halvtaligt

Halvtaligt is a Danish mathematical term describing half-integers—numbers that lie halfway between integers. In English, the same concept is usually called half-integers. These are the numbers of the form n + 1/2, where n is an integer.

The word hallmarks its etymology: halvtal means half-number, and the suffix -igt is used to form adjectives. Thus halvtaligt literally means “half-numbered” or “half-integer.”

As a set, halvtaligt numbers form an additive coset of the integers. They can be listed as ..., -3/2, -1/2, 1/2, 3/2, 5/2, .... Each element has fractional part 1/2, and the set is an arithmetic progression with common difference 1. Halvtaligt numbers are a subset of rational numbers and, more generally, of real numbers.

In geometry and combinatorics, halvtaligt coordinates arise naturally. For example, the centers of unit squares on the integer lattice Z^2 have coordinates (k + 1/2, l + 1/2), which are halvtaligt in each coordinate. In modular arithmetic, halvtaligt numbers are precisely those congruent to 1/2 modulo 1.

Regarding arithmetic, the sum of two halvtaligt numbers is always an integer: (n + 1/2) + (m + 1/2) = n + m + 1. The product of two halvtaligt numbers is generally not halvtaligt, since (n + 1/2)(m + 1/2) yields a fractional part that is typically 1/4 or 3/4.

Halvtaligt is primarily used in Danish-language mathematical texts; in English, the standard term is half-integer. See also integers, rational numbers, lattice, and modular arithmetic.