A complex number is defined as z = a + bi. Where a, b ∈ ℝ, a is the real part (ℜ) and b is the imaginary part (ℑ).
i = √−1
i2 = −1
z = a + bi is the Cartesian form. z = r(cosθ+isinθ) is the polar form where r is the modulus and θ is the argument also sometimes stated as z = r cisθ.
Modulus r the absolute distance from the origin to the point.
Argument θ the angle between the x-axis and the line connecting the origin and the point.
Instead of working in (x, y) coordinates, polar coordinates use the distance from the origin to the point (r, modulus) and the angle between the x-axis and the modulus (argument).
The conjugate of a complex number z̄ or z∗, is defined as
z = a + bi ⇒ z̄ = a − bi
1.5.1 Complex numbers in the Cartesian form
(2+3i) + (4+9i) = 2 + 4 + 3i + 9i = 6+12i
Division, however, is slightly more complex. Conjugates play a big role here, since a complex number multiplied by its conjugate is always equal to a real number.
- Convert the denominator into a real number by multiplying it with its conjugate.
2. Expand the brackets and simplify, remember that i2 = −1.
−2 + 2i
1.5.2 Complex numbers in the Polar form
Polar form allows us to do some operations quicker and more efficient, such as multiplication and division of complex numbers. The formulas can be shown for the following two complex numbers z1 = r1 cis(θ1)and z2 = r2 cis(θ2).
Note: cis x = cos x + i sin x .
Example:
Euler’s and De Moivre’s theorem
These two theorems state the relationship between the trigonometric functions and the complex exponential function. This allows us to convert between Cartesian and Polar forms.
Euler’s Theorem
eix = cosx + isinx
De Moivre’s theorem
zn = {r(cosx + isinx)}n =rn {cos(nx) + isin(nx)}
De Moivre’s theorem can be derived from Euler’s through the exponential law for integer powers.
(eix)n = eix n = zn
De Moivre’s theorem: proof by induction
Having seen the method of induction, we will now apply it to De Moivre’s theorem.
1.5.3 Nth roots of a complex number
To find nth root of a complex number (in polar form, z = r cis(θ)) you need to use the following formula:
for k = 0, 1, 2, . . . , n − 1.
- Rewrite the complex number in polar form.
