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   1# Complex & Imaginary Numbers
   2
   3## Imaginary numbers
   4
   5$i^2 = -1$
   6
   7$\therefore i = \sqrt {-1}$
   8
   9### Simplifying negative surds
  10
  11$\sqrt{-2} = \sqrt{-1 \times 2}$  
  12$= \sqrt{2}i$
  13
  14
  15## Complex numbers
  16
  17$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
  18
  19General form: $z=a+bi$
  20- $\operatorname{Re}(z) = a$
  21- $\operatorname{Im}(z) = b$
  22
  23### Addition
  24
  25If $z_1 = a+bi$ and $z_2=c+di$, then  
  26$z_1+z_2 = (a+c)+(b+d)i$
  27
  28### Subtraction
  29
  30If $z_1=a+bi$ and $z_2=c+di$, then $z_1−z_2=(a−c)+(b−d)i$
  31
  32### Multiplication by a real constant
  33
  34If $z=a+bi$ and $k \in \mathbb{R}$, then $kz=ka+kbi$
  35
  36### Powers of $i$
  37$i^0=1$
  38$i^1=i$
  39$i^2=-1$
  40$i^3=-i$
  41$i^4=1$
  42$\dots$
  43
  44Therefore..
  45
  46- $i^{4n} = 1$
  47- $i^{4n+1} = i$
  48- $i^{4n+2} = -1$
  49- $i^{4n+3} = -i$
  50
  51Divide by 4 and take remainder.
  52
  53### Multiplying complex expressions
  54
  55If $z_1 = a+bi$ and $z_2=c+di$, then  
  56$z_1 \times z_2 = (ac-bd)+(ad+bc)i$
  57
  58### Conjugates
  59
  60If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
  61
  62Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2$
  63
  64- Multiplication and addition are associative
  65
  66### Modulus
  67
  68Distance from origin.
  69$|{z}|=\sqrt{a^2+b^2}$
  70
  71$\therefore z \overline{z} = |z|^2$
  72
  73### Multiplicative inverse
  74
  75$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
  76
  77### Dividing complex numbers
  78
  79${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$
  80
  81(using multiplicative inverse)
  82
  83In practice, rationalise denominator:
  84${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
  85
  86## Argand planes
  87
  88- Geometric representation of $\mathbb{C}$
  89- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
  90- Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
  91
  92## Solving complex quadratics
  93
  94To solve $z^2+a^2=0$ (sum of two squares):
  95
  96$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
  97
  98## Polar form
  99
 100General form:
 101$z=r \operatorname{cis} \theta$
 102$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$
 103
 104where
 105- $z=a+bi$
 106- $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
 107- $\theta$ is the argument of $z$, CCW from origin
 108
 109Note each complex number has multiple polar representations:
 110$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
 111
 112### Multiplication and division in polar form
 113
 114$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
 115
 116${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
 117
 118## de Moivres' Theorum
 119
 120$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$