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Complex numbers

Extend the number system to complex numbers: arithmetic and Cartesian form, polar and exponential form, De Moivre's theorem, roots of unity, and loci in the Argand plane.

Complex numbers · MEX-12-02

Sub-topics

Arithmetic and Cartesian form

i

i^2 = -1

; add, subtract, multiply and divide complex numbers in the form

a+bi

; find the conjugate and modulus; plot complex numbers on the Argand diagram.

Polar and exponential form

Express complex numbers in modulus–argument form

r(\cos\theta+i\sin\theta)

and Euler's form

re^{i\theta}

; multiply and divide using polar form; apply De Moivre's theorem to compute powers.

Roots and identities

n

th roots of a complex number; identify roots of unity and their properties; use De Moivre's theorem to derive trigonometric identities for

\cos n\theta

\sin n\theta

Curves and regions (loci)

Identify and sketch loci in the Argand plane defined by modulus and argument conditions; interpret regions defined by modulus inequalities; express equations of circles and lines in complex form.

Open Math

Complex numbers

Complex numbers · MEX-12-02

Name:

Date:

Arithmetic and Cartesian form

Q1Straightforward

Find the imaginary part of

(3+5i)+(2-8i)

.

Q2Straightforward

Find the real part of

(4+3i)(2+i)

.

Q3Straightforward

Find

|3+4i|

.

Q4Straightforward

The complex number

z = 5-2i

. Find

\text{Re}(z)+\text{Im}(z)

.

Q5Straightforward

The complex number

z = -2+3i

is plotted on the Argand diagram with the real axis as the

x

-axis and the imaginary axis as the

y

-axis. Give the coordinates

(x, y)

of the point.

Q6Moderate

Find

(2+5i)(2-5i)

.

Q7Moderate

Find the real part of

\dfrac{3+i}{1+2i}

.

Q8Moderate

Find the imaginary part of

(1+2i)^2

.

Q9Moderate

Solve

z^2 = -9

for

z \in \mathbb{C}

. Find the imaginary part of the solution with positive imaginary part.

Q10Moderate

Given

z_1 = 2+3i

and

z_2 = 1-i

, find

|z_1 z_2|^2

.

Q11Challenging

Find all complex numbers of the form

z = a+bi

(where

a, b \in \mathbb{R}

) satisfying

z^2 = -5+12i

. Enter the positive value of

a

.

Q12Challenging

Find

|z|

where

z = \dfrac{5+12i}{3+4i}

. Give your answer as a decimal.

Q13Challenging

Let

z = a+bi

where

a, b \in \mathbb{R}

and

\bar{z} = a-bi

is its conjugate.

(a) Prove that

z + \bar{z}

is always real.
(b) Prove that

z - \bar{z}

is always purely imaginary (or zero).

Polar and exponential form

Q14Straightforward

Find the modulus of

z = \sqrt{3}+i

.

Q15Straightforward

z_1 = 4(\cos 20° + i\sin 20°)

and

z_2 = 3(\cos 70° + i\sin 70°)

. Find

|z_1 z_2|

.

Q16Straightforward

Find the real part of

z = 2(\cos 90°+i\sin 90°)

.

Q17Moderate

Find the argument of

z = \sqrt{3}+i

in degrees.

Q18Moderate

z_1 = 3(\cos 40°+i\sin 40°)

and

z_2 = 2(\cos 80°+i\sin 80°)

. Find

\arg(z_1 z_2)

in degrees.

Q19Moderate

Find the real part of

(1+i)^8

using De Moivre's theorem.

Q20Moderate

Find the argument of

z = -\sqrt{3}+i

in degrees.

Q21Moderate

z_1 = 10(\cos 60°+i\sin 60°)

and

z_2 = 2(\cos 10°+i\sin 10°)

. Find

\left|\dfrac{z_1}{z_2}\right|

.

Q22Challenging

Find the real part of

(\cos 18°+i\sin 18°)^{10}

using De Moivre's theorem.

Q23Challenging

Find the real part of

(1+i)^{12}

using De Moivre's theorem.

Q24Challenging

Find the imaginary part of

(1-i)^{10}

using De Moivre's theorem.

Q25Challenging

z = 2e^{i\pi/6}

and

w = 3e^{i\pi/3}

. Find

\text{Im}(zw)

.

Q26Challenging

Use De Moivre's theorem to expand

(\cos\theta+i\sin\theta)^3

and hence prove that:

\cos 3\theta = 4\cos^3\theta - 3\cos\theta

Roots and identities

Q27Straightforward

How many solutions does

z^5 = 1

have in

\mathbb{C}

?

Q28Straightforward

Find

\left|e^{2\pi i/7}\right|

.

Q29Straightforward

The 4th roots of unity are

1, i, -1, -i

. Find the sum

1+i+(-1)+(-i)

.

Q30Moderate

Find the argument of

\omega = e^{2\pi i/3}

in degrees.

Q31Moderate

The equation

z^2 = i

has two solutions. Find the argument (in degrees) of the solution with the smallest positive argument.

Q32Moderate

The cube roots of

8

all have the same modulus

r

. Find

r

.

Q33Moderate

Find the sum of all cube roots of

8

.

Q34Moderate

De Moivre's theorem states

(\cos\theta+i\sin\theta)^2 = \cos 2\theta+i\sin 2\theta

. Which expression gives

\cos 2\theta

?

A. $\sin^2\theta - \cos^2\theta$
B. $\cos^2\theta - \sin^2\theta$
C. $2\sin\theta\cos\theta$
D. $1+2\cos^2\theta$

Q35Challenging

Use De Moivre's theorem to expand

(\cos\theta+i\sin\theta)^3

and hence express

\cos 3\theta

as a polynomial in

\cos\theta

.

Find the coefficient of

\cos\theta

in this polynomial (include the sign).

Q36Challenging

The cube roots of

-8

include

-2

and two non-real roots. Find the imaginary part of the cube root with argument between

0°

and

180°

. Give your answer correct to 2 decimal places.

Q37Challenging

The 4th roots of

-16

have the form

z_k = 2e^{i\theta_k}

for

k = 0, 1, 2, 3

. Find the real part of the root with argument

\theta = \frac{\pi}{4}

. Give your answer correct to 2 decimal places.

Q38Challenging

Use De Moivre's theorem to prove that:

\sin 3\theta = 3\sin\theta - 4\sin^3\theta

Q39Challenging

Let

\omega = e^{2\pi i/n}

be a primitive

n

th root of unity, where

n \geq 2

is a positive integer. Prove that the sum of all

n

th roots of unity is zero:

1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0

Curves and regions (loci)

Q40Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle with centre

(a, b)

and radius

r

. State the value of

a

(the

x

-coordinate of the centre).

Q41Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle. State the radius.

Q42Straightforward

Let

z = 5 + 12i

. Calculate

|z|

.

Q43Straightforward

The locus defined by

\operatorname{Im}(z) = -2

is a horizontal line in the Argand plane. What is the

y

-coordinate of every point on this locus?

Q44Moderate

The locus

|z - 2| = |z - 8|

is the perpendicular bisector of the segment joining

2 + 0i

and

8 + 0i

on the real axis. State the

x

-coordinate of every point on this locus.

Q45Moderate

The locus

|z - 3i| = |z + 3i|

is the perpendicular bisector of the segment joining

3i

and

-3i

. State the

y

-coordinate of every point on this locus.

Q46Moderate

A point

z

satisfies

\arg(z) = \dfrac{\pi}{4}

and

\operatorname{Re}(z) = 3

. Find

\operatorname{Im}(z)

.

Q47Moderate

The circle

|z - 1 - 2i| = 3

has centre

(1, 2)

and radius

3

. What is the

y

-coordinate of the highest point on this circle?

Q48Moderate

The circle

|z + 3 - 2i| = 5

has centre

(-3, 2)

and radius

5

. Find the positive

x

-coordinate where this circle crosses the real axis (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Q49Challenging

The locus

|z - 2 - 3i| \le 4

defines a circular region. Using the triangle inequality

|z| \le |z - w| + |w|

, find the maximum value of

|z|

for points in this region. Give your answer to two decimal places.

Q50Challenging

The locus

|z - 4| = 2|z + 1|

is an Apollonius circle. By squaring both sides and expanding, show that this simplifies to the form

(x + a)^2 + y^2 = R^2

. State the radius

R

. Give your answer to two decimal places.

Q51Challenging

The locus

|z - i| + |z + i| = 4

is an ellipse with foci at

i

and

-i

. Find the positive

x

-intercept of this ellipse (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Worked solutions and answers at openmath.au/year-12/extension-2/complex-numbers