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| A '''unit fraction''' is a [[rational number]] written as a [[Fraction (mathematics)|fraction]] where the [[numerator]] is [[1 (number)|one]] and the [[denominator]] is a positive [[integer]]. A unit fraction is therefore the [[Multiplicative inverse|reciprocal]] of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4 etc.
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| == Elementary arithmetic ==
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| [[Multiplication|Multiplying]] any two unit fractions results in a product that is another unit fraction:
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| :<math>\frac1x \times \frac1y = \frac1{xy}.</math>
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| However, [[Addition|adding]], [[Subtraction|subtracting]], or [[Division (mathematics)|dividing]] two unit fractions produces a result that is generally not a unit fraction:
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| :<math>\frac1x + \frac1y = \frac{x+y}{xy}</math>
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| :<math>\frac1x - \frac1y = \frac{y-x}{xy}</math>
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| :<math>\frac1x \div \frac1y = \frac{y}{x}.</math>
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| == Modular arithmetic ==
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| 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 divisor]]s we may find ''a'' and ''b'' such that
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| :<math>\displaystyle ax + by = 1,</math> | |
| from which it follows that
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| :<math>\displaystyle ax \equiv 1 \pmod y,</math>
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| or equivalently
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| :<math>a \equiv \frac1x \pmod y.</math>
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| Thus, to divide by ''x'' (modulo ''y'') we need merely instead multiply by ''a''.
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| == Finite sums of unit fractions ==
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| Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,
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| :<math>\frac45=\frac12+\frac14+\frac1{20}=\frac13+\frac15+\frac16+\frac1{10}.</math>
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| 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.<ref>{{citation
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| | last = Guy | first = Richard K. | author-link = Richard K. Guy
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| | contribution = D11. Egyptian Fractions
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| | edition = 3rd
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| | isbn = 978-0-387-20860-2
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| | page = 252–262
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| | publisher = Springer-Verlag
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| | title = Unsolved problems in number theory
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| | year = 2004}}.</ref> 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 number]]s.
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| 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.
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| == Series of unit fractions ==
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| Many well-known [[Series (mathematics)|infinite series]] have terms that are unit fractions. These include:
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| * The [[harmonic series (mathematics)|harmonic series]], the sum of all positive unit fractions. This sum diverges, and its partial sums
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| :: <math>\frac11+\frac12+\frac13+\cdots+\frac1n</math> | |
| : closely approximate [[natural logarithm|ln]] ''n'' + [[Euler-Mascheroni constant|γ]] as ''n'' increases.
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| * The [[Basel problem]] concerns the sum of the square unit fractions, which converges to [[Pi|π]]<sup>2</sup>/6
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| * [[Apéry's constant]] is the sum of the cubed unit fractions.
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| * The binary [[geometric series]], which adds to 2, and the [[reciprocal Fibonacci constant]] are additional examples of a series composed of unit fractions.
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| == Matrices of unit fractions ==
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| The [[Hilbert matrix]] is the matrix with elements
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| :<math>B_{i,j} = \frac1{i+j-1}.</math>
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| It has the unusual property that all elements in its [[matrix inverse|inverse matrix]] are integers.<ref>{{citation
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| | last = Choi | first = Man Duen
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| | doi = 10.2307/2975779
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| | mr = 701570
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| | issue = 5
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| | journal = The American Mathematical Monthly
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| | pages = 301–312
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| | title = Tricks or treats with the Hilbert matrix
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| | volume = 90
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| | year = 1983}}.</ref> Similarly, {{harvtxt|Richardson|2001}} defined a matrix with elements
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| :<math>C_{i,j} = \frac1{F_{i+j-1}},</math> | |
| where ''F''<sub>i</sub> denotes the ''i''th [[Fibonacci number]]. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.<ref>{{citation
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| | last = Richardson | first = Thomas M.
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| | title = The Filbert matrix
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| | journal = [[Fibonacci Quarterly]]
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| | volume = 39
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| | issue = 3
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| | year = 2001
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| | pages = 268–275
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| | arxiv = math.RA/9905079
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| | bibcode = 1999math......5079R
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| | url = http://www.fq.math.ca/Scanned/39-3/richardson.pdf}}</ref>
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| ==Adjacent fractions==
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| Two fractions are called '''adjacent''' if their difference is a unit fraction.<ref>{{PlanetMath|urlname=AdjacentFraction|title=Adjacent Fraction}}</ref><ref>{{MathWorld |title=Adjacent Fraction |id=AdjacentFraction}}</ref>
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| == Unit fractions in probability and statistics ==
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| In a [[Uniform distribution (discrete)|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.<ref>{{citation|page=66|title=Aspects of statistical inference|volume=246|series=Wiley Series in Probability and Statistics|first=Alan H.|last=Welsh|publisher=John Wiley and Sons|year=1996|isbn=978-0-471-11591-5}}.</ref> 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''.<ref>{{citation|title=Theory of Zipf's Law and Beyond|volume=632|series=Lecture Notes in Economics and Mathematical Systems|first1=Alexander|last1=Saichev|first2=Yannick|last2=Malevergne|first3=Didier|last3=Sornette|publisher=Springer-Verlag|year=2009|isbn=978-3-642-02945-5}}.</ref>
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| == Unit fractions in physics ==
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| The energy levels of [[photon]]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 [[Atomic orbital|electron orbitals]] in a [[hydrogen atom]] are inversely proportional to square unit fractions, and the energy of a photon is [[Quantization (physics)|quantized]] to the difference between two levels.<ref>{{citation|pages=81–86|title=Modern Atomic and Nuclear Physics|first1=Fujia|last1=Yang|first2=Joseph H.|last2=Hamilton|publisher=World Scientific|year=2009|isbn=978-981-283-678-6}}.</ref>
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| [[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.<ref>{{citation|title=Eddington's search for a fundamental theory: a key to the universe|first=Clive William|last=Kilmister|publisher=Cambridge University Press|year=1994|isbn=978-0-521-37165-0}}.</ref>
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| == References ==
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| {{reflist}}
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| ==External links==
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| *{{Mathworld | title = Unit Fraction | urlname = UnitFraction}}
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| [[Category:Fractions]]
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| [[Category:One]]
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| [[Category:Elementary arithmetic]]
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