Halfinteger

A halfinteger, also called a half-integer, is a real number of the form n + 1/2 where n is an integer. Equivalently, it is any number with fractional part 0.5. The set of halfintegers can be written as Z + 1/2, and they lie halfway between consecutive integers on the number line.

Key properties. Halfintegers form an arithmetic progression with common difference 1. They obey: the sum of two halfintegers is an integer, and the difference of two halfintegers is an integer. Adding a halfinteger to an integer yields another halfinteger, while products do not, in general, stay within the halfinteger set. In modular terms, halfintegers form the coset 1/2 + Z, meaning their fractional part is always 1/2.

Examples. Typical halfintegers include -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, and so on. They are precisely the numbers of the form (2k+1)/2 for integers k.

Relation to other concepts. Halfintegers are the integers shifted by one half. They arise where positions are defined on a grid with half-step offsets, such as certain lattice or combinatorial problems. They also connect to the idea of fractional parts and to the broader study of arithmetic progressions and cosets in real numbers.

In science, the term half-integer is used in contexts such as half-integer spin in quantum mechanics, though this usage refers to discrete quantum values rather than the numerical property alone. See also fractional part, Z, and lattice-based problems.