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In mathematics, a half-integer is a number of the form $n+\frac{1}{2},$ where $n$ is an integer. For example, $4\frac{1}{2},7/2,-\frac{13}{2},8.5$ are all half-integers. The name "half-integer" is perhaps misleading, as each integer $n$ is itself half of the integer $2n$. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.

Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two).^[1]

The set of all half-integers is often denoted $\mathbb{Z}+\frac{1}{2}=\left(\frac{1}{2}\mathbb{Z}\right)\setminus \mathbb{Z}.$ The integers and half-integers together form a group under the addition operation, which may be denoted^[2] $\frac{1}{2}\mathbb{Z}.$ However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\notin \frac{1}{2}\mathbb{Z}.$^[3] The smallest ring containing them is $\mathbb{Z}\left[\frac{1}{2}\right]$, the ring of dyadic rationals.

• The sum of $n$ half-integers is a half-integer if and only if $n$ is odd. This includes $n=0$ since the empty sum 0 is not half-integer.
• The negative of a half-integer is a half-integer.
• The cardinality of the set of half-integers is equal to that of the integers. This is due to the existence of a bijection from the integers to the half-integers: $f:x\to x+0.5$, where $x$ is an integer.