Unit fraction

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, 1/42 etc.

Elementary arithmetic

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

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

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

${\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}$

${\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}$

${\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}$

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

$\displaystyle ax+by=1,$

from which it follows that

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

or equivalently

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

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,

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

The ancient Egyptians 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. 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

${\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}$

closely approximate log_e(n)+γ as n increases.

• The Basel problem concerns the sum of the square unit fractions, which converges to π²/6

• Apéry's constant is the sum of the cubed unit fractions.

• The binary geometric series, which adds to 2, is another example of a series composed of unit fractions.

Matrices of unit fractions

The Hilbert matrix is the matrix with elements

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

It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson defined a matrix with elements

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

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

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. 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.

Unit fractions in physics

The energy levels of the Bohr model of electron orbits in a Hydrogen atom are proportional to square unit fractions, and therefore the energy levels of photons that can be absorbed or emitted by a Hydrogen atom according to this model are similarly proportional to the differences of two such fractions. It was believed for some time that the Eddington number, or fine structure constant, was exactly a unit fraction, 1/137, but this is now known to be false.

References

• Melik, Jani. "Unit fraction". PlanetMath. Retrieved 2006-09-09.

• Richardson, Thomas M. (2001). "The Filbert matrix". Fibonacci Quart. 39 (3): 268–275.

• Weisstein, Eric W. "Unit Fraction". MathWorld–A Wolfram Web Resource. Retrieved 2006-09-09.