# 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

The magnitude, \(|z|\), and argument, \(\arg(z)\), of the complex number \(z\) are given by

\(x\) 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
\[
z^\ast = x - iy
\]