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 $\left({\C_{\ne 0}, \times}\right)$ is the complex number $1 + 0 i$.


Proof

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


and similarly:

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

$\blacksquare$


Sources