Complex Numbers Summary
Academic Skills Advice
What does a complex number mean?
A complex number has a ‘real’ part and an ‘imaginary’ part (the imaginary part involves the
square root of a negative number).
e.g. Z = a + 𝒊b
We use Z to denote a complex number:
Real
Example:
Imaginary
You might see the 𝑖 before or after it’s
number - it doesn’t matter which.
Z = 4 + 3i
Re(Z) = 4 Im(Z) = 3
Sometimes (especially in engineering)
a j is used instead of 𝑖 – they mean the
same thing.
Powers of i
𝑖 stands for √−1
so:
For any power of 𝑖 take out as many 𝑖
and they will all end up as ±𝑖 or ±1.
Example:
2
𝑖 2 = (√−1) = -1
𝑖 4 = (𝑖 2)2 = (-1)2 = 1
4’s
and 𝑖
2’s
as possible
Summary:
𝒊 = √−𝟏
𝒊𝟐 = −𝟏
𝒊𝟒 = 𝟏
𝒊−𝟐 = −𝟏
𝒊−𝟒 = 𝟏
𝑖 11 = (𝑖 4 )2 𝑖 2 𝑖 = 12 × −1 × 𝑖 = −𝑖
Adding & Subtracting
This is easy – just add or subtract the real part and add or subtract the imaginary parts:
Examples:
(4 + 3𝑖 ) + (2 + 6𝑖 ) = (6 + 9𝑖 )
(3 + 7𝑖 ) – (1 – 3𝑖 ) = (2 + 10𝑖 )
Multiplying
Multiply out the 2 brackets.
Example:
(3 + 5𝑖 )(4 – 2𝑖 ) = 12 – 6𝑖 +20𝑖 – 10𝑖 2 = 12 + 14𝑖 – 10 (-1) = 22 + 14𝑖
Complex Conjugate
The conjugate is exactly the same as the complex number but with the opposite sign in the
middle. When multiplied together they always produce a real number because the middle
terms disappear (like the difference of 2 squares with quadratics).
Example:
(4 + 6𝑖 )(4 – 6𝑖 ) = 16 – 24𝑖 + 24𝑖 – 36𝑖 2 = 16 – 36(-1) = 16 + 36 = 52
© H Jackson 2010 / Academic Skills
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Dividing
Dividing by a real number:
divide the real part and divide the imaginary part.
Dividing by a complex number:
Multiply top and bottom of the fraction by the complex
conjugate of the denominator so that it becomes real, then
do as above.
Examples:
3+4𝑖
2
4−5𝑖
3+2𝑖
3
=
2
4
+ 𝑖 = 1.5 + 2𝑖
2
4−5𝑖
=
3+2𝑖
×
3−2𝑖
12−8𝑖−15𝑖+10𝑖 2
=
3−2𝑖
9−6𝑖+6𝑖−4𝑖 2
=
12−23𝑖+10(−1)
9−4(−1)
=
2−23𝑖
13
=
𝟐
𝟏𝟑
−
𝟐𝟑
𝟏𝟑
𝒊
Graphical Representation
A complex number can be represented on an Argand diagram by plotting the real part on
the 𝑥-axis and the imaginary part on the y-axis.
Example:
Imaginary
P (𝑧
= 𝑥 + 𝑦𝑖)
y
0
Nb Tan(θ) =
𝑦
𝑥
𝜃
𝑥
Real
is written as |𝒛| and is the length of OP, therefore |𝑧| = √𝑥 2 + 𝑦 2
is the angle θ that is made with the horizontal axis (denoted by ∠).
Modulus:
Argument:
Properties of the Modulus and Argument
𝑧1 =
Let
𝑧2
Then |𝑧1 | =
These let you find
the modulus and the
argument when one
complex number is
divided by another.
𝜋
Examples:
|𝑧2 |
Then |𝑧1 | = |𝑧2 | × |𝑧3 |
|𝑧3 |
∠𝑧1 = ∠𝑧2 − ∠𝑧3
And
𝜋
And
𝑧
if 𝑧2 = 5𝑒 2 𝑖 , 𝑧3 = 3𝑒 3 𝑖 , 𝑧1 = 𝑧2
3
Then |𝑧1 | =
And
|𝑧2 |
|𝑧3 |
5
∠𝑧1 = ∠𝑧2 − ∠𝑧3
𝜋
𝜋
5
𝜋
∴ 𝑧1 = 3 𝑒 6 𝑖
© H Jackson 2010 / Academic Skills
∠𝑧1 = ∠𝑧2 + ∠𝑧3
𝜋
if
These let you find the
modulus and the
argument when one
complex number is
multiplied by another.
𝜋
𝑧2 = 5𝑒 2 𝑖 , 𝑧3 = 3𝑒 3 𝑖 , 𝑧1 = 𝑧2 𝑧3
Then |𝑧1 | = |𝑧2 | × |𝑧3 | = 15
=3
=2−3 =
𝑧1 = 𝑧2 𝑧3
Let
𝑧3
𝜋
6
And
∠𝑧1 = ∠𝑧2 + ∠𝑧3
𝜋
𝜋
=2+3
=
5𝜋
6
5𝜋
∴ 𝑧1 = 15𝑒 6 𝑖
2
Polar Form
Once you have shown a complex number on an Argand diagram then it’s easy to find the
length of the line and the angle it makes with the 𝑥-axis.
To find the length: use Pythagoras
To find the angle: use tan 𝜽
Once you have this info you can write the complex number in polar form:
𝒛 = 𝒓𝒄𝒐𝒔𝜽 + 𝒊𝒓𝒔𝒊𝒏𝜽
𝒛 = 𝒓(𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽)
Example:
Express 𝑧 = 3 + 4𝑖 in polar form
imaginary
Pythagoras: 𝑟 = √32 + 42 = √16 + 9 = √25 = 5
𝑟
0
Where 𝑟 is the length of the line and 𝜃
is the angle it makes with the 𝑥-axis.
4
𝜃
3
4
𝜃 = 53.1o
Tan 𝜃:
tan𝜃 = 3
Polar form:
𝑧 = 5(cos 53.1 + 𝑖𝑠𝑖𝑛53.1)
real
Nb if, when you draw the complex number, it’s in a different quadrant adjust the angle as necessary.
Exponential Form
𝒛 = 𝒓𝒆𝒊𝜽
The 3 forms:
(where 𝑟 and 𝜃 are the same as in polar form but 𝜃 must be in radians)
Basic
𝒛 = 𝒂 + 𝒊𝒃
Polar
𝒛 = 𝒓(𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽)
Exponential
𝒛 = 𝒓𝒆𝒊𝜽
De Moivres Theorem:
Is used for raising a complex number to a power.
𝒛𝒏 = 𝒓𝒏 (𝒄𝒐𝒔𝒏𝜽 + 𝒊𝒔𝒊𝒏𝒏𝜽)
𝜋
e.g
e.g.2: (1 + 𝑖)100
If 𝑧 = 3𝑒 3 𝑖
then 𝑧 5 = 35 (𝑐𝑜𝑠
© H Jackson 2010 / Academic Skills
5𝜋
3
+ 𝑖𝑠𝑖𝑛
5𝜋
)
3
We could use De Moivres or:
(1 + 𝑖)100 = ((1 + 𝑖)2 )50
3