Unit fraction
Encyclopedia
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 etc.
any two unit fractions results in a product that is another unit fraction:
However, adding
, subtracting
, or dividing
two unit fractions produces a result that is generally not a unit fraction:
, 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 divisor
s we may find a and b such that
from which it follows that
or equivalently
Thus, to divide by x (modulo y) we need merely instead multiply by a.
The ancient Egyptians used sums of distinct unit fractions in their notation for more general rational number
s, 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 group
s 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.
have terms that are unit fractions. These include:
It has the unusual property that all elements in its inverse matrix are integers. Similarly, defined a matrix with elements
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.
, 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.
s 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, 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.
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
written as a fraction where the numerator is one and the denominator is a positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 etc.
Elementary arithmetic
MultiplyingMultiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
any two unit fractions results in a product that is another unit fraction:
However, adding
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtracting
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, or dividing
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
two unit fractions produces a result that is generally not a unit fraction:
Modular arithmetic
Unit fractions play an important role in modular arithmeticModular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, 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
Extended Euclidean algorithm
The extended Euclidean algorithm is an extension to the Euclidean algorithm. Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y that satisfy Bézout's identityThe extended Euclidean algorithm is particularly useful when a...
for greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
s we may find a and b such that
from which it follows that
or equivalently
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,The ancient Egyptians used sums of distinct unit fractions in their notation for more general rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, 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
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture
Erdos–Straus conjecture
In number theory, the Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdős and Ernst G...
concern sums of unit fractions, as does the definition of Ore's harmonic numbers.
In geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
s 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 seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
have terms that are unit fractions. These include:
- The harmonic seriesHarmonic series (mathematics)In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
, the sum of all positive unit fractions. This sum diverges, and its partial sums
-
-
- closely approximate lnNatural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
n + γEuler-Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
as n increases.
- The Basel problemBasel problemThe Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...
concerns the sum of the square unit fractions, which converges to πPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
2/6
- Apéry's constantApéry's constantIn mathematics, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics...
is the sum of the cubed unit fractions.
- The binary geometric series, which adds to 2, and the reciprocal Fibonacci constant are additional examples of a series composed of unit fractions.
Matrices of unit fractions
The Hilbert matrix is the matrix with elementsIt has the unusual property that all elements in its inverse matrix are integers. Similarly, defined a matrix with elements
where Fi denotes the ith Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.
Adjacent fractions
Two fractions are called adjacent if their difference is a unit fraction.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 indifferencePrinciple of indifference
The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
, 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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s that can be absorbed or emitted by a hydrogen atom are, according to the Rydberg formula
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...
, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by the Bohr model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
, according to which the energy levels of electron orbitals
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
in a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
are inversely proportional to square unit fractions, and the energy of a photon is quantized
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
to the difference between two levels.
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.