Unit 7 of 10

Complex Polar Form and Polar Coordinates

  • 7 lessons
  • Parts 169 to 182

Polar coordinates locate a point by its distance from the origin and its angle instead of by horizontal and vertical steps. This unit plots points in polar form, converts between polar and rectangular coordinates, and uses the same idea to write complex numbers by modulus and argument.

In polar form, multiplying complex numbers means multiplying lengths and adding angles, which leads to De Moivre’s theorem and the roots of unity. The unit ends with polar graphs, including limaçons, cardioids, roses and lemniscates.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Polar Coordinates

  1. Introduction to Polar Coordinates

    How to name a point by its distance from a fixed center and an angle, what a negative radius means, and why every point has infinitely many polar names.

    Theory part 169Practice part 170

  2. Converting Between Polar and Rectangular Coordinates

    How to move points and equations between polar and rectangular form, how to choose the angle in the right quadrant, and how to check that a conversion neither adds nor loses points.

    Theory part 171Practice part 172

Complex Numbers in Polar Form

  1. Polar Form of Complex Numbers

    How to write a nonzero complex number by its distance from zero and an angle, why that angle is not unique, which angle is the principal one, and how to convert back to rectangular form.

    Theory part 173Practice part 174

  2. Products and Quotients in Polar Form

    Why multiplying complex numbers multiplies their moduli and adds their arguments, why dividing by a nonzero number divides and subtracts them, and how to see multiplication as a rotation and a scaling.

    Theory part 175Practice part 176

  3. De Moivre’s Theorem and Roots of Unity

    How to raise a complex number in polar form to any integer power, how to find all n of its nth roots, and why those roots sit evenly spaced at the corners of a regular polygon.

    Theory part 177Practice part 178

Polar Graphs

  1. Polar Graphs of Limacons and Cardioids

    How to recognize a limaçon from its equation, predict whether it has an inner loop, a cusp or a dent, use symmetry tests to halve the work, and find its farthest point from the pole.

    Theory part 179Practice part 180

  2. Polar Graphs of Roses and Lemniscates

    How the petal count of a rose depends on whether its angular frequency is odd or even, how negative radii draw petals on the far side, and which angles give real points on a lemniscate.

    Theory part 181Practice part 182