# 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
$z^\ast = x - iy$

### Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

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