Complex Number

A number of the form \(z = x + i y\) where \(i = \sqrt{-1}\).
\(x\) is known as the real part of \(z\), or \(\Re(z)\), and \(y\) as the imaginary part, or \(\Im(z)\).
To add a pair of complex numbers \(z_0 = x_0 + i y_0\) and \(z_1 = x_1 + i y_1\) the real and imaginary parts of each are added separately to yield
\[ z_0 + z_1 = (x_0 + x_1) + i (y_0 + y_1) \]
To multiply them we first calculate the products of each of the four pairs of terms and then add them
\[ \begin{align*} z_0 \times z_1 &= (x_0 \times x_1) + (x_0 \times i y_1) + (x_1 \times i y_0) + (i y_0 \times i y_1)\\ &= x_0 x_1 + i x_0 y_1 + i x_1 y_0 + i^2 y_0 y_1\\ &= (x_0 x_1 - y_0 y_1) + i (x_0 y_1 + x_1 y_0) \end{align*} \]
since \(i^2 = -1\).

The magnitude, \(|z|\), and argument, \(\arg(z)\), of the complex number \(z\) are given by
\[ \begin{align*} |z| &= \sqrt{x^2 + y^2}\\ \arg(z) &= \arctan\left(\frac{y}{x}\right)\\ \end{align*} \]
and can be used to represent it as
\[ z = |z| \cos(\arg(z)) + i |z| \sin(\arg(z)) \]
The conjugate of \(z\), written \(z^\ast\), is the complex number with the negation of its imaginary term
\[ z^\ast = x - iy \]