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Unit fraction

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From Wikipedia, the free encyclopedia

One over a whole number

For fractions of a measurement unit, seeUnit prefix.

Aunit fraction is a positivefraction with one as itsnumerator, 1/n. It is themultiplicative inverse (reciprocal) of thedenominator of the fraction, which must be a positivenatural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole.

Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Everyrational number can be represented as a sum of distinct unit fractions; these representations are calledEgyptian fractions based on their use inancient Egyptian mathematics. Many infinite sums of unit fractions are meaningful mathematically.

In geometry, unit fractions can be used to characterize the curvature oftriangle groups and the tangencies ofFord circles. Unit fractions are commonly used infair division, and this familiar application is used inmathematics education as an early step toward the understanding of other fractions. Unit fractions are common inprobability theory due to theprinciple of indifference. They also have applications incombinatorial optimization and in analyzing the pattern of frequencies in thehydrogen spectral series.

Arithmetic

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The unit fractions are therational numbers that can be written in the form ${\frac {1}{n}},$ where $n {\displaystyle n}$ can be any positivenatural number. They are thus themultiplicative inverses of the positive integers. When something is divided into $n {\displaystyle n}$ equal parts, each part is a $1/n$ fraction of the whole.^[1]

Elementary arithmetic

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Multiplying any two unit fractions results in a product that is another unit fraction:^[2] ${\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.$ However,adding,^[3]subtracting,^[3] ordividing two unit fractions produces a result that is generally not a unit fraction: ${\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}$

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

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

As the last of these formulas shows, every fraction can be expressed as a quotient of two unit fractions.^[4]

Modular arithmetic

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Inmodular arithmetic, any unit fraction can be converted into an equivalent whole number using theextended Euclidean algorithm.^[5]^[6] This conversion can be used to perform modular division: dividing by a number $x {\displaystyle x}$ , modulo $y {\displaystyle y}$ , can be performed by converting the unit fraction $1/x$ into an equivalent whole number modulo $y {\displaystyle y}$ , and then multiplying by that number.^[7]

In more detail, suppose that $x {\displaystyle x}$ isrelatively prime to $y {\displaystyle y}$ (otherwise, division by $x {\displaystyle x}$ is not defined modulo $y {\displaystyle y}$ ). The extended Euclidean algorithm for thegreatest common divisor can be used to find integers $a {\displaystyle a}$ and $b {\displaystyle b}$ such thatBézout's identity is satisfied: $\displaystyle ax+by=\gcd(x,y)=1.$ In modulo- $y {\displaystyle y}$ arithmetic, the term $b y {\displaystyle by}$ can be eliminated as it is zero modulo $y {\displaystyle y}$ . This leaves $\displaystyle ax\equiv 1{\pmod {y}}.$ That is, $a {\displaystyle a}$ is the modular inverse of $x {\displaystyle x}$ , the number that when multiplied by $x {\displaystyle x}$ produces one. Equivalently,^[5]^[6] $a\equiv {\frac {1}{x}}{\pmod {y}}.$ Thus division by $x {\displaystyle x}$ (modulo $y {\displaystyle y}$ ) can instead be performed by multiplying by the integer $a {\displaystyle a}$ .^[7]

Combinations

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Several constructions in mathematics involve combining multiple unit fractions together, often by adding them.

Finite sums

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Any positive rational number can be written as the sum of distinct 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}}.

These sums are calledEgyptian fractions, because the ancient Egyptian civilisations used them as notation for more generalrational numbers. 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.^[8] The topic of Egyptian fractions has also seen interest in modernnumber theory; for instance, theErdős–Graham problem^[9] and theErdős–Straus conjecture^[10] concern sums of unit fractions, as does the definition ofOre's harmonic numbers.^[11]

A pattern of spherical triangles with reflection symmetry across each triangle edge. Spherical reflection patterns like this with $2x$ , $2y$ , and $2z$ triangles at each vertex (here, $x,y,z=2,3,5$ ) only exist when ${\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}>1$ .

{\displaystyle 2x} — A pattern of spherical triangles with reflection symmetry across each triangle edge. Spherical reflection patterns like this with $2x$ , $2y$ , and $2z$ triangles at each vertex (here, $x,y,z=2,3,5$ ) only exist when ${\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}>1$ .

Ingeometric 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.^[12]

Infinite series

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Many well-knowninfinite series have terms that are unit fractions. These include:

Theharmonic 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 thenatural logarithm of $n {\displaystyle n}$ plus theEuler–Mascheroni constant.^[13] Changing every other addition to a subtraction produces the alternating harmonic series, which sums to thenatural logarithm of 2:^[14] $\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.$
TheLeibniz formula for π is^[15] $1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.$
TheBasel problem concerns the sum of the square unit fractions:^[16] $1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots ={\frac {\pi ^{2}}{6}}.$ Similarly,Apéry's constant is anirrational number, the sum of the cubed unit fractions.^[17]
The binarygeometric series is^[18] $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.$

Matrices

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AHilbert matrix is asquare matrix in which the elements on the $i {\displaystyle i}$ thantidiagonal all equal the unit fraction $1/i$ . That is, it has elements $B_{i,j}={\frac {1}{i+j-1}}.$ For example, the matrix ${\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\end{bmatrix}}$ is a Hilbert matrix. It has the unusual property that all elements in itsinverse matrix are integers.^[19] Similarly,Richardson (2001) defined a matrix whose elements are unit fractions whose denominators areFibonacci numbers: $C_{i,j}={\frac {1}{F_{i+j-1}}},$ where $F_{i}$ denotes the $i {\displaystyle i}$ th Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.^[20]

Adjacency and Ford circles

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Fractions with tangentFord circles differ by a unit fraction

Two fractions $a/b$ and $c/d$ (in lowest terms) are calledadjacent if $ad-bc=\pm 1,$ which implies that they differ from each other by a unit fraction: $\left|{\frac {1}{a}}-{\frac {1}{b}}\right|={\frac {|ad-bc|}{bd}}={\frac {1}{bd}}.$ For instance, ${\tfrac {1}{2}}$ and ${\tfrac {3}{5}}$ are adjacent: $1\cdot 5-2\cdot 3=-1$ and ${\tfrac {3}{5}}-{\tfrac {1}{2}}={\tfrac {1}{10}}$ . However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance, ${\tfrac {1}{3}}$ and ${\tfrac {2}{3}}$ differ by a unit fraction, but are not adjacent, because for them $ad-bc=3$ .^[21]

This terminology comes from the study ofFord circles. These are a system of circles that are tangent to thenumber line at a given fraction and have the squared denominator of the fraction as their diameter. Fractions $a/b$ and $c/d$ are adjacent if and only if their Ford circles aretangent circles.^[21]

Applications

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Fair division and mathematics education

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Inmathematics education, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole.^[22]^[23] A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort offair division are a standard classroom example in teaching students to work with unit fractions.^[24]

Probability and statistics

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A six-sided die has probability 1/6 of landing on each side

In auniform distribution on a discrete space, all probabilities are equal unit fractions. Due to theprinciple of indifference, probabilities of this form arise frequently in statistical calculations.^[25]

Unequal probabilities related to unit fractions arise inZipf's law. This states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the $n {\displaystyle n}$ th item is selected is proportional to the unit fraction $1/n$ .^[26]

Combinatorial optimization

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In the study ofcombinatorial optimization problems,bin packing problems involve an input sequence of items with fractional sizes, which must be placed into bins whose capacity (the total size of items placed into each bin) is one. Research into these problems has included the study of restricted bin packing problems where the item sizes are unit fractions.^[27]^[28]

One motivation for this is as a test case for more general bin packing methods. Another involves a form ofpinwheel scheduling, in which a collection of messages of equal length must each be repeatedly broadcast on a limited number of communication channels, with each message having a maximum delay between the start times of its repeated broadcasts. An item whose delay is $k {\displaystyle k}$ times the length of a message must occupy a fraction of at least $1/k$ of the time slots on the channel it is assigned to, so a solution to the scheduling problem can only come from a solution to the unit fraction bin packing problem with the channels as bins and the fractions $1/k$ as item sizes.^[27]

Even for bin packing problems with arbitrary item sizes, it can be helpful to round each item size up to the next larger unit fraction, and then apply a bin packing algorithm specialized for unit fraction sizes. In particular, theharmonic bin packing method does exactly this, and then packs each bin using items of only a single rounded unit fraction size.^[28]

Physics

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Thehydrogen spectral series, on a logarithmic scale. The frequencies of the emission lines are proportional to differences of pairs of unit fractions.

The energy levels ofphotons that can be absorbed or emitted by a hydrogen atom are, according to theRydberg formula, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by theBohr model, according to which the energy levels ofelectron orbitals in ahydrogen atom are inversely proportional to square unit fractions, and the energy of a photon isquantized to the difference between two levels.^[29]

Arthur Eddington argued that thefine-structure constant was a unit fraction. He initially thought it to be 1/136 and later changed his theory to 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.^[30]

References

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^Solomon, Pearl Gold (2007),The Math We Need to Know and Do in Grades 6 9: Concepts, Skills, Standards, and Assessments, Corwin Press, p. 157,ISBN 978-1-4129-1726-1
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^Humenberger, Hans (Fall 2014), "Egyptian fractions – representations as sums of unit fractions",Mathematics and Computer Education,48 (3):268–283,ProQuest 1622317875
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^Guy, Richard K. (2004), "D11. Egyptian Fractions",Unsolved problems in number theory (3rd ed.), Springer-Verlag, pp. 252–262,ISBN 978-0-387-20860-2
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^Elsholtz, Christian;Tao, Terence (2013),"Counting the number of solutions to the Erdős–Straus equation on unit fractions"(PDF),Journal of the Australian Mathematical Society,94 (1):50–105,arXiv:1107.1010,doi:10.1017/S1446788712000468,MR 3101397,S2CID 17233943
^Ore, Øystein (1948), "On the averages of the divisors of a number",The American Mathematical Monthly,55 (10):615–619,doi:10.2307/2305616,JSTOR 2305616
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^Boas, R. P. Jr.;Wrench, J. W. Jr. (1971), "Partial sums of the harmonic series",The American Mathematical Monthly,78 (8):864–870,doi:10.1080/00029890.1971.11992881,JSTOR 2316476,MR 0289994
^Freniche, Francisco J. (2010),"On Riemann's rearrangement theorem for the alternating harmonic series"(PDF),The American Mathematical Monthly,117 (5):442–448,doi:10.4169/000298910X485969,JSTOR 10.4169/000298910x485969,MR 2663251,S2CID 20575373
^Roy, Ranjan (1990),"The discovery of the series formula forπ by Leibniz, Gregory and Nilakantha"(PDF),Mathematics Magazine,63 (5):291–306,doi:10.1080/0025570X.1990.11977541, archived fromthe original(PDF) on 2023-03-14, retrieved2023-03-22
^Ayoub, Raymond (1974),"Euler and the zeta function",The American Mathematical Monthly,81 (10):1067–86,doi:10.2307/2319041,JSTOR 2319041, archived fromthe original on 2019-08-14, retrieved2023-03-22
^van der Poorten, Alfred (1979),"A proof that Euler missed ... Apéry's proof of the irrationality of $\zeta (3)$ "(PDF),The Mathematical Intelligencer,1 (4):195–203,doi:10.1007/BF03028234,S2CID 121589323, archived fromthe original(PDF) on 2011-07-06
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v t e Fractions andratios
	Division and ratio	Dividend ÷Divisor =Quotient
	Fraction	⁠Numerator/Denominator⁠ = Quotient
	Algebraic Aspect Binary Continued Decimal Dyadic Egyptian Golden Silver Integer Irreducible Reduction Just intonation LCD Musical interval Paper size Percentage Unit