}}.}$

Thus, to divide by x (modulo y) we need merely instead multiply by a.

## Finite sums of unit fractions

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

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

The ancient Egyptian civilisations used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. 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.[1] The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture 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.

## Series of unit fractions

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:
${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.}$

• 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 of unit fractions

The Hilbert matrix is the matrix with elements

${\displaystyle B_{i,j}={\frac {1}{i+j-1}}.}$

It has the unusual property that all elements in its inverse matrix are integers.[2] Similarly, Richardson (2001) defined a matrix with elements

${\displaystyle C_{i,j}={\frac {1}{F_{i+j-1}}},}$

where Fi denotes the ith Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.[3]

Fractions with tangent Ford circles differ by a unit fraction

Two fractions ${\displaystyle a/b}$  and ${\displaystyle c/d}$  (in lowest terms) are called adjacent if ${\displaystyle ad-bc=\pm 1}$ , which implies that their difference ${\displaystyle |ad-bc|/bd}$  is a unit fraction. For instance, ${\displaystyle {\tfrac {1}{2}}}$  and ${\displaystyle {\tfrac {3}{5}}}$  are adjacent: ${\displaystyle 1\cdot 5-2\cdot 3=-1}$  and ${\displaystyle {\tfrac {3}{5}}-{\tfrac {1}{2}}={\tfrac {1}{10}}}$ . However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance, ${\displaystyle {\tfrac {1}{3}}}$  and ${\displaystyle {\tfrac {2}{3}}}$  differ by a unit fraction, but are not adjacent, because for them ${\displaystyle ad-bc=3}$ . The terminology comes from the study of Ford circles, 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.[4]

## Unit fractions in 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.[5] Additionally, Zipf's law states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the nth item is selected is proportional to the unit fraction 1/n.[6]

## Unit fractions in 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.[7]

Arthur Eddington argued that the fine-structure constant was a unit fraction, first 1/136 then 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.[8]