Half-integer

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a half-integer is a <a href="/facts/Number/auQSAE2m">number</a> of the form

n
        +
        
          
            
              1
              2
            
          
        
        ,
      
    
    {\displaystyle n+{\tfrac {1}{2}},}

where 
  
    
      
        n
      
    
    {\displaystyle n}
  
 is an integer. For example,

4
        
          
            
              1
              2
            
          
        
        ,
        
        7
        
          /
        
        2
        ,
        
        −
        
          
            
              13
              2
            
          
        
        ,
        
        8.5
      
    
    {\displaystyle 4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5}

are all half-integers. The name "half-integer" is perhaps misleading, as each integer 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is itself half of the integer 
 
 
 
 2
 n
 
 
 {\displaystyle 2n}
 
. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> that a distinct term is convenient.
Note that halving an integer does not always produce a half-integer; this is only true for <a href="/facts/Odd_integer/7zq2UM4V">odd integers</a>. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the <a href="/facts/Dyadic_rational/LPwRYKaD">dyadic rationals</a> (numbers produced by dividing an integer by a <a href="/facts/Power_of_two/MA6WEpTt">power of two</a>).

Half-integer open-in-new

Half-integer