Complex & Imaginary Numbers

Imaginary numbers

i² = −1

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

Simplifying negative surds

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

          $= \sqrt{2}i$

Complex numbers

ℂ = {a + bi : a, b ∈ ℝ}

General form: z = a + bi - Re(z)=a - Im(z)=b

Addition

If z₁ = a + bi and z₂ = c + di, then

            z₁ + z₂ = (a + c)+(b + d)i

Subtraction

If z₁ = a + bi and z₂ = c + di, then

           z₁ − z₂ = (a − c)+(b − d)i

Multiplication by a real constant

If z = a + bi and k ∈ ℝ, then

           kz = ka + kbi

Powers of i

i⁰ = 1 i¹ = i i² = −1 i³ = −i i⁴ = 1 …

Therefore.. - i⁴ⁿ = 1 - i^4n + 1 = i - i^4n + 2 = −1 - i^4n + 3 = −i

Multiplying complex expressions

If z₁ = a + bi and z₂ = c + di, then            z₁ × z₂ = (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 ℂ
• Horizontal =Re(z); vertical =Im(z)
• Multiplication by i results in an anticlockwise rotation of $\pi \over 2$

Solving complex quadratics

To solve z² + a² = 0 (sum of two squares):

z² + a² = z² − (ai)²               =(z + ai)(z − ai)

Polar form

General form: z = rcisθ =rcosθ + rsinθi

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

Note each complex number has multiple polar representations: z = rcisθ = rcis(θ + 2nπ) where n is integer number of revolutions

Multiplication and division in polar form

z₁z₂ = r₁r₂cis(θ₁ + θ₂) (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

(rcisθ)ⁿ = rⁿcis(nθ)