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) .
Irreducible fraction
Natural number
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
, etc.
As a word
One half is one of the few fractions which are commonly expressed in natural
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
nil
0
{\displaystyle 0}
and
unity
1
{\displaystyle 1}
(which are the elementary
,
1
2
{\displaystyle {\tfrac {1}{2}}}
. It has two different
base ten
, the familiar
0.5
{\displaystyle 0.5}
and the
recurring
0.4
9
¯
{\displaystyle 0.4{\overline {9}}}
, with a similar pair of expansions in any even
representation, it has only a single representation with a repeating fractional component (such as
0.
1
¯
{\displaystyle 0.{\overline {1}}}
in
ternary and
0.
2
¯
{\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
n
{\displaystyle n}
raised to the power of one half is equal to the square root of
n
{\displaystyle n}
,
n
1
2
=
n
.
{\displaystyle n^{\tfrac {1}{2}}={\sqrt {n}}.}
Properties
A
abundancy index
:
σ
(
n
)
n
=
k
2
,
{\displaystyle {\frac {\sigma (n)}{n}}={\frac {k}{2}},}
where
k
{\displaystyle k}
is odd , and
σ
(
n
)
{\displaystyle \sigma (n)}
is the
sum-of-divisors function. The first three hemiperfect numbers are
2 ,
24 , and 4320.
[1]
The area
T
{\displaystyle T}
of a triangle with base
b
{\displaystyle b}
and altitude
h
{\displaystyle h}
is computed as,
T
=
b
2
×
h
.
{\displaystyle T={\frac {b}{2}}\times h.}
Ed Pegg Jr. noted that the length
d
{\displaystyle d}
equal to
1
2
1
30
(
61421
−
23
5831385
)
{\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
n
{\displaystyle n}
-th
triangular number :
P
2
(
n
)
=
n
(
n
+
1
)
2
;
{\displaystyle P_{2}(n)={\frac {n(n+1)}{2}};}
and in the formula for computing magic constants for magic squares ,
M
2
(
n
)
=
n
2
(
n
2
+
1
)
.
{\displaystyle M_{2}(n)={\frac {n}{2}}\left(n^{2}+1\right).}
Successive natural numbers yield the
n
{\displaystyle n}
-th metallic mean
M
{\displaystyle M}
by the equation,
M
(
n
)
=
n
+
n
2
+
4
2
.
{\displaystyle M_{(n)}={\frac {n+{\sqrt {n^{2}+4}}}{2}}.}
In the study of finite groups , alternating groups have order
n
!
2
.
{\displaystyle {\frac {n!}{2}}.}
By
π
2
=
∑
n
=
1
∞
(
−
1
)
ε
(
n
)
n
=
1
+
1
2
−
1
3
+
1
4
+
1
5
−
1
6
−
1
7
+
⋯
,
{\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
ε
(
n
)
{\displaystyle \varepsilon (n)}
is the number of
prime factors
of the form
p
≡
3
(
m
o
d
4
)
{\displaystyle p\equiv 3\,(\mathrm {mod} \,4)}
of
n
{\displaystyle n}
(see
modular arithmetic ).
modular discriminant
|
τ
|
≥
1
{\displaystyle |\tau |\geq 1}
and
−
1
2
<
R
(
τ
)
≤
1
2
{\displaystyle -{\tfrac {1}{2}}<{\mathfrak {R}}(\tau )\leq {\tfrac {1}{2}}}
, where
−
1
2
<
R
(
τ
)
<
0
⇒
|
τ
|
>
1.
{\displaystyle -{\tfrac {1}{2}}<{\mathfrak {R}}(\tau )<0\Rightarrow |\tau |>1.}
For the gamma function , a non-integer argument of one half yields,
Γ
(
1
2
)
=
π
;
{\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]
ζ
(
3
)
=
−
1
2
Γ
‴
(
1
)
+
3
2
Γ
′
(
1
)
Γ
″
(
1
)
−
(
Γ
′
(
1
)
)
3
=
−
1
2
ψ
(
2
)
(
1
)
;
{\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
ψ
(
m
)
(
z
)
{\displaystyle \psi ^{(m)}(z)}
the polygamma function of order
m
{\displaystyle m}
on the complex numbers
C
{\displaystyle \mathbb {C} }
.
The upper half-plane
H
{\displaystyle {\mathcal {H}}}
is the set of points
(
x
,
y
)
{\displaystyle (x,y)}
in the
Cartesian plane
with
y
>
0
{\displaystyle y>0}
. In the context of complex numbers, the upper half-plane is defined as
H
:=
{
x
+
i
y
∣
y
>
0
;
x
,
y
∈
R
}
.
{\displaystyle {\mathcal {H}}:=\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}.}
In
.
For
n
{\displaystyle n}
equal to
1
{\displaystyle 1}
,
Bernouilli numbers
B
n
{\displaystyle B_{n}}
hold a value of
±
1
2
{\displaystyle \pm {\tfrac {1}{2}}}
. In the
Riemann hypothesis , every nontrivial
complex root of the
Riemann zeta function has a real part equal to
1
2
{\displaystyle {\tfrac {1}{2}}}
.
Computer characters
½ In Unicode U+00BD Different from ¼, ¾
One-half has its own
C1 Controls and Latin-1 Supplement block and a cross-reference in the
Number Forms block, rendering as
½ .
[7] The
HTML entity is
½
,
[8] and its
PC entry is
Alt +
0 1 8 9 .
[9] The single-precision
floating-point for ½ is 3F000000
16 .
In
fractions
).
See also
Postal stamp, Ireland, 1940: one halfpenny postage due.
References
Division and ratio Fraction
Numerator / Denominator = Quotient