Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: A complex number a + bi can be written in polar form r(cos θ + i sin θ), where r is its distance from the origin and θ is its angle from the positive real axis. In polar form, multiplication multiplies moduli and adds angles, and powers follow De Moivre’s theorem.
Standards: N-CN.B.4, N-CN.B.5, N-CN.B.6, N-CN.A.3 · Level: Grade 12 Precalculus
How do you read a complex number on the complex plane?
The complex plane has a horizontal real axis and a vertical imaginary axis. The number z = a + bi sits at the point (a, b). Its distance from the origin is the modulus, and the angle from the positive real axis, measured counterclockwise, is the argument.
|z| = r = √(a2 + b2). The argument θ satisfies tan θ = ba with θ placed in the correct quadrant. The conjugate is z̄ = a − bi, the mirror image across the real axis, and z·z̄ = |z|2.
How do you switch between rectangular and polar form?
z = r(cos θ + i sin θ), so a = r cos θ and b = r sin θ.
- Find r = √(a2 + b2).
- Find the reference angle tan⁻¹(|b| ÷ |a|).
- Use the signs of a and b to pick the quadrant and get θ.
- Write r(cos θ + i sin θ).
For −1 + √3i: r = √(1 + 3) = 2. The reference angle is tan⁻¹(√3) = 60°, and Quadrant II gives θ = 180° − 60° = 120°. So −1 + √3i = 2(cos 120° + i sin 120°).
How do you multiply, divide and take powers in polar form?
| Operation | Rule for z₁ = r₁∠θ₁ and z₂ = r₂∠θ₂ |
|---|---|
| Product | r₁r₂ ∠ (θ₁ + θ₂) |
| Quotient | r₁r₂ ∠ (θ₁ − θ₂) |
| Power (De Moivre) | rn ∠ nθ |
| Conjugate | r ∠ (−θ) |
Adding the moduli when multiplying. Moduli multiply; only the angles add.
How do you find distance and midpoint between two complex numbers?
For z₁ = a + bi and z₂ = c + di, the distance is |z₂ − z₁| = √((c − a)2 + (d − b)2), and the midpoint is z₁ + z₂2. For A = −3 + i and B = 2 + 4i, the distance is √(25 + 9) = √34 ≈ 5.83, and the midpoint is −12 + 52i.
Think of multiplying by r∠θ as a stretch by r followed by a turn of θ. Draw the arrow first, then check your algebra against it.
- Modulus: |a + bi| = √(a2 + b2); conjugate: a − bi.
- Polar form: z = r(cos θ + i sin θ) with a = r cos θ and b = r sin θ.
- Product: multiply moduli, add arguments. Quotient: divide moduli, subtract arguments.
- De Moivre: [r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ).
- Distance between two points in the complex plane is the modulus of their difference.
Polar Form and Powers of Complex Numbers: frequently asked questions
What is the difference between the modulus and the argument?
The modulus is the length of the arrow from the origin to the number, a non-negative real value. The argument is the angle of that arrow, measured counterclockwise from the positive real axis. Together they locate the number just like coordinates do.
Why do the angles add when two complex numbers are multiplied?
Multiplying cos α + i sin α by cos β + i sin β and using the sum formulas for sine and cosine gives cos(α + β) + i sin(α + β). Angles add because each factor rotates the arrow.
Can the argument of a complex number be negative?
Yes. An argument of −60° points to the same direction as 300°. Any two arguments that differ by a multiple of 360° describe the same number, so a final answer should name the range used.
What is De Moivre’s theorem used for?
It turns a power of a complex number in polar form into a single step: raise the modulus to the power and multiply the angle by the exponent. It also leads to the n distinct nth roots of any nonzero complex number.
How do I find the argument in Quadrant II or III?
Compute the reference angle with the inverse tangent of |b| ÷ |a|. Then subtract it from 180° in Quadrant II, add it to 180° in Quadrant III, or subtract it from 360° in Quadrant IV.
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