Complex Multiplication Identity is One

Theorem

Let $\C_{\ne 0}$ be the set of complex numbers without zero.

The identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.

Proof

 $\displaystyle \paren {x + i y} \paren {1 + 0 i}$ $=$ $\displaystyle \paren {x \cdot 1 - y \cdot 0} + i \paren {x \cdot 0 + y \cdot 1}$ $\displaystyle$ $=$ $\displaystyle \paren {x + i y}$

and similarly:

 $\displaystyle \paren {1 + 0 i} \paren {x + i y}$ $=$ $\displaystyle \paren {1 \cdot x - 0 \cdot y} + i \paren {0 \cdot x + 1 \cdot y}$ $\displaystyle$ $=$ $\displaystyle \paren {x + i y}$

$\blacksquare$