Complex Multiplication Identity is One

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Theorem

Let $\C$ be the set of complex numbers.

The identity element of $\left({\C^*, \times}\right)$ is the complex number $1 + 0 i$.

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

and similarly:

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

$\blacksquare$

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

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