Arithmetic and Cartesian form

Define

i

where

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.

Worked examples

Multiplying complex numbers

Straightforward

Problem

Find

(2+3i)(1-5i)

, expressing the answer in the form

a+bi

Dividing complex numbers

Moderate

Problem

Express

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

in the form

a+bi

Finding a complex square root

Challenging

Problem

Find all

z \in \mathbb{C}

satisfying

z^2 = 3+4i

Practise

Q1·Straightforward

Find the imaginary part of

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

Q2·Straightforward

Find the real part of

(4+3i)(2+i)

Q3·Straightforward

Find

|3+4i|

Q4·Straightforward

The complex number

z = 5-2i

. Find

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

Q5·Straightforward

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.

Q6·Moderate

Find

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

Q7·Moderate

Find the real part of

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

Q8·Moderate

Find the imaginary part of

(1+2i)^2

Q9·Moderate

Solve

z^2 = -9

for

z \in \mathbb{C}

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

Q10·Moderate

Given

z_1 = 2+3i

and

z_2 = 1-i

, find

|z_1 z_2|^2

Q11·Challenging

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

Q12·Challenging

Find

|z|

where

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

. Give your answer as a decimal.

Q13·Challenging

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).

✎ Work this one through on paper — proofs are self-assessed.