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
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
0
{\displaystyle 0}
and
unity
1
{\displaystyle 1}
(which are the elementary
additive and
multiplicative
identities ) as the
quotient of the first two non-zero
integers ,
1
2
{\displaystyle {\tfrac {1}{2}}}
. It has two different
decimal representations in
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
base ; while in odd bases, one half has no
terminating 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
hemiperfect number is a positive
integer with a half-integer
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
Euler , a classical
formula involving pi , and yielding a simple expression:
[4]
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 ).
Fundamental region of the modular
j-invariant in the
upper half-plane (shaded gray ), with
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
differential geometry , this is the
universal covering space of surfaces with constant negative
Gaussian curvature , by the
uniformization theorem .
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
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.
References
^
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 .
^
Ed Pegg Jr. (July 2000).
"Commentary on weekly puzzles" . Mathpuzzle . Retrieved 2023-08-17 .
^
Weisstein, Eric W.
"Almost integer" .
MathWorld -- A
WolframAlpha Resource . Retrieved 2023-08-17 .
^
Euler, Leonhard (1748).
Introductio in analysin infinitorum (in Latin). Vol. 1. apud Marcum-Michaelem Bousquet & socios. p. 244.
^ 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).
^ Bloch, Spencer; Masha, Vlasenko.
"Gamma functions, monodromy and Apéry constants" (PDF) . University of Chicago (Paper). pp. 1–34.
S2CID
126076513 .
^
"Latin-1 Supplement" . SYMBL . Retrieved 2023-07-18 .
Division and ratio Fraction
Numerator / Denominator = Quotient