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MathidiaMathematics Academy
Complex numbers
Lesson 4 of 75
Challenge

Modulus and argument

Objective. Convert between Cartesian and polar information.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Every complex number z = a + bi can be written in modulus-argument (polar) form z = r(cos θ + i sin θ), where r = |z| = √(a²+b²) and θ = arg(z) is the angle such that cos θ = a/r and sin θ = b/r. This form is powerful because multiplying two complex numbers multiplies their moduli and adds their arguments, which is far easier than expanding brackets. For z = 1 + i, r = √2 and θ = π/4, so z = √2(cos(π/4) + i sin(π/4)). Converting between Cartesian and polar form requires checking which quadrant z lies in before quoting θ.

Worked example

Write z = −√3 + i in modulus-argument form, giving the argument in radians between −π and π.

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Illustrated card showing 3
  1. 1r = |z| = √((−√3)² + 1²) = √(3+1) = √4 = 2
  2. 2Reference angle: arctan(1/√3) = π/6
  3. 3Real part negative, imaginary part positive, so z is in the second quadrant: θ = π − π/6 = 5π/6
  4. 4z = 2(cos(5π/6) + i sin(5π/6))

Answer: z = 2(cos(5π/6) + i sin(5π/6))

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Common mistake. Students compute θ = arctan(b/a) without checking the quadrant, giving a wrong angle for negative a. For z = −1 + i, arctan(1/−1) gives −π/4 on a calculator, but the true argument is 3π/4.

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