Complex Multiplication Identity is One

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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

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


and similarly:

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

$\blacksquare$


Sources