Complex Numbers in Polar Form

Develop modulus and argument, de moivre and complex roots.

Lesson 3 of 32 parts Practice Flashcards

1/2

Modulus and Argument

Objective: Explain modulus and argument, apply the rule, and justify relevant restrictions.

For z=x+iy, modulus is |z|=√(x²+y²), and a nonzero z can be written r(cos θ+i sin θ) where θ is an argument. Arguments differ by 2πk; a principal argument chooses one specified interval. The zero complex number has modulus zero and no unique argument. Real and imaginary components are coordinates, not two separate moduli.

Formulas

$z=r(cos θ+i sin θ); r=√(x²+y²)$

Worked examples

Try itFind the modulus of z=21+28i.Show answer

Step 1: Use |z|=√((Re z)²+(Im z)²). Step 2: Compute √(1225). Step 3: The modulus is 35.

Swipe sideways to move between sections

Modulus and Argument

Objective: Explain modulus and argument, apply the rule, and justify relevant restrictions.

Formulas

$z=r(cos θ+i sin θ); r=√(x²+y²)$

Worked examples

Try itFind the modulus of z=21+28i.Show answer

Step 1: Use |z|=√((Re z)²+(Im z)²). Step 2: Compute √(1225). Step 3: The modulus is 35.

De Moivre and Complex Roots

Objective: Explain de moivre and complex roots, apply the rule, and justify relevant restrictions.

For an integer n and z=r(cos θ+i sin θ), De Moivre gives zⁿ=rⁿ(cos nθ+i sin nθ), with z≠0 required for negative n. The n distinct nth roots of a nonzero number have modulus r^(1/n) and arguments (θ+2πk)/n for k=0,…,n−1. Multiplication adds arguments; powers multiply them. Root angles are evenly spaced around a circle.

Formulas

$[r(cos θ+i sin θ)]ⁿ=rⁿ(cos nθ+i sin nθ)$

Worked examples

Try itFor z=3(cos θ+i sin θ), find the modulus of z^7.Show answer

Step 1: Use De Moivre: zⁿ=rⁿ(cos nθ+i sin nθ). Step 2: The new modulus is 3^7. Step 3: This equals 2187; the angle is multiplied by 7.

You finished the reading!

Lock it in with a few practice questions. This is the last lesson in the chapter.

Practice

Previous lessonParametric Equations and Motion

Last lesson