How to Write Complex Numbers in Polar Form with Worked Examples

Complex numbers become far easier to interpret when they are viewed geometrically. Instead of treating a number such as 3 + 4i only as an algebraic expression, polar form describes it by its distance from the origin and its angle from the positive real axis. This approach is especially useful in trigonometry, electrical engineering, signal processing, and any calculation involving roots, powers, or rotations.

What Is Polar Form?

A complex number is usually written in rectangular form as:

z = a + bi

Here, a is the real part, b is the imaginary part, and i is defined by i² = -1. In polar form, the same number is written as:

z = r(cos θ + i sin θ)

or, more compactly, as:

z = r cis θ

where cis θ means cos θ + i sin θ. In advanced settings, you may also see Euler’s form:

z = re^iθ

All three forms describe the same complex number. The difference is that polar form emphasizes geometry: r measures the number’s distance from zero, while θ measures its direction.

The Two Key Ingredients: Modulus and Argument

To convert z = a + bi into polar form, you need two values:

• Modulus: the distance from the origin to the point (a, b).
• Argument: the angle made with the positive real axis.

The modulus is calculated using the Pythagorean theorem:

r = √(a² + b²)

The argument is based on the tangent ratio:

tan θ = b/a

However, a serious caution is necessary: the tangent ratio alone does not always identify the correct quadrant. For example, both 45° and 225° have the same tangent value, but they point in opposite directions on the complex plane. Always check the signs of a and b.

Quadrants Matter

The point (a, b) lies in one of four quadrants:

• Quadrant I: a > 0, b > 0
• Quadrant II: a < 0, b > 0
• Quadrant III: a < 0, b < 0
• Quadrant IV: a > 0, b < 0

When using a calculator, the safest method is to use the atan2 function if available. It takes both coordinates into account and returns the angle in the correct quadrant. If you only use tan⁻¹(b/a), you may need to add 180° or π radians, depending on the quadrant.

Step-by-Step Method

To convert a complex number into polar form, follow this reliable process:

1. Identify a and b from z = a + bi.
2. Compute the modulus using r = √(a² + b²).
3. Find a reference angle using tan⁻¹(|b/a|), when a ≠ 0.
4. Determine the correct quadrant from the signs of a and b.
5. Write the final answer as r(cos θ + i sin θ).

Worked Example 1: Convert 3 + 4i

Let:

z = 3 + 4i

Here, a = 3 and b = 4. First calculate the modulus:

r = √(3² + 4²) = √(9 + 16) = √25 = 5

Now find the angle:

tan θ = 4/3

θ = tan⁻¹(4/3) ≈ 53.13°

Since both the real and imaginary parts are positive, the number lies in Quadrant I, so the angle is already correct.

Therefore:

3 + 4i = 5(cos 53.13° + i sin 53.13°)

In radians, this is approximately:

3 + 4i = 5(cos 0.9273 + i sin 0.9273)

Worked Example 2: Convert -1 + √3i

Let:

z = -1 + √3i

Here, a = -1 and b = √3. Calculate the modulus:

r = √((-1)² + (√3)²) = √(1 + 3) = 2

Now examine the angle. The reference angle satisfies:

tan α = √3/1 = √3

So:

α = 60°

But the complex number has a negative real part and a positive imaginary part, so it lies in Quadrant II. The Quadrant II angle is:

θ = 180° – 60° = 120°

Therefore:

-1 + √3i = 2(cos 120° + i sin 120°)

In radians:

-1 + √3i = 2(cos 2π/3 + i sin 2π/3)

Worked Example 3: Convert -2 – 2i

Let:

z = -2 – 2i

Here, a = -2 and b = -2. The modulus is:

r = √((-2)² + (-2)²) = √(4 + 4) = √8 = 2√2

The reference angle is:

α = tan⁻¹(|-2/-2|) = tan⁻¹(1) = 45°

Since both parts are negative, the point is in Quadrant III. Therefore:

θ = 180° + 45° = 225°

So the polar form is:

-2 – 2i = 2√2(cos 225° + i sin 225°)

In radians:

-2 – 2i = 2√2(cos 5π/4 + i sin 5π/4)

Special Cases to Know

Some complex numbers lie directly on an axis. These cases are straightforward but should be handled carefully:

• Positive real axis: If z = a and a > 0, then θ = 0°.
• Negative real axis: If z = a and a < 0, then θ = 180° or π.
• Positive imaginary axis: If z = bi and b > 0, then θ = 90° or π/2.
• Negative imaginary axis: If z = bi and b < 0, then θ = 270° or 3π/2.

The complex number 0 + 0i is a special exception. Its modulus is 0, but its argument is undefined because the origin has no direction.

Why Polar Form Is Useful

Polar form is not just a different notation; it changes how calculations are performed. When multiplying complex numbers in polar form, you multiply the moduli and add the arguments:

r₁ cis θ₁ · r₂ cis θ₂ = r₁r₂ cis(θ₁ + θ₂)

For division, you divide the moduli and subtract the arguments:

(r₁ cis θ₁)/(r₂ cis θ₂) = (r₁/r₂) cis(θ₁ – θ₂)

This is much cleaner than expanding products in rectangular form, especially when working with repeated powers or roots. For example, calculating (1 + i)⁸ is significantly easier after converting 1 + i into polar form.

Common Mistakes to Avoid

• Ignoring the quadrant: This is the most common error. Always check the signs of a and b.
• Confusing degrees and radians: Keep one unit throughout the problem unless conversion is required.
• Forgetting the modulus: Polar form requires both r and θ, not just the angle.
• Using tangent carelessly when a = 0: If the real part is zero, determine the angle from the axis directly.

Final Check

A good way to verify your answer is to convert it back into rectangular form. If z = r(cos θ + i sin θ), then:

a = r cos θ

b = r sin θ

For example, from the first worked example:

5 cos 53.13° ≈ 3

5 sin 53.13° ≈ 4

So the rectangular form is again 3 + 4i, confirming that the polar conversion is correct.

Writing complex numbers in polar form is ultimately a disciplined process: compute the modulus, determine the correct angle, and express the result clearly. Once the quadrant issue is handled properly, polar form becomes one of the most powerful and practical ways to work with complex numbers.