UNIT FRACTION
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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.
Multiplying 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:
:
:
:
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
:
or equivalently
:
Thus, to divide by ''x'' (modulo ''y'') we need merely instead multiply by ''a''.
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.
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 log''e''(''n'')+γ as ''n'' increases.
★ The Basel problem concerns the sum of the square unit fractions, which converges to Ï€2/6
★ 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.
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
:
where ''F''i denotes the ''i''th Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.
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 ''n''th item is selected is proportional to the unit fraction 1/''n''.
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.
★ Unit fraction Melik, Jani
★ The Filbert matrix, Richardson, Thomas M., , , Fibonacci Quart., 2001
★ Unit Fraction Weisstein, Eric W
Elementary arithmetic
Multiplying 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:
:
:
:
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
:
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 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 log''e''(''n'')+γ as ''n'' increases.
★ The Basel problem concerns the sum of the square unit fractions, which converges to Ï€2/6
★ 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
:
It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson defined a matrix with elements
:
where ''F''i 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 ''n''th 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
★ Unit fraction Melik, Jani
★ The Filbert matrix, Richardson, Thomas M., , , Fibonacci Quart., 2001
★ Unit Fraction Weisstein, Eric W
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