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

Result R5673 on D1207: Complex number

Complex arithmetic expression for the multiplicative inverse of a complex number

Formulation 0

Let $z = (x, y) \in \mathbb{C}$ be a D1207: Complex number such that

(i)	\begin{equation} z \neq (0, 0) \end{equation}
(ii)	\begin{equation} w : = \frac{(x, -y)}{x^2 + y^2} \end{equation}

Then \begin{equation} z w = w z = 1 \end{equation}

Formulation 1

Let $z = x + i y \in \mathbb{C}$ be a D1207: Complex number such that

(i)	\begin{equation} z \neq 0 \end{equation}
(ii)	\begin{equation} w : = \frac{x - i y}{x^2 + y^2} \end{equation}

Then \begin{equation} z w = w z = 1 \end{equation}

Proofs

Proof 0

Let $z = x + i y \in \mathbb{C}$ be a D1207: Complex number such that

(i)	\begin{equation} z \neq 0 \end{equation}
(ii)	\begin{equation} w : = \frac{x - i y}{x^2 + y^2} \end{equation}

We have \begin{equation} z w = \frac{(x + i y) (x - i y)}{x^2 + y^2} = \frac{(x^2 - i x y + i x y - i^2 y^2)}{x^2 + y^2} = \frac{x^2 + y^2}{x^2 + y^2} = 1 \end{equation} and an equivalent result holds for $z w$. $\square$