Complex & Imaginary Numbers

Imaginary numbers

$i^2 = -1$

$\therefore i = \sqrt {-1}$

Simplifying negative surds

$\sqrt{-2} = \sqrt{-1 \times 2}$

          $= \sqrt{2}i$

Complex numbers

$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$

General form: $z=a+bi$

• $\operatorname{Re}(z) = a$
• $\operatorname{Im}(z) = b$

Addition

If $z_1 = a+bi$ and $z_2=c+di$, then

            $z_1+z_2 = (a+c)+(b+d)i$

Subtraction

If $z_1=a+bi$ and $z_2=c+di$, then

           $z_1−z_2=(a−c)+(b−d)i$

Multiplication by a real constant

If $z=a+bi$ and $k \in \mathbb{R}$, then

           $kz=ka+kbi$

Powers of $i$

$i^0=1$
$i^1=i$
$i^2=-1$
$i^3=-i$
$i^4=1$
$\dots$

Therefore…

• $i^{4n} = 1$
• $i^{4n+1} = i$
• $i^{4n+2} = -1$
• $i^{4n+3} = -i$

Multiplying complex expressions

If $z_1 = a+bi$ and $z_2=c+di$, then
           $z_1 \times z_2 = (ac-bd)+(ad+bc)i$

Conjugates

If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)

Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2$

• Multiplication and addition are associative

Modulus

Distance from origin.
$|{z}|=\sqrt{a^2+b^2}$

$\therefore z \overline{z} = |z|^2$

Multiplicative inverse

$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$

Dividing complex numbers

${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$

(using multiplicative inverse)

In practice, rationalise denominator:
${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$

Argand planes

• Geometric representation of $\mathbb{C}$
• Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
• Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$

Solving complex quadratics

To solve $z^2+a^2=0$ (sum of two squares):

$z^2+a^2=z^2-(ai)^2$
              $=(z+ai)(z-ai)$

Polar form

General form:
$z=r \operatorname{cis} \theta$
$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$

where

• $z=a+bi$
• $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
• $\theta$ is the argument of $z$, CCW from origin

Note each complex number has multiple polar representations:
$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions

Multiplication and division in polar form

$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)

${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)

de Moivres’ Theorum

$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$