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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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 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
[ edit ]
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
[ edit ]
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
[ edit ]
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 .
The Bernoulli number
B
1
{\displaystyle B_{1}}
has the value
±
1
2
{\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
1
2
{\displaystyle {\tfrac {1}{2}}}
.
Computer characters
[ edit ]
The "one-half" symbol has its own code point as a precomposed character in the Number Forms block of Unicode , rendering as ½ .
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
[ edit ]
Postal stamp, Ireland, 1940: one halfpenny postage due.
References
[ edit ]
^ 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 .
t
e
Division and ratio
Fraction
Numerator / Denominator = Quotient
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=One_half&oldid=1234787825 "
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