One half

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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

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
and unity (which are the elementary
multiplicative identities) as the quotient of the first two non-zero integers
, . It has two different
base ten
, the familiar and the
recurring
, 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 in ternary and 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 raised to the power of one half is equal to the square root of ,

Properties

A

abundancy index
:

where is odd, and is the

sum-of-divisors function. The first three hemiperfect numbers are 2, 24, and 4320.[1]

The area of a triangle with base and altitude is computed as,

Ed Pegg Jr. noted that the length equal to is almost an integer, approximately 7.0000000857.[2][3]

One half figures in the formula for calculating

figurate numbers
, such as the -th triangular number:

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

Successive natural numbers yield the -th metallic mean by the equation,

In the study of finite groups, alternating groups have order

By

Euler, a classical formula involving pi, and yielding a simple expression:[4]

where is the number of

prime factors
of the form of (see
modular arithmetic).

modular discriminant
and , where

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

while inside Apéry's constant, which represents the sum of the reciprocals of all positive cubes, there is[5][6]

with the polygamma function of order on the complex numbers .

The upper half-plane is the set of points in the

Cartesian plane
with . In the context of complex numbers, the upper half-plane is defined as

In

universal covering space of surfaces with constant negative Gaussian curvature, by the uniformization theorem
.

For equal to ,

Bernouilli numbers
hold a value of . In the
Riemann hypothesis, every nontrivial complex root of the Riemann zeta function has a real part equal to .

Computer characters

½
vulgar fraction one half
In UnicodeU+00BD
Different from
Different from¼, 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+0189.[9] The single-precision floating-point
for ½ is 3F00000016.

In

fractions
).

See also

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

References

  1. ^ 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.
  2. ^ Ed Pegg Jr. (July 2000). "Commentary on weekly puzzles". Mathpuzzle. Retrieved 2023-08-17.
  3. ^ Weisstein, Eric W. "Almost integer". MathWorld -- A WolframAlpha Resource. Retrieved 2023-08-17.
  4. Euler, Leonhard (1748). Introductio in analysin infinitorum
    (in Latin). Vol. 1. apud Marcum-Michaelem Bousquet & socios. p. 244.
  5. ^ 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).
  6. S2CID 126076513
    .
  7. ^ "Latin-1 Supplement". SYMBL. Retrieved 2023-07-18.
  8. ^ "HTML Character Entity References". SYMBL. Retrieved 2023-07-18.
  9. ^ "Alt Codes". Alt-Codes. Retrieved 2023-07-18.
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