27/01/2025
Complex Numbers
▪ A number represented by 𝒛 = 𝑥 + 𝑖𝑦 is a complex number where 𝑥 and 𝑦
are two real numbers and 𝑖 =
−1
▪ This representation is defined rectangular form
Imaginary Axis
COMPLEX NUMBERS IN MATLAB
(x, y)
z
▪ A complex number 𝒛 can be expressed as a vector
r
𝒛 = (𝑥, 𝑦) on a plane.

𝑥 = 𝑅𝑒 𝒛
𝑦 = 𝐼𝑚(𝒛)
Real Axis
2
1
2
Complex Numbers in Polar Form
▪ The Polar Form represents an alternate way to express Complex numbers.
𝑟 = 𝒛 = 𝑎𝑏𝑠 𝒛 =
𝑥2 + 𝑦 2
𝜃 = ∠ 𝒛 = 𝑎𝑟𝑔 𝒛 = tan −1
In matlab, enter the following commands:
>> z=3-5i
Imaginary Axis
(x, y)
z
r
▪ If we have 𝑟 and 𝜃, then 𝑥 and 𝑦
are obtained by:
>> real (z)
>> imag(z)
>> conj(z)

Real Axis
𝑥 = 𝑟𝑐𝑜𝑠𝜃 = 𝑅𝑒 𝒛
𝑦 = 𝑟𝑠𝑖𝑛𝜃 = 𝐼𝑚 𝒛
4
3
3
and
𝑦
𝑥
4
1
27/01/2025
Euler’s Formula
▪
Representation of Complex Numbers in MATLAB
Euler's formula, determined by the Swiss Leonhard Euler, establishes a
mathematical relationship between trigonometric functions and the complex
exponential function.
▪
Complex numbers can be expressed both in rectangular and polar form
•
The Rectangular form is described by a simple addition of a real and
imaginary part:
𝒛 = 𝑎 + 𝑖𝑏
𝑒𝑖𝜃 = 𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃
▪
▪ Note: Since i in electronics is often used to describe current, some
times j is used. In MATLAB, they are equivalent
Therefore,
𝒛 = 𝑟𝑐𝑜𝑠𝜃 + 𝑖𝑟𝑠𝑖𝑛𝜃 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)
𝒛 = 𝑟𝑒𝑖𝜃
▪
The Polar form can be described in terms of Euler’s formula;
𝒛 = 𝑟 ∗ exp(𝑖 ∗ 𝑡ℎ𝑒𝑡𝑎)
▪
Common engineering notation of the polar form:
𝑟𝑒𝑖𝜃
≡ 𝑟∠𝜃
5
5
6
6
Representation of Complex Numbers in MATLAB
▪
Example:
Example: convert the following complex number to polar form.
▪
Example: Convert the following complex number to rectangular form.
𝒛 = 4 + 𝑖3
𝒛 = 4𝑒𝑖2
𝑥2 + 𝑦 2 = 4 2 + 3 2 = 5
3
𝜃 = tan−1 = 36.87 ° 0.6435 𝑟𝑎𝑑
4
𝒛 = 5∠36.87 °
𝑟=
𝑥 = 4𝑐𝑜𝑠2 = −1.6646
>> z=4+3i;
>> r=abs(z)
r=5
>> theta_rad=angle(z)
heta_rad=0.6435
>> theta_degree=theta_rad*(180/pi)
theta_degree=36.87
>> a=r*exp(i*theta_rad)
a=4.0000 + 3.0000i
▪
In MATLAB:
>> z=4*exp(2i)
z =
-1.6646 + 3.6372i
8
7
7
𝑦 = 4𝑠𝑖𝑛2 = 3.6372
𝒛 = −1.6646 + 𝑖3.6372
8
2