Editing Half-integer (section)

==Notation and algebraic structure==
The [[Set (mathematics)|set]] of all half-integers is often denoted 
<math display=block>\mathbb Z + \tfrac{1}{2} \quad = \quad \left( \tfrac{1}{2} \mathbb Z \right) \smallsetminus \mathbb Z ~.</math>
The integers and half-integers together form a [[group (mathematics)|group]] under the addition operation, which may be denoted<ref>{{cite book |first=Vladimir G. |last=Turaev |year=2010 |title=Quantum Invariants of Knots and 3-Manifolds |edition=2nd |series=De Gruyter Studies in Mathematics |volume=18 |publisher=Walter de Gruyter |isbn=9783110221848 |page=390}}</ref>
<math display=block>\tfrac{1}{2} \mathbb Z ~.</math>
However, these numbers do not form a [[ring (mathematics)|ring]] because the product of two half-integers is not a half-integer; e.g. <math>~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~.</math><ref>{{cite book |first1=George |last1=Boolos |first2=John P. |last2=Burgess |first3=Richard C. |last3=Jeffrey |year=2002 |title=Computability and Logic |page=105 |publisher=Cambridge University Press |isbn=9780521007580 |url=https://books.google.com/books?id=0LpsXQV2kXAC&pg=PA105}}</ref> The [[subring|smallest ring containing]] them is <math>\Z\left[\tfrac12\right]</math>, the ring of [[dyadic rational]]s.