ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4261 on D993: Real number
Subresult of R2788: Real binomial theorem

Real binomial theorem for exponent two

Formulation 0

Let $a, b \in \mathbb{R}$ each be a D993: Real number.

Then \begin{equation} (a + b)^2 = a^2 + 2 a b + b^2 \end{equation}

Proofs

Proof 0

Let $a, b \in \mathbb{R}$ each be a D993: Real number.

By direct computation \begin{equation} \begin{split} a^2 + 2 a b + b^2 & = a a + 2 a b + b b \\ & = a a + a b + a b + b b \\ & = a a + a b + b a + b b \\ & = a (a + b) + b (a + b) \\ & = (a + b) (a + b) \\ & = (a + b)^2 \end{split} \end{equation} $\square$

Proof 1

Let $a, b \in \mathbb{R}$ each be a D993: Real number.

This result is a particular case of R2788: Real binomial theorem. $\square$