Editing Unit fraction

{{Short description|One over a whole number}}
{{Good article}}
{{Use mdy dates|cs1-dates=ly|date=March 2023}}
{{Use list-defined references|date=March 2023}}
{{for|fractions of a measurement unit|Unit prefix}}
[[File:Pizza-3007395.jpg|thumb|Slices of approximately 1/8 of a pizza]]

A '''unit fraction''' is a positive [[fraction]] with one as its [[numerator]], 1/{{mvar|n}}. It is the [[multiplicative inverse]] (reciprocal) of the [[denominator]] of the fraction, which must be a positive [[natural 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. Every [[rational number]] can be represented as a sum of distinct unit fractions; these representations are called [[Egyptian fraction]]s based on their use in [[ancient Egyptian mathematics]]. Many infinite sums of unit fractions are meaningful mathematically.

In geometry, unit fractions can be used to characterize the curvature of [[triangle group]]s and the tangencies of [[Ford circle]]s. Unit fractions are commonly used in [[fair division]], and this familiar application is used in [[mathematics education]] as an early step toward the understanding of other fractions. Unit fractions are common in [[probability theory]] due to the [[principle of indifference]]. They also have applications in [[combinatorial optimization]] and in analyzing the pattern of frequencies in the [[hydrogen spectral series]].

==Arithmetic==
The unit fractions are the [[rational number]]s that can be written in the form <math display=block>\frac1n,</math> where <math>n</math> can be any positive [[natural number]]. They are thus the [[multiplicative inverse]]s of the positive integers. When something is divided into <math>n</math> equal parts, each part is a <math>1/n</math> fraction of the whole.{{r|cavkin}}

===Elementary arithmetic===
[[Multiplication|Multiplying]] any two unit fractions results in a product that is another unit fraction:{{r|solomon}}
<math display=block>\frac1x \times \frac1y = \frac1{xy}.</math>
However, [[Addition|adding]],{{r|betz}} [[Subtraction|subtracting]],{{r|betz}} or [[Division (mathematics)|dividing]] two unit fractions produces a result that is generally not a unit fraction:
<math display=block>\frac1x + \frac1y = \frac{x+y}{xy}</math>

<math display=block>\frac1x - \frac1y = \frac{y-x}{xy}</math>

<math display=block>\frac1x \div \frac1y = \frac{y}{x}.</math>

As the last of these formulas shows, every fraction can be expressed as a quotient of two unit fractions.{{r|humenberger}}

===Modular arithmetic===
In [[modular arithmetic]], any unit fraction can be converted into an equivalent whole number using the [[extended Euclidean algorithm]].{{r|modlin|modinv}} This conversion can be used to perform modular division: dividing by a number <math>x</math>, modulo <math>y</math>, can be performed by converting the unit fraction <math>1/x</math> into an equivalent whole number modulo <math>y</math>, and then multiplying by that number.{{r|brent}}

In more detail, suppose that <math>x</math> is [[relatively prime]] to <math>y</math> (otherwise, division by <math>x</math> is not defined modulo <math>y</math>). The extended Euclidean algorithm for the [[greatest common divisor]] can be used to find integers <math>a</math> and <math>b</math> such that [[Bézout's identity]] is satisfied:
<math display=block>\displaystyle ax + by = \gcd(x,y)=1.</math>
In modulo-<math>y</math> arithmetic, the term <math>by</math> can be eliminated as it is zero modulo <math>y</math>. This leaves
<math display=block>\displaystyle ax \equiv 1 \pmod y.</math>
That is, <math>a</math> is the modular inverse of <math>x</math>, the number that when multiplied by <math>x</math> produces one. Equivalently,{{r|modlin|modinv}}
<math display=block>a \equiv \frac1x \pmod y.</math>
Thus division by <math>x</math> (modulo <math>y</math>) can instead be performed by multiplying by the integer <math>a</math>.{{r|brent}}

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

===Finite sums===
{{See also|List of sums of reciprocals#Finitely many terms}}

Any positive rational number can be written as the sum of distinct unit fractions, in multiple ways.  For example,
:<math>\frac45=\frac12+\frac14+\frac1{20}=\frac13+\frac15+\frac16+\frac1{10}.</math>
These sums are called [[Egyptian fraction]]s, because the ancient Egyptian civilisations used them as notation for more general [[rational number]]s.  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.{{r|guy}} The topic of Egyptian fractions has also seen interest in modern [[number theory]]; for instance, the [[Erdős–Graham problem]]{{r|croot}} and the [[Erdős–Straus conjecture]]{{r|eltao}} concern sums of unit fractions, as does the definition of [[Ore's harmonic number]]s.{{r|ore}}

[[File:Icosahedral reflection domains.png|thumb|A pattern of spherical triangles with reflection symmetry across each triangle edge. Spherical reflection patterns like this with <math>2x</math>, <math>2y</math>, and <math>2z</math> triangles at each vertex (here, <math>x,y,z=2,3,5</math>) only exist when <math>\tfrac1x+\tfrac1y+\tfrac1z>1</math>.]]
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.{{r|magnus}}

===Infinite series===
{{See also|List of sums of reciprocals#Infinitely many terms}}

Many well-known [[Series (mathematics)|infinite series]] have terms that are unit fractions. These include:
* The [[harmonic series (mathematics)|harmonic series]], the sum of all positive unit fractions. This sum diverges, and its partial sums <math display=block>\frac11 + \frac12 + \frac13 + \cdots + \frac1n</math> closely approximate the [[natural logarithm]] of <math>n</math> plus the [[Euler–Mascheroni constant]].{{r|boawre}} Changing every other addition to a subtraction produces the alternating harmonic series, which sums to the [[natural logarithm of 2]]:{{r|freniche}} <math display=block>\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.</math>
* The [[Leibniz formula for π]] is{{r|roy}} <math display=block>1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4}.</math>
* The [[Basel problem]] concerns the sum of the square unit fractions:{{r|ayoub}} <math display=block>1 + \frac14 + \frac19 + \frac1{16} + \cdots = \frac{\pi^2}{6}.</math> Similarly, [[Apéry's constant]] is an [[irrational number]], the sum of the cubed unit fractions.{{r|vdp}}
* The binary [[geometric series]] is{{r|euler}} <math display=block>1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \cdots = 2.</math>

===Matrices===
A [[Hilbert matrix]] is a [[square matrix]] in which the elements on the {{nowrap|<math>i</math>th}} [[antidiagonal]] all equal the unit fraction <math>1/i</math>. That is, it has elements
<math display=block>B_{i,j} = \frac1{i+j-1}.</math>
For example, the matrix
<math display=block>\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}</math>
is a Hilbert matrix. It has the unusual property that all elements in its [[matrix inverse|inverse matrix]] are integers.{{r|choi}} Similarly, {{harvtxt|Richardson|2001}} defined a matrix whose elements are unit fractions whose denominators are [[Fibonacci number]]s:
<math display=block>C_{i,j} = \frac1{F_{i+j-1}},</math>
where <math>F_i</math> denotes the {{nowrap|<math>i</math>th}} Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.{{r|richardson}}

===Adjacency and Ford circles===
[[File:Ford circles colour.svg|thumb|Fractions with tangent [[Ford circle]]s differ by a unit fraction]]
Two fractions <math>a/b</math> and <math>c/d</math> (in lowest terms) are called '''adjacent''' if <math display=block>ad-bc=\pm1,</math> which implies that they differ from each other by a unit fraction:
<math display=block>\left|\frac{1}{a}-\frac{1}{b}\right|=\frac{|ad-bc|}{bd}=\frac{1}{bd}.</math>
For instance, <math>\tfrac12</math> and <math>\tfrac35</math> are adjacent: <math>1\cdot 5-2\cdot 3=-1</math> and <math>\tfrac35-\tfrac12=\tfrac1{10}</math>. However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance, <math>\tfrac13</math> and <math>\tfrac23</math> differ by a unit fraction, but are not adjacent, because for them <math>ad-bc=3</math>.{{r|ford}}

This terminology comes from the study of [[Ford circle]]s. These are a system of circles that are tangent to the [[number line]] at a given fraction and have the squared denominator of the fraction as their diameter. Fractions <math>a/b</math> and <math>c/d</math> are adjacent if and only if their Ford circles are [[tangent circles]].{{r|ford}}

==Applications==
===Fair division and mathematics education===
In [[mathematics 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.{{r|polkinghorne|superheroes}} A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of [[fair division]] are a standard classroom example in teaching students to work with unit fractions.{{r|fair}}

===Probability and statistics===
[[File:Dice 2005.jpg|thumb|upright|A six-sided die has probability 1/6 of landing on each side]]
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.{{r|welsh}}

Unequal probabilities related to unit fractions arise in [[Zipf's law]]. This states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the {{nowrap|<math>n</math>th}} item is selected is proportional to the unit fraction <math>1/n</math>.{{r|zipf}}

===Combinatorial optimization===
In the study of [[combinatorial 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.{{r|blt|harmony}}

One motivation for this is as a test case for more general bin packing methods. Another involves a form of [[pinwheel 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 <math>k</math> times the length of a message must occupy a fraction of at least <math>1/k</math> 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 <math>1/k</math> as item sizes.{{r|blt}}

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, the [[harmonic bin packing]] method does exactly this, and then packs each bin using items of only a single rounded unit fraction size.{{r|harmony}}

===Physics===
[[File:Hydrogen spectrum.svg|thumb|upright=1.2|The [[hydrogen spectral series]], on a logarithmic scale. The frequencies of the emission lines are proportional to differences of pairs of unit fractions.]]
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.{{r|yang}}

[[Arthur Eddington]] argued that the [[fine-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.{{r|eddington}}
{{-}}

== See also ==
* [[17-animal inheritance puzzle]], a puzzle involving fair division into unit fractions
* [[Submultiple]], a number that produces a unit fraction when used as the numerator with a given denominator
* [[Superparticular ratio]], one plus a unit fraction, important in musical harmony

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{{Fractions and ratios}}

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