Difference of Two Squares - Proof

Proof

The proof is straightforward. Starting from the right-hand side, apply the distributive law to get

,

and set

as an application of the commutative law. The resulting identity is one of the most commonly used in mathematics.

The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring R.

Conversely, if this identity holds in a ring R for all pairs of elements a and b of the ring, then R is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get

and for this to be equal to, we must have

for all pairs a, b of elements of R, so the ring R is commutative.