A mixed fraction representing a positive number expresses a nonintegral improper fraction 
in the form 
, where 
is a positive integer
and 
is a proper fraction, with 
and 
positive integers satisfying
. If 
is an integer, the fractional
part is zero and no mixed notation is needed. In common
usage such as cooking recipes, 
is often written as 
(e.g., 1 
),
much to the chagrin of mathematicians, to whom 
means 
,
which is quite a different beast from 
.
For a negative mixed fraction, the minus sign applies to both the whole and fractional parts. Thus 
denotes 
,
not 
.
(The author of this work discovered this fact early in his mathematical career after having points marked off a calculus exam for using the recipe-like notation. Future mathematicians are therefore encouraged to avoid mixed fractions, except perhaps in the kitchen.)
