A unit fraction
is a rational number
written as a fraction
where the numerator
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.
any two unit fractions results in a product that is another unit fraction:
, or dividing
two unit fractions produces a result that is generally not a unit fraction:
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
from which it follows that
Thus, to divide by x
) 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,
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
- closely approximate ln n + γ as n increases.
Matrices of unit fractions
The Hilbert matrix is the matrix with elements
It has the unusual property that all elements in its inverse matrix
are integers. Similarly, Richardson defined a matrix with elements
denotes the i
th 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 orbitals
in a hydrogen atom
are proportional to square unit fractions. Therefore the energy levels of photons
that can be absorbed or emitted by a hydrogen atom are, according to this model, proportional to the differences of two unit fractions. 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.