The Complex Numbers

Let $\mathbb{R}$ be the set of real numbers, with its addition and multiplication.

The complex numbers are a field $\mathbb{C}$, whose addition and multiplication are written $+$ and $\cdot$ (the product $z\cdot w$ being also written $zw$), together with a distinguished element $i\in\mathbb{C}$, subject to the following three conditions.

1. (The real numbers are a subfield.) $\mathbb{R}\subseteq\mathbb{C}$, and for all $a,b\in\mathbb{R}$ the sum $a+b$ and the product $ab$ formed in $\mathbb{C}$ coincide with the sum and product of $a$ and $b$ formed in $\mathbb{R}$.

2. (Imaginary unit.) $i\cdot i=-1$, where $1$ denotes the multiplicative identity of $\mathbb{C}$ and $-1$ its additive inverse in $\mathbb{C}$.

3. (Generation.) For every $z\in\mathbb{C}$ there exist $a,b\in\mathbb{R}$ with $z=a+b\cdot i$.

The element $i$ is called the imaginary unit. A structure satisfying conditions 1-3 exists, and any two such structures are related by a unique isomorphism that fixes every real number and carries one imaginary unit to the other, by Existence and Uniqueness of the Complex Numbers; the symbol $\mathbb{C}$ denotes a fixed such field and $i$ its distinguished imaginary unit.