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Complex numbers in polar form: introduction

Summary

This lesson opens the unit devoted to complex numbers in polar form. It presents its six lessons and the thread that unites them: describing each complex number by its distance from the origin and by its direction turns the product into a rotation composed with a dilation, powers into multiples of an angle and roots into regular polygons. It reconstructs the historical background of these ideas, from the formula for the cubic published by Cardano in 1545 and Bombelli’s rules of calculation to the geometric representation of Wessel, Argand and Gauss and Euler’s formula; it poses the twofold problem that gave rise to them, algebraic and geometric, which the example of the mean proportional between 11 and −1-1 sums up, and points out where they are used later, in this course, in other courses on the site and in science and technology.

Learning objectives

  1. Describe the content of each lesson of the unit and explain why they follow one another in that order.
  2. Recount, with its protagonists and their approximate dates, the transition of complex numbers from a computational device to a geometric object, from Cardano and Bombelli to Wessel, Argand, Gauss and Euler.
  3. State the algebraic problem and the geometric problem that gave rise to the polar form, and interpret as rotations the product by ii and the mean proportionals between 11 and −1-1.
  4. Identify the applications of the polar form in the following unit, in other courses on the site and in physics and engineering.

Entry assessment

Before working through the unit, it is advisable to measure the starting point. This self-assessment does not grade: it serves to show what has been mastered and where to slow down. The first ten questions cover the six content lessons of the unit: one the lesson on the polar form and another the one on geometry with complex numbers; two each the lessons on De Moivre’s formula, on roots and on polynomial equations; and two the one on the complex exponential, the second of them a modeling problem, which asks for a situation in context to be set up with phasors. The same skills are measured again at the end, in the synthesis lesson. The last two check what the unit takes as known from Course I and from the earlier units: the rectangular form, the conjugate and the modulus, the special values and the sum and difference formulas.

Entry assessment

Before starting the unit, measure your starting point. It is a short self-assessment: it is not graded and its result is stored only in your browser.

  1. Question 1What is the principal argument of z=−1−i3z = -1 - i\sqrt{3}?

  2. Question 2Compute (3+i)6\left( \sqrt{3} + i \right)^6.

  3. Question 3What is the linearization of sin⁡2θcos⁡2θ\sin^2 \theta \cos^2 \theta?

  4. Question 4Which of these numbers is a fourth root of −16-16?

  5. Question 5How many primitive ninth roots of unity are there?

  6. Question 6Which complex numbers zz satisfy ez=−ee^z = -e?

  7. Question 7Two loudspeakers emit the same pure tone. At the place where the listener is, the pressure variations they produce are 3cos⁡(ωt)3 \cos(\omega t) and 8cos⁡(ωt+2π3)8 \cos\left( \omega t + \frac{2\pi}{3} \right), in millipascals. What is the amplitude of the total pressure variation, in millipascals?

  8. Question 8What is the image of the point 3+i3 + i under the quarter-turn rotation, in the positive direction, about the point 1+i1 + i?

  9. Question 9Let z1z_1, z2z_2 and z3z_3 be the roots of z3−2z2+5z−1z^3 - 2z^2 + 5z - 1. Compute z12+z22+z32z_1^2 + z_2^2 + z_3^2.

  10. Question 10What are the roots of the equation t3−6t+2=0t^3 - 6t + 2 = 0 like?

  11. Question 11From earlier unitsWhat is the inverse of 3−4i3 - 4i?

  12. Question 12From earlier unitsWhat is the exact value of sin⁡π12\sin \frac{\pi}{12}?