What is Complex numbers?

The set of complex numbers is

where $\iu$ is a symbol subject to the single rule $\iu^2 = -1$; addition and multiplication are performed as for real polynomials in $\iu$, reducing $\iu^2$ to $-1$. The reals $a$ and $b$ are the real part $\Rea(z)$ and imaginary part $\Ima(z)$ of $z = a + \iu b$; this expression is the algebraic form of $z$, and it is unique: $a + \iu b = a' + \iu b'$ if and only if $a = a'$ and $b = b'$.