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   1# Complex & Imaginary Numbers
   2
   3## Imaginary numbers
   4
   5$i^2 = -1 \quad \therefore i = \sqrt {-1}$
   6
   7### Simplifying negative surds
   8
   9$\sqrt{-2} = \sqrt{-1 \times 2}$  
  10$= \sqrt{2}i$
  11
  12
  13## Complex numbers
  14
  15$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
  16
  17General form: $z=a+bi$  
  18$\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$
  19
  20### Addition
  21
  22If $z_1 = a+bi$ and $z_2=c+di$, then  
  23$z_1+z_2 = (a+c)+(b+d)i$
  24
  25### Subtraction
  26
  27If $z_1=a+bi$ and $z_2=c+di$, then $z_1−z_2=(a−c)+(b−d)i$
  28
  29### Multiplication by a real constant
  30
  31If $z=a+bi$ and $k \in \mathbb{R}$, then $kz=ka+kbi$
  32
  33### Powers of $i$
  34$i^0=1$
  35$i^1=i$
  36$i^2=-1$
  37$i^3=-i$
  38$i^4=1$
  39$\dots$
  40
  41Therefore..
  42
  43- $i^{4n} = 1$
  44- $i^{4n+1} = i$
  45- $i^{4n+2} = -1$
  46- $i^{4n+3} = -i$
  47
  48Divide by 4 and take remainder.
  49
  50### Multiplying complex expressions
  51
  52If $z_1 = a+bi$ and $z_2=c+di$, then  
  53$z_1 \times z_2 = (ac-bd)+(ad+bc)i$
  54
  55### Conjugates
  56
  57If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
  58
  59Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2 = |z|^2$
  60
  61- Multiplication and addition are associative
  62
  63#### Properties
  64
  65- $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
  66- $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
  67- $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$
  68- $z \overline{z} = |z|^2$
  69- $z + \overline{z} = 2 \operatorname{Re}(z)$
  70
  71
  72### Modulus
  73
  74Distance from origin.
  75$|{z}|=\sqrt{a^2+b^2}$
  76
  77$\therefore z \overline{z} = |z|^2$
  78
  79#### Properties
  80
  81- $|z_1 z_2| = |z_1| |z_2|$
  82- $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$
  83- $|z_1 + z_2| \le |z_1 + |z_2|$
  84
  85### Multiplicative inverse
  86
  87$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
  88
  89### Dividing complex numbers
  90
  91${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$
  92
  93(using multiplicative inverse)
  94
  95In practice, rationalise denominator:
  96${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
  97
  98## Argand planes
  99
 100- Geometric representation of $\mathbb{C}$
 101- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
 102- Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
 103
 104## Solving complex quadratics
 105
 106To solve $z^2+a^2=0$ (sum of two squares):
 107
 108$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
 109
 110## Polar form
 111
 112General form:
 113$z=r \operatorname{cis} \theta$
 114$= r(\operatorname{cos}\theta+i \operatorname{sin}\theta)$
 115
 116$z=a+bi$  
 117$z=r\operatorname{cis}\theta$  
 118
 119
 120- $z=a+bi$
 121- $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
 122- $\theta$ is the argument of $z$, CCW from origin
 123
 124Note each complex number has multiple polar representations:
 125$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
 126
 127### Conjugate in polar form
 128
 129$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
 130
 131### Multiplication and division in polar form
 132
 133$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
 134
 135${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
 136
 137## de Moivres' Theorem
 138
 139$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$