Polar Form: Complex Numbers as Rotation and Scaling

Every complex number can be described two ways:

Rectangular form: z = a + bi (go right a, up b) Polar form: z = r(cos θ + i sin θ) (go distance r at angle θ)

Rectangular is natural for addition. Polar is natural for multiplication.

Switch between them depending on what you're doing. The insight: complex multiplication is rotation combined with scaling.

The Polar Coordinates

For z = a + bi:

r = |z| = √(a² + b²) — the modulus, distance from origin

θ = arg(z) = arctan(b/a) — the argument, angle from positive real axis

Polar form:

z = r(cos θ + i sin θ)

Also written z = r cis θ or z = r∠θ (the angle notation).

Converting Between Forms

Rectangular to polar: Given z = a + bi:

• r = √(a² + b²)
• θ = arctan(b/a), adjusted for quadrant

Polar to rectangular: Given z = r cis θ:

• a = r cos θ
• b = r sin θ
• z = r cos θ + ir sin θ

Example: Convert 1 + i to polar.

• r = √(1² + 1²) = √2
• θ = arctan(1/1) = π/4
• z = √2 cis(π/4) = √2(cos π/4 + i sin π/4)

Example: Convert 2 cis(π/3) to rectangular.

• a = 2 cos(π/3) = 2 · (1/2) = 1
• b = 2 sin(π/3) = 2 · (√3/2) = √3
• z = 1 + √3 i

Why Polar Form Matters

Multiplication becomes simple.

For z₁ = r₁ cis θ₁ and z₂ = r₂ cis θ₂:

z₁ · z₂ = r₁r₂ cis(θ₁ + θ₂)

Multiply the moduli. Add the arguments.

In rectangular form, (a + bi)(c + di) = (ac - bd) + (ad + bc)i. Four multiplications, messy.

In polar form: multiply two numbers, add two angles. Clean.

Proof of the Multiplication Rule

z₁z₂ = r₁(cos θ₁ + i sin θ₁) · r₂(cos θ₂ + i sin θ₂)

= r₁r₂[(cos θ₁ cos θ₂ - sin θ₁ sin θ₂) + i(sin θ₁ cos θ₂ + cos θ₁ sin θ₂)]

= r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

= r₁r₂ cis(θ₁ + θ₂)

The angle addition formulas from trigonometry do the work.

Division in Polar Form

z₁ / z₂ = (r₁/r₂) cis(θ₁ - θ₂)

Divide the moduli. Subtract the arguments.

Example: (2 cis π/3) / (4 cis π/6) = (2/4) cis(π/3 - π/6) = (1/2) cis(π/6).

No conjugate multiplication needed.

Powers: De Moivre's Theorem

For any complex number z = r cis θ and integer n:

zⁿ = rⁿ cis(nθ)

Raise the modulus to the power. Multiply the argument by n.

This is De Moivre's Theorem.

Example: (1 + i)⁸

First convert: 1 + i = √2 cis(π/4)

(1 + i)⁸ = (√2)⁸ cis(8 · π/4) = 16 cis(2π) = 16 cis(0) = 16

Verify: computing (1+i)² = 2i, (2i)² = -4, (-4)² = 16. ✓

De Moivre makes high powers trivial.

Roots in Polar Form

The nth roots of z = r cis θ are:

ⁿ√z = ⁿ√r · cis((θ + 2πk)/n) for k = 0, 1, 2, ..., n-1

There are n distinct nth roots, evenly spaced around a circle.

Example: Find the cube roots of 8.

8 = 8 cis(0).

Cube roots: ³√8 · cis((0 + 2πk)/3) = 2 cis(2πk/3) for k = 0, 1, 2.

• k = 0: 2 cis(0) = 2
• k = 1: 2 cis(2π/3) = 2(-1/2 + i√3/2) = -1 + i√3
• k = 2: 2 cis(4π/3) = 2(-1/2 - i√3/2) = -1 - i√3

Three cube roots, forming an equilateral triangle.

The Geometric Picture

Multiplying z by w = r cis θ:

• Scales z by factor r
• Rotates z counterclockwise by angle θ

This is why i² = -1:

• i = cis(π/2) — angle 90°
• i² = cis(π) — angle 180°
• cis(π) = -1

Multiplying by i is a quarter turn.

Special Angles

Memorizing these helps with quick calculations.

Complex Exponential Preview

Euler's formula says:

e^(iθ) = cos θ + i sin θ

So polar form becomes:

z = r · e^(iθ)

This is the exponential form — even cleaner than cis notation.

Multiplication: r₁e^(iθ₁) · r₂e^(iθ₂) = r₁r₂ · e^(i(θ₁+θ₂))

The exponent rules apply directly. Angles add because exponents add.

Applications

Signal processing: Signals are represented as A·e^(iωt), where A is amplitude and ω is frequency. Operations become simple.

AC circuits: Impedance is complex. Polar form gives magnitude (resistance) and phase directly.

Quantum mechanics: Wave functions have phase. Polar form separates magnitude (probability amplitude) from phase.

Rotations: Any 2D rotation is multiplication by e^(iθ). Rotation matrices reduce to complex multiplication.

The Principal Argument

The argument θ is only defined up to multiples of 2π. Adding 2π gives the same point.

The principal argument Arg(z) restricts θ to a standard range:

• Usually (-π, π] or [0, 2π)

This makes θ unique (except for z = 0, which has no well-defined argument).

Summary

Polar form: z = r cis θ = r(cos θ + i sin θ) = re^(iθ)

Modulus: r = |z| = distance from origin Argument: θ = arg(z) = angle from positive real axis

Operations:

• Multiply: multiply moduli, add arguments
• Divide: divide moduli, subtract arguments
• Power: raise modulus to power, multiply argument by power
• Roots: take root of modulus, divide argument by root degree (n roots)

Polar form reveals the geometry. Multiplication is rotation and scaling.

Further Reading

• Needham, T. Visual Complex Analysis. Geometric foundations.
• Nahin, P. An Imaginary Tale. Historical development.
• Marsden, J. Basic Complex Analysis. Undergraduate text.

This is Part 4 of the Complex Numbers series. Next: "Euler's Formula" — the equation that unifies exponentials and rotation.

Part 4 of the Complex Numbers series.

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