IB

# Complex numbers

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

i = −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

Adding and subtracting complex numbers in Cartesian form is fairly straight forward. Add real and imaginary parts to each other:

(2+3i) + (4+9i) = 2 + 4 + 3i + 9i = 6+12i

Multiplying complex numbers is like multiplying two parentheses:

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.

Rewriting of a fraction with complex numbers
1. Convert the denominator into a real number by multiplying it with its conjugate.

2. Expand the brackets and simplify, remember that i2 = −1.

3. Write in a + bi form.

−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:

Multiplication:      z1 × z2 = r1 × r1 cis(θ1 + θ2)
Division:

### 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.

Example. Prove: zn = (cos(x)isin(x))n = cos(nx) + isin(nx).

## 1.5.3 Nth roots of a complex number

nth root of a complex number z    is a number ω such that ωn = z.

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.

Finding complex roots
Find 3 roots of z3 = 4 + 4√3i and draw them on the complex plane.
1. Rewrite the complex number in polar form.
2. Insert values into formula.
3. Draw the roots on the complex plane, they should be equally spaced out with the same length.
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