⧉ Complex numbers §

= an extension to real numbers

Notations §

Cartesian coordinate notation §

$z=a+bi$

• $z$: the complex number $=Re(z)+Im(z)i$
• $a$: the real part $=Re(z)$
• $b$: the imaginary part $=Im(z)$
• $i$: imaginary unit, spanning the second dimension of the complex number space
  • special meaning: $i^2=-1$

Polar coordinate notation §

$z=A(cos\varphi +i\sin \varphi )$

• $a=A\cos \varphi$
• $b=A\sin \varphi$
• with:
  • $A$: length of the vector
  • $\varphi$: angle relative to the real axis

Properties §

Euler’s formula §

The following relation holds:

• $exp(i\varphi )=e^{i\varphi }=cos(\varphi )+i\sin (\varphi )$
• → complex sum of sine and cosine can be expressed using a single exponential function
• → $z=A{e}^{i\varphi }$