A reduced binary quadratic form is a binary quadratic form 
with integer coefficients
satisfying one of the following reduction conditions, using the binary
quadratic form discriminant convention 
.
1. If 
and 
,
then 
is positive definite, also called
imaginary, and is reduced iff 
and 
whenever 
or 
.
2. If 
,
then 
is indefinite, also called real. For
nonsquare 
,
it is reduced iff 
.
Under the action of the special linear group 
,
every positive definite binary
quadratic form with integer coefficients
is equivalent to a unique reduced form. Every indefinite binary quadratic form with integer coefficients whose binary
quadratic form discriminant is positive and nonsquare is equivalent to a finite,
nonempty set of reduced forms.
