Complex Numbers #2
By DarthVader
Date: 2023-04-03
Topic: 192 see comments
Post views: 792
Complex Numbers
- A complex number is made up of two parts, a real part and an imaginary part containing ‘i’. This means a complex number can be used to carry two pieces of information in one expression.
- Complex numbers are used extensively in engineering and electronics.
- In electronics, complex numbers can be used to describe sine waves.
i = √−1
i2 = −1
Cartesian & Polar Coordinates + Exponential form
A complex number in cartesian form is:
a + bi
In polar form:
r(cos(θ) + isin(θ) ) or; rcos(θ) + risin(θ)
And in compact polar form:
r∠θ
In exponential form:
reiθ
Where:
r = the modulus.
θ = the argument.
Multiplication & Division In Polar Form
For two complex numbers in polar form:
- z = r(cos(θ) + isin(θ) )
and:
- w = s(cos(ρ) + isin(ρ) )
We can multiply them together using the formula:
zw = rs(cos(θ + ρ) + isin(θ + ρ) )
and divide them to give:
z/w = (r/s)(cos(θ − ρ) + isin(θ − ρ) )
Similarly in compact form, if we have two complex numbers:
- z = r∠θ
and,
- w = s∠ρ
Then their product is:
zw = rs∠(θ + ρ)
and division gives:
z/w = r/s∠(θ − ρ)
De Moivre's Formula:
De Moivre's formula is used when calculating powers of complex numbers. For z = r(cos(θ) + isin(θ) ), it can be written as:
zn = rn(cos(nθ) + isin(nθ) )
or in compact form as:
zn = rn∠nθ
To find a power of a complex number in cartesian form:
- Convert to polar form.
- Find the power using De Moivre's formula.
- Convert back to cartesian form.
Euler's Formula
We can define Euler's formula as:
eiθ = cos(θ) + isin(θ)
where i2 = −1 and θ is a real number
Multiplying & Dividing Complex Numbers In Exponential Form
For two complex numbers in exponential form,
z = reiθ - and: w = seiρ
where in exponential form reiθ:
r = Modulus.
θ = Argument.
we can multiply them together to give:
zw = rsei(θ + ρ)
and divide them to get:
z/w = (r/s)ei(θ − ρ)
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