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.^[2][3]

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.

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