Complex Numbers

Basics:

• real part + imaginary part
• fundamental to quantum computing

1) Cartesian form

$z=x+iy$

where $x,y\in \mathbb{R}$ and $i$ is the imaginary unit with property $i^2=-1$.

This is known as the cartesian form because $z$ corresponds to a point or a vector in the two dimensional coordinate system known as the cartesian plane.

In the complex plane, $z=x+iy=e^{i\theta }$, with $x$ along the real axis ($\mathrm{Re}$), $y$ along the imaginary axis, $r$ the length of the vector and $\theta$ its angle.

The number sets nest as $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{R}\subset \mathbb{C}$.

Relationships

• Real part $\Rightarrow x=\mathrm{Re}(z)$
• Imaginary part $\Rightarrow y=\mathrm{Im}(z)$
• Conjugate $=z^{\ast }=x-iy$
• mod (abs) $\Rightarrow r=|z|$
• Imaginary unit $\Rightarrow i=\sqrt{-1}$ or $i^2=1$
• $\tan (\theta )=\frac{y}{x}$, $\theta =\arctan (y,x)$
• Polar to cartesian $\Rightarrow x=r\cos \theta$ and $y=r\sin \theta$
• Polar form $\Rightarrow z=r(\cos \theta +i\sin \theta )$

E.g.

$z=1+\sqrt{3}i$

$z^{\ast }=1-\sqrt{3}i$

$|z|=\sqrt{1^2+(\sqrt{3}{)}^2}=2$

$\theta =\arctan \left(\frac{\sqrt{3}}{1}\right)=\frac{\pi }{3}$