Complex Numbers

Question 1

Types of numbers (4)

Answer 1

Real number
Imaginary numbers (i = √-1)
Complex numbers z = a + bi
Conjugate of a complex number z*
z = a - bi

Question 2

Powers of i

Answer 2

i = √-1
i² = -1
i³ = -i
i⁴ = 1

Question 3

To solve cubic etc for complex numbers

Answer 3

Trial and error for first real root
usually x = 1

Question 4

Square root of a complex number

Answer 4

Must also be complex

Square both sides

Equate real to real and imaginary to imaginary from either side of the equation

Solve simultaneously

In exponential or polar form:
√z = √r(cosθ + i sinθ)

square both sides then square root r and half theta values

eg
2θ = π/3 + 2kπ
k = 0 and k = 1

for cubic set k = 2

Question 4

Modulus and argument of line in argand plane

Answer 4

|z|= √x² + (y+i)²
arg = tan⁻¹(b/a)

x can be substitutes for a
y can be substituted for b

Question 5

Equation forms (3)

Answer 5

Rectangular
z = x + iy

Polar form
z = r(cosθ + i sinθ)

Exponential form
z = r cisθ = reᶦ*

(*) θ

Question 6

De Moivre’s theorem

Answer 6

zⁿ = rⁿ (cos nθ + i sin nθ)

Question 7

Circle loci

Answer 7

|z - z₀|= k

(z) variable complex number [a number on the circle
(z₀) fixed complex number [centre of circle|
(k) radius

Question 8

Angle loci

Answer 8

arg(z - z₀) = ∞

where ∞ is the angle

Question 9

Perpendicular bisector loci

Answer 9

|z - z₀|=|z - z₁|

(z) variable complex number
(z₀ & z₁) fixed complex number

Question 10

Standard equation of a circle loci

Answer 10

|z - z₀|= a|z - z₁|

eg.
centre of circle (2, 3)
radius of circle (4)

|z - (-2 - 3i)|= 4