Canonical Form and Arithmetic of Complex Numbers

Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$, and let $\mathbb{R}$ be the set of real numbers. Then the following hold.

1. (Identities and inverses of real numbers) The additive identity of $\mathbb{C}$ is the real number $0$ and the multiplicative identity of $\mathbb{C}$ is the real number $1$. For every $a\in\mathbb{R}$, the additive inverse of $a$ in $\mathbb{C}$ is the real number $-a$; and for every $a\in\mathbb{R}$ with $a\neq0$, the multiplicative inverse of $a$ in $\mathbb{C}$ is the real number $1/a$.

2. (The imaginary unit is not real) $i\notin\mathbb{R}$.

3. (Canonical form) For every $z\in\mathbb{C}$ there is exactly one pair of real numbers $a,b$ with $z=a+bi$.

4. (Arithmetic in canonical form) For all real numbers $a,b,c,d$,

where the sums, differences and products inside the parentheses on the right-hand sides are those of $\mathbb{R}$.