Formulas · Mathematics

Complex numbers

4 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 11

Modulus of a complex number

∣ z ∣ = a^{2} + b^{2}

What each letter means
$∣ z ∣$	the modulus of z = a + ib: its distance from 0 in the Argand plane
$a$	the real part of z
$b$	the imaginary part of z (the number beside i)

The modulus of z = a + ib is √(a² + b²): the distance of the point (a, b) from 0 in the Argand plane.

Why it works

Plot a + ib as the point (a, b). Its distance from the origin is √(a² + b²) by Pythagoras. It is also the square root of z times its conjugate, since (a + ib)(a − ib) = a² + b².

When to use it

The size of a complex number, dividing by a complex number (the square of the modulus goes underneath), and the multiplicative inverse.

How to use it

Read the real part a and the imaginary part b (b is the number beside i, without the i).
Work out √(a² + b²).

To remember: The modulus is the distance from 0.

Other forms

Same formula, another form $∣ z ∣^{2} = a^{2} + b^{2}$ The square of the modulus, which is z times its conjugate.
To remember itThe modulus of a product is the product of the moduli: |z₁z₂| = |z₁| |z₂|.

Worked examples

Story

A robot moves 5 m east and 12 m north, written 5 + 12i. How far is it from where it started?

$25 + 144 = 13$

Answer: 13

Picture

Show that 1 + i, 1 − i, −1 + i and −1 − i are all the same distance from 0.

Each has a = 1 or −1 and b = 1 or −1, so a² + b² = 2 each time:
$1 + 1 = 2$
All four lie on the circle of radius √2 round 0.

Answer: Each is √2 from 0

Direct

Find the modulus of 3 + 4i.

$3^{2} + 4^{2} = 5$

Answer: 5

Reverse

The modulus of a + 8i is 10, with a positive. Find a.

Try it first, then show the working

$6^{2} + 8^{2} = 10$

Answer: 6

Exam

Find the modulus of (1 + i) ÷ (1 − i).

Try it first, then show the working

Multiply the top and bottom by 1 + i: (1 + i)² ÷ 2 = 2i ÷ 2 = i.
So a = 0 and b = 1:
$0^{2} + 1^{2} = 1$

Answer: 1

Common mistake: Squaring the i as well: for 3 + 4i, b is 4, and b² is 16, not (4i)² = −16.

Class 11

Multiplying complex numbers

(a + ib) (c + i d) = (a c - b d) + i (a d + b c)

What each letter means
$a$	the real part of the first number
$b$	the imaginary part of the first number
$c$	the real part of the second number
$d$	the imaginary part of the second number

To multiply two complex numbers, multiply out the brackets as usual and replace i² by −1.

Why it works

(a + ib)(c + id) = ac + iad + ibc + i²bd. Since i² = −1, the last term is −bd, a real number. Gathering the real parts and the i parts gives (ac − bd) + i(ad + bc).

When to use it

Multiplying and squaring complex numbers, and checking a complex root of a quadratic.

How to use it

Multiply each part of the first bracket by each part of the second.
Replace i² by −1.
Collect the real parts and the i parts.

To remember: Multiply out, then i² becomes −1.

Other forms

Special case $(a + ib)^{2} = a^{2} - b^{2} + 2 iab$ When the two numbers are the same (c = a, d = b).
To remember itThe powers of i repeat in fours: i, −1, −i, 1, then i again.

Worked examples

Story

In a drawing app, multiplying by i turns a point a quarter turn anticlockwise round 0. Where does the point 3 + 2i go?

$i (3 + 2 i) = 3 i + 2 i^{2}$
$3 i + 2 i^{2} = - 2 + 3 i$
The point (3, 2) has turned to (−2, 3).

Answer: −2 + 3i

Picture

Find (1 + i)² and (1 + i)⁴.

$(1 + i)^{2} = 1 + 2 i + i^{2}$
$1 + 2 i + i^{2} = 2 i$
$(2 i)^{2} = - 4$
So (1 + i)⁴ = −4: each squaring doubles the angle, and four eighth turns make a half turn.

Answer: 2i and −4

Direct

Multiply (2 + 3i)(4 − i).

$(2 + 3 i) (4 - i) = 8 - 2 i + 12 i - 3 i^{2}$
$8 - 2 i + 12 i + 3 = 11 + 10 i$

Answer: 11 + 10i

Reverse

Find real x and y if (x + iy)(2 − 3i) = 4 + i.

Try it first, then show the working

Divide: x + iy = (4 + i) ÷ (2 − 3i). Multiply the top and bottom by 2 + 3i.
$(4 + i) (2 + 3 i) = 5 + 14 i$
$(2 - 3 i) (2 + 3 i) = 13$
So x = 5/13 and y = 14/13.

Answer: x = 5/13, y = 14/13

Exam

Show that (1 − i)⁴ is a real number, and find it.

Try it first, then show the working

$(1 - i)^{2} = - 2 i$
$(- 2 i)^{2} = - 4$

Answer: -4

Common mistake: Leaving i² in the answer, or writing + bd (forgetting that i² = −1).

Class 11

A complex number times its conjugate

(a + ib) (a - ib) = a^{2} + b^{2}

What each letter means
$a$	the real part
$b$	the imaginary part

A complex number times its conjugate is a real number: (a + ib)(a − ib) = a² + b², the square of its modulus.

Why it works

It is (x + y)(x − y) = x² − y² with x = a and y = ib: a² − i²b² = a² + b², since i² = −1.

When to use it

Dividing by a complex number (multiply the top and bottom by the conjugate of the bottom to make the bottom real), and finding the modulus.

How to use it

Change the sign of the imaginary part to get the conjugate.
Multiply: the answer is a² + b², with no i left.

To remember: z times z-bar is the modulus squared.

Other forms

Special case $(1 + ib) (1 - ib) = 1 + b^{2}$ When the real part is 1.
To remember itz z̄ = |z|²: the product is the square of the distance from 0.

Worked examples

Story

An engineer must work out 10 ÷ (3 + i). Make the bottom real first.

Multiply the top and bottom by 3 − i.
$(3 + i) (3 - i) = 10$
$\frac{10 ( 3 - i )}{10} = 3 - i$

Answer: 3 − i

Picture

The point 5 + 12i and its mirror image in the real axis, 5 − 12i, are multiplied. What is the product, and what does it mean?

$(5 + 12 i) (5 - 12 i) = 25 + 144$
$25 + 144 = 169$
That is 13², the square of the distance of each point from 0.

Answer: 169

Direct

Multiply (3 + 4i)(3 − 4i).

$(3 + 4 i) (3 - 4 i) = 9 - 16 i^{2}$
$9 - 16 i^{2} = 25$

Answer: 25

Reverse

The product of a + 2i and its conjugate is 13, with a positive. Find a.

Try it first, then show the working

Here a² + 4 = 13, and so a² = 9 and a = 3.
$(3 + 2 i) (3 - 2 i) = 13$

Answer: a = 3

Exam

Find the multiplicative inverse of 2 − 3i.

Try it first, then show the working

Multiply the top and bottom of 1 ÷ (2 − 3i) by 2 + 3i:
$(2 - 3 i) (2 + 3 i) = 13$
So the inverse is (2 + 3i) ÷ 13, which is 2/13 + (3/13)i.

Answer: 2/13 + (3/13)i

Common mistake: Writing a² − b²: the minus from the conjugate and the minus from i² cancel out.

Class 11

Dividing complex numbers

\frac{a + ib}{c + i d} = \frac{( a c + b d ) + i ( b c - a d )}{c ^{2} + d ^{2}}

What each letter means
$a$	the real part of the top number
$b$	the imaginary part of the top number
$c$	the real part of the bottom number (c and d not both 0)
$d$	the imaginary part of the bottom number

To divide by c + id, multiply the top and bottom by its conjugate c − id; the bottom becomes the real number c² + d².

Why it works

(c + id)(c − id) = c² + d², which has no i. The top becomes (a + ib)(c − id) = (ac + bd) + i(bc − ad). Multiplying the top and bottom by the same number does not change the fraction.

When to use it

Dividing complex numbers, writing a quotient in the form x + iy, and finding a multiplicative inverse (a = 1, b = 0).

How to use it

Multiply the top and bottom by the conjugate of the bottom.
Multiply out the top, using i² = −1.
Divide the real part and the i part by c² + d².

To remember: Make the bottom real with its conjugate.

Other forms

Special case $\frac{1}{c + i d} = \frac{c - i d}{c ^{2} + d ^{2}}$ When a = 1 and b = 0: the multiplicative inverse.

Worked examples

Story

In an electric circuit, the current is the voltage divided by the impedance. Work out 10 ÷ (4 + 3i).

Multiply the top and bottom by 4 − 3i.
$(4 + 3 i) (4 - 3 i) = 25$
$\frac{40 - 30 i}{25} = 1.6 - 1.2 i$

Answer: 1.6 − 1.2i

Picture

Dividing by i turns a point a quarter turn clockwise round 0. Where does 2 + 5i go?

The conjugate of i is −i, and i times −i is 1.
$(2 + 5 i) (- i) = 5 - 2 i$
The point (2, 5) moves to (5, −2).

Answer: 5 − 2i

Direct

Write (5 + i) ÷ (2 − 3i) in the form x + iy.

Multiply the top and bottom by 2 + 3i.
$(5 + i) (2 + 3 i) = 7 + 17 i$
$(2 - 3 i) (2 + 3 i) = 13$

Answer: 7/13 + (17/13)i

Reverse

Find z if (1 + i)z = 3 − i.

Try it first, then show the working

Divide: z = (3 − i) ÷ (1 + i). Multiply the top and bottom by 1 − i.
$(3 - i) (1 - i) = 2 - 4 i$
$\frac{2 - 4 i}{2} = 1 - 2 i$

Answer: 1 − 2i

Exam

Write (3 + i√5)(3 − i√5) ÷ ((√3 + √2 i) − (√3 − i√2)) in the form x + iy.

Try it first, then show the working

The top is 9 + 5 = 14; the bottom is 2√2 i.
Dividing by i is multiplying by −i, so the answer is −14i ÷ (2√2).
$\frac{14}{2 2} = \frac{7 2}{2}$

Answer: 0 − (7√2/2)i

Common mistake: Multiplying only the bottom by the conjugate, or using the conjugate of the top instead of the bottom.