One half

For the computer virus, see OneHalf.

"Half" redirects here; for other uses that do not relate to "one half" as a number (½), see Half (disambiguation).

← −0.5 0.5 1.5 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	one half
Ordinal	1⁄2th (halfth)
Binary	0.12
Ternary	0.11111111113
Senary	0.36
Octal	0.48
Duodecimal	0.612
Hexadecimal	0.816
Greek	∠
Roman numerals	S
Egyptian hieroglyph	𓐛
Hebrew	חֵצִ
Malayalam	൴
Chinese	半
Tibetan	༪

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.

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

As a word

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

One half is a rational number that lies midway between nil ${\displaystyle 0}$ and unity ${\displaystyle 1}$ (which are the elementary additive and multiplicative identities) as the quotient of the first two non-zero integers, ${\displaystyle {\tfrac {1}{2}}}$. It has two different decimal representations in base ten, the familiar ${\displaystyle 0.5}$ and the recurring ${\displaystyle 0.4{\overline {9}}}$, 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 ${\displaystyle 0.{\overline {1}}}$ in ternary and ${\displaystyle 0.{\overline {2}}}$ 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 ${\displaystyle n}$ raised to the power of one half is equal to the square root of ${\displaystyle n}$,

${\displaystyle n^{\tfrac {1}{2}}={\sqrt {n}}.}$

Properties

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

${\displaystyle {\frac {\sigma (n)}{n}}={\frac {k}{2}},}$

where ${\displaystyle k}$ is odd, and ${\displaystyle \sigma (n)}$ is the sum-of-divisors function. The first three hemiperfect numbers are 2, 24, and 4320. ^[1]

The area ${\displaystyle T}$ of a triangle with base ${\displaystyle b}$ and altitude ${\displaystyle h}$ is computed as

${\displaystyle T={\frac {b}{2}}\times h.}$

Ed Pegg Jr. noted that the length ${\displaystyle d}$ equal to ${\textstyle {\frac {1}{2}}{\sqrt {{\frac {1}{30}}(61421-23{\sqrt {5831385}})}}}$ is almost an integer, approximately 7.0000000857.

One half figures in the formula for calculating figurate numbers, such as the ${\displaystyle n}$-th triangular number:

${\displaystyle P_{2}(n)={\frac {n(n+1)}{2}};}$

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

${\displaystyle M_{2}(n)={\frac {n}{2}}\left(n^{2}+1\right).}$

Successive natural numbers yield the ${\displaystyle n}$-th metallic mean ${\displaystyle M}$ by the equation,

${\displaystyle M_{(n)}={\frac {n+{\sqrt {n^{2}+4}}}{2}}.}$

In the study of finite groups, alternating groups have order

${\displaystyle {\frac {n!}{2}}.}$

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

${\displaystyle {\frac {\pi }{2}}=\sum _{n=1}^{\infty }{\frac {(-1)^{\varepsilon (n)}}{n}}=1+{\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}-{\frac {1}{7}}+\cdots ,{\text{ }}}$

where ${\displaystyle \varepsilon (n)}$ is the number of prime factors of the form ${\displaystyle p\equiv 3\,(\mathrm {mod} \,4)}$ of ${\displaystyle n}$ (see modular arithmetic).

Fundamental region of the modular j-invariant in the upper half-plane (shaded gray), with modular discriminant ${\displaystyle |\tau |\geq 1}$ and ${\displaystyle -{\tfrac {1}{2}}<{\mathfrak {R}}(\tau )\leq {\tfrac {1}{2}}}$, where ${\displaystyle -{\tfrac {1}{2}}<{\mathfrak {R}}(\tau )<0\Rightarrow |\tau |>1.}$

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

${\displaystyle \Gamma ({\tfrac {1}{2}})={\sqrt {\pi }};}$

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

${\displaystyle \zeta (3)=-{\tfrac {1}{2}}\Gamma '''(1)+{\tfrac {3}{2}}\Gamma '(1)\Gamma ''(1)-{\big (}\Gamma '(1){\big )}^{3}=-{\tfrac {1}{2}}\psi ^{(2)}(1);{\text{ }}}$

with ${\displaystyle \psi ^{(m)}(z)}$ the polygamma function of order ${\displaystyle m}$ on the complex numbers ${\displaystyle \mathbb {C} }$.

The upper half-plane ${\displaystyle {\mathcal {H}}}$ is the set of points ${\displaystyle (x,y)}$ in the Cartesian plane with ${\displaystyle y>0}$. In the context of complex numbers, the upper half-plane is defined as

${\displaystyle {\mathcal {H}}:=\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}.}$

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

The Bernoulli number ${\displaystyle B_{1}}$ has the value ${\displaystyle \pm {\tfrac {1}{2}}}$ (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 ${\displaystyle {\tfrac {1}{2}}}$.

Computer characters

½
vulgar fraction one half
In  Unicode	U+00BD½VULGAR FRACTION ONE HALF
Related
See also	U+00BC¼ VULGAR FRACTION ONE QUARTER U+00BE¾ 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. ^[7]

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

See also

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

• Division by two

References

1. Sloane, N. J. A. (ed.). "SequenceA159907(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.
7. "Latin-1 Supplement". SYMBL. Retrieved 2023-07-18.