One half

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

Short description: Irreducible fraction

Short description: Natural number

← −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
Quaternary	0.24
Quinary	0.22222222225
Senary	0.36
Octal	0.48
Duodecimal	0.612
Hexadecimal	016
Vigesimal	0.A20
Base 36	0.I36
Greek	∠
Egyptian hieroglyph	𓐛
Hebrew	חֵצִ
Malayalam	൴
Chinese	半
Tibetan	༪

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

[math]\displaystyle{ n^{\tfrac {1}{2}} = \sqrt {n}. }[/math]

Properties

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

[math]\displaystyle{ \frac {\sigma(n)}{n} = \frac {k}{2}, }[/math]

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

The area [math]\displaystyle{ T }[/math] of a triangle with base [math]\displaystyle{ b }[/math] and altitude [math]\displaystyle{ h }[/math] is computed as,

[math]\displaystyle{ T = \frac {b}{2} \times h. }[/math]

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

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

[math]\displaystyle{ P_2(n) = \frac{n(n+1)}{2}; }[/math]

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

[math]\displaystyle{ M_2(n) = \frac{n}{2} \left(n^{2} + 1\right). }[/math]

Successive natural numbers yield the [math]\displaystyle{ n }[/math]-th metallic mean [math]\displaystyle{ M }[/math] by the equation,

[math]\displaystyle{ M_{(n)} = \frac{n+\sqrt{n^2+4}}{2}. }[/math]

In the study of finite groups, alternating groups have order

[math]\displaystyle{ \frac {n!}{2}. }[/math]

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

[math]\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 { } }[/math]

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

Fundamental region of the modular j-invariant in the upper half-plane (shaded gray), with modular discriminant [math]\displaystyle{ |\tau| \ge 1 }[/math] and [math]\displaystyle{ -\tfrac{1}{2} \lt \mathfrak{R}(\tau) \le \tfrac{1}{2} }[/math], where [math]\displaystyle{ -\tfrac{1}{2} \lt \mathfrak{R}(\tau) \lt 0 \Rightarrow |\tau| \gt 1. }[/math]

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

[math]\displaystyle{ \Gamma (\tfrac {1}{2})=\sqrt {\pi}; }[/math]

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

[math]\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 { } }[/math]

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

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

[math]\displaystyle{ \mathcal{H} := \{x + iy \mid y \gt 0;\ x, y \in \mathbb{R} \}. }[/math]

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

For [math]\displaystyle{ n }[/math] equal to [math]\displaystyle{ 1 }[/math], Bernouilli numbers [math]\displaystyle{ B_{n} }[/math] hold a value of [math]\displaystyle{ \pm \tfrac {1}{2} }[/math]. In the Riemann hypothesis, every nontrivial complex root of the Riemann zeta function has a real part equal to [math]\displaystyle{ \tfrac {1}{2} }[/math].

Computer characters

vulgar fraction one half
In Unicode	U+00BD

One-half has its own code point in some early extensions of ASCII at 171 (AB₁₆). In Unicode, it has its own code unit at U+00BD (decimal 189) in the C1 Controls and Latin-1 Supplement block and a cross-reference in the Number Forms block, rendering as ½. The HTML entity is ½, and its PC entry is +. The single-precision floating-point for ½ is 3F000000₁₆.

In typewriters, one half is one of the few fractions to usually have a key of its own (see fractions).

See also

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

• Division by two