Polar form, de Moivre’s theorem and roots of unity · Grade 12 · 5 units
In Polar form, de Moivre’s theorem and roots of unity (Grade 12 · 5 units) you practise 6 skills: Writing a complex number in polar form · From polar form to the real and imaginary parts · Multiplying and dividing in polar form · De Moivre’s theorem · The roots of unity · Equations in which a power of a complex number equals a given number. Every complex number other than zero can be written in polar form, from two facts: its modulus, the distance of its point from the origin, and its argument, the angle from the positive real axis to the segment from the origin to the point. The argument is chosen to be at least zero degrees and less than three hundred and sixty degrees. The real part is the modulus times the cosine of the argument, and the imaginary part is the modulus times its sine.
When two complex numbers in polar form are multiplied, the moduli multiply and the arguments add; when they are divided, the moduli divide and the arguments subtract. So multiplying by the imaginary unit turns the point a quarter turn anticlockwise about the origin, without changing its distance from the origin. By de Moivre’s theorem, to raise a complex number to a whole power, raise its modulus to that power and multiply its argument by the power. It follows that an equation in which a power of a complex number equals a number other than zero has exactly as many solutions as the power. All the solutions lie on one circle, at the vertices of a regular polygon. The solutions of the equation in which the power equals one are called the roots of unity, and their sum is zero. A model proof: de Moivre’s theorem by induction on the power. The base case: when the power is one, both sides equal the number itself. The inductive step: assume the theorem holds for some power, and multiply both sides by the number itself. In a product the moduli multiply and the arguments add, so the modulus of the next power is the modulus raised to a power one greater, and its argument is the argument of the number times a power one greater. That is exactly the claim for the next power, so the claim holds for every whole power.
Tip: To multiply in polar form, multiply the moduli and add the arguments.
What you learn
Writing a complex number in polar form
For example: The complex number z = 1 − i is given. What is its modulus r ?
Answer and solution
The answer: √2
By Pythagoras r² = 1 + 1 = 2 , so r = √2 .
From polar form to the real and imaginary parts
For example: The complex number z = r(cos θ + i sin θ) is given, with r = 3 and θ = 270° . What are the coordinates of the point that represents z in the Gauss plane?
Answer and solution
The answer: (0, −3)
Here cos θ = 0 and sin θ = −1 , so the point is (0, −3) .
Multiplying and dividing in polar form
For example: The complex numbers z = 3(cos 340° + i sin 340°) and w = 3(cos 170° + i sin 170°) are given. What is the argument θ of z × w , with 0° ≤ θ < 360° ?
Answer and solution
The answer: 150 °
The sum is 340° + 170° = 510° , and taking away 360° gives 150° . The argument of z × w is 150° .
De Moivre’s theorem
For example: What is the argument θ of (cos 20° + i sin 20°)⁹ , with 0° ≤ θ < 360° ?
Answer and solution
The answer: 180 °
The angle is 9 × 20° = 180° . The argument is 180° .
The roots of unity
For example: What is the angle between two neighbouring roots of z⁴ = 1 ?
Answer and solution
The answer: 90 °
The roots split the circle into equal parts, so the angle is 360° ÷ 4 = 90° .
Equations in which a power of a complex number equals a given number
For example: How many solutions does z⁶ = 1 have?
Answer and solution
The answer: 6
The equation zⁿ = w with w ≠ 0 has exactly n solutions, their arguments 360° ÷ n apart. Here n = 6 , so there are 6 solutions.
Not on the site yet
Parts of this topic that are not built yet: Euler’s formula (enrichment) · The logarithmic spiral.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).