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The fraction of a unit (aliquot fraction) is a rational number in the form of a fraction , the numerator of which is equal to unity, and the denominator is a positive integer . The unit fraction is thus the inverse of a positive integer, 1 / n . Examples are 1/1, 1/2, 1/3, 1/4, etc.

Content

1 Elementary arithmetic
2 Modular Arithmetic
3 The final amount of fractions of a unit
4 Sequences of unit fractions
5 Matrices from fractions of a unit
6 Adjacent fractions
7 Unit fractions in probability theory and statistics
8 Unit fractions in physics
9 See also
10 notes
11 Literature
12 Links

Elementary Arithmetic

Multiplying any two fractions of a unit gives a fraction of a unit:

{\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.

{\ displaystyle {\ frac {1} {x}} \ times {\ frac {1} {y}} = {\ frac {1} {xy}}.}

However, adding , subtracting or dividing two parts of a unit gives in the general case a result different from the parts of a unit:

{\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}

{\ displaystyle {\ frac {1} {x}} + {\ frac {1} {y}} = {\ frac {x + y} {xy}}}

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

{\ displaystyle {\ frac {1} {x}} - {\ frac {1} {y}} = {\ frac {yx} {xy}}}

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

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

Modular Arithmetic

Unit fractions play an important role in modulo comparisons because they can be used to reduce modular division to calculate the largest common divisor. In particular, suppose we want to calculate the result of dividing by x modulo y . For division by x to be defined modulo y , x and y must be coprime . Then, using the extended Euclidean algorithm for finding the greatest common factor, we can find a and b such that

\displaystyle ax+by=1,

{\ displaystyle \ displaystyle ax + by = 1,}

whence follows

\displaystyle ax\equiv 1{\pmod {y}},

{\ displaystyle \ displaystyle ax \ equiv 1 {\ pmod {y}},}

which is equivalent

a\equiv {\frac {1}{x}}{\pmod {y}}.

{\ displaystyle a \ equiv {\ frac {1} {x}} {\ pmod {y}}.}

Thus, to divide by x (modulo y ) you just need to multiply by a .

Final Amount of Unit

Any positive rational number can be represented as a sum of fractions of a unit in several 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}}.

{\ displaystyle {\ frac {4} {5}} = {\ frac {1} {2}} + {\ frac {1} {4}} + {\ frac {1} {20}} = {\ frac {1} {3}} + {\ frac {1} {5}} + {\ frac {1} {6}} + {\ frac {1} {10}}.}

The ancient Egyptians used the sums of various fractions of a unit in the record of rational numbers, and such sums are often called Egyptian fractions . There is still an interest in the analysis of methods used by the ancients to select possible representations and calculate such representations ^[1] . The theme of Egyptian fractions is of interest for modern number theory . For example, the Erdos – Graham hypothesis and the Erdos – Strauss hypothesis concern sums of fractions of units, as well as the definition of .

In the geometric theory of groups, groups of triangles are classified as Euclidean, spherical, and hyperbolic depending on whether the sum of fractions of one associated with them is equal to one, less than one, or more than one.

Unit Sequence

Many well-known infinite series have terms in the form of fractions of a unit. Among them:

A harmonic series is the sum of all positive fractions of a unit. This amount diverges and its partial amounts

{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}

{\ displaystyle {\ frac {1} {1}} + {\ frac {1} {2}} + {\ frac {1} {3}} + \ cdots + {\ frac {1} {n}}}

are closely approximated by ln n + γ with increasing n .

The Basel problem considers the sum of squares of fractions of a unit that converges to π 2/6
Aperi constant - the sum of cubes of fractions of a unit.
Geometric progressions and the are additional examples of units of units.

Matrices of fractions of a unit

The Hilbert matrix has as elements the numbers

B_{i,j}={\frac {1}{i+j-1}}.

{\ displaystyle B_ {i, j} = {\ frac {1} {i + j-1}}.}

It has an unusual property - all elements of its inverse matrix are integers ^[2] . In a similar way, Richardson ^[3] defined a matrix with elements

C_{i,j}={\frac {1}{F_{i+j-1}}},

{\ displaystyle C_ {i, j} = {\ frac {1} {F_ {i + j-1}}},}

where F _i denotes the i- th Fibonacci number . He called this matrix “Philbert's matrix” and it has the same property ^[4] .

Adjacent Fractions

Two fractions are called adjacent if their difference is a fraction of unity ^[5] ^[6] .

Unit fractions in probability theory and statistics

In a discrete uniform distribution, all probabilities are equal to a fraction of unity. According to the probabilities of this type often arise in statistical calculations ^[7] . In addition, Zipf’s law states that for many observed events, including the selection of objects from an ordered sequence, the probability that the nth object will be selected is proportional to the fraction of 1 / n ^[8] .

Unit Fractions in Physics

The energy levels of photons that can be absorbed or emitted by a hydrogen atom, according to Rydberg's formula , are proportional to the difference of two fractions of one. This phenomenon is explained by the Bohr model , according to which the energy levels of electronic orbitals in a hydrogen atom are inversely proportional to the square of fractions of a unit, and the photon energy is quantized by the level difference ^[9] .

Arthur Eddington claimed that the fine structure constant is equal to a fraction of unity, first 1/136, and then 1/137. This statement turned out to be false and the current estimate of the fine structure constant is (with an accuracy of 6 digits) 1 / 137.036 ^[10] .

Notes

↑ Guy, 2004 , p. 252–262.
↑ Choi, 1983 , p. 301-312.
↑ Richardson, 2001 .
↑ Richardson, 2001 , p. 268-275.
↑ Adjacent Fraction on the PlanetMath website .
↑ Weisstein, Eric W. Adjacent Fraction on the Wolfram MathWorld website.
↑ Welsh, 1996 , p. 66.
↑ Saichev, Malevergne, Sornette, 2009 .
↑ Yang, Hamilton, 2009 , p. 81–86.
↑ Kilmister, 1994 .

Literature

Richard K. Guy. Unsolved problems in number theory. - 3rd. - Springer-Verlag, 2004. - ISBN 978-0-387-20860-2 .
Man Duen Choi. Tricks or treats with the Hilbert matrix // The American Mathematical Monthly. - 1983 .-- T. 90 , no. 5 . - DOI : 10.2307 / 2975779 .
Thomas M. Richardson. The Filbert matrix // Fibonacci Quarterly . - 2001. - T. 39 , no. 3 . - . - arXiv : math.RA / 9905079 .
Fujia Yang, Joseph H. Hamilton. Modern Atomic and Nuclear Physics. - World Scientific, 2009. - ISBN 978-981-283-678-6 .
Clive William Kilmister. Eddington's search for a fundamental theory: a key to the universe. - Cambridge University Press, 1994. - ISBN 978-0-521-37165-0 .
Alan H. Welsh. Aspects of statistical inference. - John Wiley and Sons, 1996. - T. 246. - (Wiley Series in Probability and Statistics). - ISBN 978-0-471-11591-5 .
Alexander Saichev, Yannick Malevergne, Didier Sornette. Theory of Zipf's Law and Beyond. - Springer-Verlag, 2009. - T. 632. - (Lecture Notes in Economics and Mathematical Systems). - ISBN 978-3-642-02945-5 .

Links

Weisstein, Eric W. Unit Fraction ( Wolfram ) at Wolfram MathWorld .

[_5ea70925eed37ff4-1] Guy, 2004 , p. 252–262.

[_c1754c116a8e3a1a-2] Choi, 1983 , p. 301-312.

[_6b0b8e7478687f73-3] Richardson, 2001 .

[_94b187db8ca98f18-4] Richardson, 2001 , p. 268-275.

[5] Adjacent Fraction on the PlanetMath website .

[6] Weisstein, Eric W. Adjacent Fraction on the Wolfram MathWorld website.

[_34d9a3841117ee97-7] Welsh, 1996 , p. 66.

[_cc0eaed4e6c52459-8] Saichev, Malevergne, Sornette, 2009 .

[_d29b40f8053fc545-9] Yang, Hamilton, 2009 , p. 81–86.

[_c425d2b8c168bc34-10] Kilmister, 1994 .