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An integral binary quadratic form $[a,b,c]$ with $a>0$ and negative discriminant of a binary quadratic form is reduced whenwith the additional convention $b\ge 0$ whenever $|b|=a$ or $a=c$. These inequalities bound the polynomial coefficients without changing the represented integers.

Every proper equivalence class of positive definite forms has exactly one reduced representative with this boundary convention. For primitive binary quadratic forms, this is the classical reduction theorem; the general case follows by dividing by the common content, which is invariant under proper equivalence.^[ref]

The sign rule removes actual duplicates. A shear takes $[a,a,c]$ to $[a,-a,c]$, and the determinant-one rotation $(x,y)\mapsto (-y,x)$ takes $[a,b,a]$ to $[a,-b,a]$. Keeping the nonnegative middle polynomial coefficient chooses one boundary copy. For example, $[1,2,2]$ is not reduced; the reduction algorithm for a positive definite binary quadratic form takes it back to $[1,0,1]$.

Seth Viren Neel authored notes on binary quadratic forms and the ideal class group. The notes develop equivalence, reduction and genus theory, showing how bounded coefficients permit finite enumeration of reduced positive definite forms. This gives a concrete computational view of class numbers while retaining the distinction between a form and its equivalence class.^[ref]