One half

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One half is the irreducible fraction resulting from dividing one (1) by two (2), or the fraction resulting from dividing any number by its double.

← −0.5 0.5 1.5 →
−1 0 1 2 3 4 5 6 7 8 9
Cardinalone half
Ordinal12th (halfth)
Binary0.12
Ternary0.11111111113
Senary0.36
Octal0.48
Duodecimal0.612
Hexadecimal0.816
Greek
Roman numeralsS
Egyptian hieroglyph𓐛
Hebrewחֵצִ
Malayalam
Chinese
Tibetan

It often appears in mathematical equations, recipes, measurements, etc.

As a word

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One half is one of the few fractions which are commonly expressed in natural languages by suppletion rather than regular derivation. In English, for example, compare the compound "one half" with other regular formations like "one-sixth".

A half can also be said to be one part of something divided into two equal parts. It is acceptable to write one half as a hyphenated word, one-half.

Mathematics

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One half is a rational number that lies midway between nil   and unity   (which are the elementary additive and multiplicative identities) as the quotient of the first two non-zero integers,  . It has two different decimal representations in base ten, the familiar   and the recurring  , with a similar pair of expansions in any even base; while in odd bases, one half has no terminating representation, it has only a single representation with a repeating fractional component (such as   in ternary and   in quinary).

Multiplication by one half is equivalent to division by two, or "halving"; conversely, division by one half is equivalent to multiplication by two, or "doubling".

 
A square of side length one, here dissected into rectangles whose areas are successive powers of one half.

A number   raised to the power of one half is equal to the square root of  ,

 

Properties

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A hemiperfect number is a positive integer with a half-integer abundancy index:

 

where   is odd, and   is the sum-of-divisors function. The first three hemiperfect numbers are 2, 24, and 4320.[1]

The area   of a triangle with base   and altitude   is computed as

 
 
Ed Pegg Jr. noted that the length   equal to   is almost an integer, approximately 7.0000000857.[2][3]

One half figures in the formula for calculating figurate numbers, such as the  -th triangular number:

 

and in the formula for computing magic constants for magic squares,

 

Successive natural numbers yield the  -th metallic mean   by the equation,

 

In the study of finite groups, alternating groups have order

 

By Euler, a classical formula involving pi, and yielding a simple expression:[4]

 

where   is the number of prime factors of the form   of   (see modular arithmetic).

 
Fundamental region of the modular j-invariant in the upper half-plane (shaded gray), with modular discriminant   and  , where  

For the gamma function, a non-integer argument of one half yields,

 

while inside Apéry's constant, which represents the sum of the reciprocals of all positive cubes, there is[5][6]

 

with   the polygamma function of order   on the complex numbers  .

The upper half-plane   is the set of points   in the Cartesian plane with  . In the context of complex numbers, the upper half-plane is defined as

 

In differential geometry, this is the universal covering space of surfaces with constant negative Gaussian curvature, by the uniformization theorem.

The Bernoulli number   has the value   (its sign depending on competing conventions).

The Riemann hypothesis is the conjecture that every nontrivial complex root of the Riemann zeta function has a real part equal to  .

Computer characters

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½
In UnicodeU+00BD ½ VULGAR FRACTION ONE HALF
Related
See alsoU+00BC ¼ VULGAR FRACTION ONE QUARTER
U+00BE ¾ VULGAR FRACTION THREE QUARTERS

The "one-half" symbol has its own code point as a precomposed character in the Number Forms block of Unicode, rendering as ½.

The reduced size of this symbol may make it illegible to readers with relatively mild visual impairment; consequently the decomposed forms 12 or 1/2 may be more appropriate.

See also

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Postal stamp, Ireland, 1940: one halfpenny postage due.

References

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  1. ^ Sloane, N. J. A. (ed.). "Sequence A159907 (Numbers n with half-integral abundancy index, sigma(n)/n equals k+1/2 with integer k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-31.
  2. ^ Ed Pegg Jr. (July 2000). "Commentary on weekly puzzles". Mathpuzzle. Retrieved 2023-08-17.
  3. ^ Weisstein, Eric W. "Almost integer". MathWorld -- A WolframAlpha Resource. Retrieved 2023-08-17.
  4. ^ Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. 1. apud Marcum-Michaelem Bousquet & socios. p. 244.
  5. ^ Evgrafov, M. A.; Bezhanov, K. A.; Sidorov, Y. V.; Fedoriuk, M. V.; Shabunin, M. I. (1972). A Collection of Problems in the Theory of Analytic Functions (in Russian). Moscow: Nauka. p. 263 (Ex. 30.10.1).
  6. ^ Bloch, Spencer; Masha, Vlasenko. "Gamma functions, monodromy and Apéry constants" (PDF). University of Chicago (Paper). pp. 1–34. S2CID 126076513.