A calculus of the absurd

4.3.3 Difference of two squares
  • Theorem 4.3.1 Let \(x, y \in \mathbb {R}\), in which case

    \begin{equation} \Big (x + y\Big )\Big (x + (-y)\Big ) \equiv x^2 - y^2 \end{equation}

    I have also seen this referred to as the “third binomial law”, but I believe this terminology is non-standard in English-speaking countries.

Proof: we use the standard method for proving identities (see Section 5.2).

\begin{align} \Big (x + y\Big )\Big (x + (-y)\Big ) &= (x+y) \times (x) + (x+y) \times (-y) \\ &= x \times x + y \times x + -(y \times x) - (y \times y) \\ &= x^2 - y^2 \end{align}

This shows up often, and if you don’t spot it (as I have done a few times) it often makes life very painful.