Half integer

Semi-integer - a number of a number

\dots ,-1{\tfrac {1}{2}},-{\tfrac {1}{2}},{\tfrac {1}{2}},1{\tfrac {1}{2}},2{\tfrac {1}{2}},\dots

{\ displaystyle \ dots, -1 {\ tfrac {1} {2}}, - {\ tfrac {1} {2}}, {\ tfrac {1} {2}}, 1 {\ tfrac {1} { 2}}, 2 {\ tfrac {1} {2}}, \ dots}

{\ displaystyle \ dots, -1 {\ tfrac {1} {2}}, - {\ tfrac {1} {2}}, {\ tfrac {1} {2}}, 1 {\ tfrac {1} { 2}}, 2 {\ tfrac {1} {2}}, \ dots}

That is the number of the form $n+1/2$ ${\ displaystyle n + 1/2}$ $n + 1/2$ where $n$ ${\ displaystyle n}$ $n$ - whole . In other words, it is a rational number with a fractional part. $1/2$ ${\ displaystyle 1/2}$ $1/2$ .

A set of half-integer numbers is usually denoted by $\mathbb {Z} +{\tfrac {1}{2}}$ ${\ displaystyle \ mathbb {Z} + {\ tfrac {1} {2}}}$ $\ mathbb {Z} + {\ tfrac {1} {2}}$ , here $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ $\ mathbb {Z}$ denotes a ring of integers.

Half-integers are used in quantum physics (in particular, the spin values of fermions are half-integers).

Properties

The union of sets of whole and half-integer numbers forms an additive group ${\tfrac {1}{2}}\mathbb {Z}$ ${\ displaystyle {\ tfrac {1} {2}} \ mathbb {Z}}$ , this group is not a ring (since the product of two half-integers in the general case does not give an integer or half-integer number).
Half-integers are a subclass of binary-rational numbers , that is, rational numbers that can be represented as a particular arbitrary integer and two as an integer.
The gamma function of the integer and semi-integer argument can be expressed in terms of elementary functions , for other classes of numbers such representations have not yet been found.

Literature

Malcolm Sabin. Analysis and Design of Univariate Subdivision Schemes // Geometry and Computing. - Springer, 2010. - Vol . 6 . - ISBN 9783642136481 .

Half integer

Properties

Literature

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