Polar Form of a Complex Number

Polar Form of a Complex Number

The polar coordinates of a complex number on the complex plane. The polar form of a complex number is written in any of the following forms: rcos θ + irsin θ, r(cos θ + isin θ), or rcis θ. In any of these forms r is called the modulus or absolute value. θ is called the argument.

Complex plane diagram showing point a+bi with modulus (radius) and argument (angle θ) from real axis labeled.

Key Formula

z = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta

Where:

$z$ = The complex number
$r$ = The modulus (absolute value), equal to √(a² + b²), where a + bi is the rectangular form
$\theta$ = The argument (angle measured counterclockwise from the positive real axis), found using θ = tan⁻¹(b/a) with quadrant adjustment
$i$ = The imaginary unit, where i² = −1

Worked Example

Problem: Convert the complex number z = 1 + i√3 to polar form.

Step 1: Identify a and b from the rectangular form a + bi.

a = 1, \quad b = \sqrt{3}

Step 2: Calculate the modulus r using the formula r = √(a² + b²).

r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2

Step 3: Find the argument θ. Since a > 0 and b > 0, the number is in Quadrant I, so use θ = tan⁻¹(b/a) directly.

\theta = \tan^{-1}\!\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}

Step 4: Write the polar form by substituting r and θ into the formula.

z = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)

Answer: z = 2(cos π/3 + i sin π/3), or equivalently 2 cis(π/3).

Another Example

This example involves a complex number in Quadrant III, requiring careful quadrant adjustment when computing the argument — a step where many students make errors.

Problem: Convert z = −3 − 3i to polar form.

Step 1: Identify a and b from the rectangular form.

a = -3, \quad b = -3

Step 2: Calculate the modulus.

r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

Step 3: Find the reference angle using tan⁻¹(|b/a|).

\text{reference angle} = \tan^{-1}\!\left(\frac{3}{3}\right) = \tan^{-1}(1) = \frac{\pi}{4}

Step 4: Since both a and b are negative, the complex number lies in Quadrant III. Add π to the reference angle to get the correct argument.

\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}

Step 5: Write the polar form.

z = 3\sqrt{2}\left(\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4}\right)

Answer: z = 3√2 (cos 5π/4 + i sin 5π/4), or 3√2 cis(5π/4).

Frequently Asked Questions

How do you convert a complex number from rectangular form to polar form?

Find the modulus r = √(a² + b²) and the argument θ = tan⁻¹(b/a), adjusting θ based on which quadrant the point (a, b) falls in. Then write the number as r(cos θ + i sin θ). The quadrant check is essential because tan⁻¹ alone only returns values between −π/2 and π/2.

Why is polar form useful for multiplying and dividing complex numbers?

When you multiply two complex numbers in polar form, you simply multiply their moduli and add their arguments: r₁ cis θ₁ · r₂ cis θ₂ = r₁r₂ cis(θ₁ + θ₂). For division, you divide the moduli and subtract the arguments. This is far simpler than expanding products using FOIL in rectangular form.

What is the difference between polar form and exponential form of a complex number?

Polar form uses trigonometric functions: r(cos θ + i sin θ). Exponential form uses Euler's formula to write the same number as re^(iθ). They represent the same thing — exponential form is just a more compact notation that follows directly from Euler's identity e^(iθ) = cos θ + i sin θ.

Polar Form vs. Rectangular Form

	Polar Form	Rectangular Form
Notation	r(cos θ + i sin θ) or r cis θ	a + bi
Components	Modulus r and argument θ	Real part a and imaginary part b
Best for	Multiplication, division, powers, and roots	Addition and subtraction
Geometric meaning	Distance from origin and angle from positive real axis	Horizontal and vertical displacement on the complex plane
Conversion	a = r cos θ, b = r sin θ	r = √(a² + b²), θ = tan⁻¹(b/a)

Why It Matters

Polar form appears throughout precalculus and introductory college math whenever you need to multiply, divide, or raise complex numbers to powers. De Moivre's Theorem, which requires polar form, lets you compute z^n or find nth roots of complex numbers efficiently. Electrical engineering and physics also rely heavily on polar representation of complex numbers to describe waves, impedance, and phasors.

Common Mistakes

Mistake: Using θ = tan⁻¹(b/a) without adjusting for the correct quadrant.

Correction: The arctangent function only returns angles in (−π/2, π/2). If the complex number is in Quadrant II or III (a < 0), you must add π to the arctangent result. If it is in Quadrant IV, you can add 2π to get a positive angle. Always plot the point to verify which quadrant it belongs to.

Mistake: Forgetting that the modulus r must always be non-negative.

Correction: The modulus is defined as r = √(a² + b²), which is always ≥ 0. If you get a negative value, you have made an arithmetic error. A negative sign belongs in the angle, not the modulus.

Related Terms

Complex Numbers — The numbers being expressed in polar form
Complex Plane — The coordinate plane where polar form is visualized
Absolute Value of a Complex Number — The modulus r in the polar form
Argument of a Complex Number — The angle θ in the polar form
Cis — Shorthand notation for cos θ + i sin θ
De Moivre's Theorem — Uses polar form to compute powers and roots
Polar Coordinates — The coordinate system polar form is based on
Polar-Rectangular Conversion Formulas — Formulas for switching between the two forms