Polar Form, Modulus, and Argument of Complex Numbers

The polar (trigonometric) form of a complex number is z = r(cos θ + i sin θ), where r = |z| is the modulus (distance from the origin) and θ = arg z is the argument (the angle from the positive real axis). In brief: for z = 1 + i, r = √2 and θ = π/4, so z = √2(cos π/4 + i sin π/4). The conversion formulas are a = r cos θ, b = r sin θ, r = √(a² + b²) and tan θ = b/a, with the quadrant of the point deciding the angle.

Polar form makes multiplication, division, powers and roots of complex numbers simple, since moduli multiply and arguments add. The key step is finding the correct argument, which requires attention to the quadrant. This guide explains the modulus and the argument, the conversion in both directions, the principal argument, gives worked examples for all four quadrants and the axes, and a table of standard values.

What is the modulus?

The modulus (absolute value) of z = a + bi is |z| = r = √(a² + b²), the distance of the point (a, b) from the origin in the complex plane. Properties: |z| ≥ 0, with |z| = 0 only for z = 0; |zw| = |z||w|; |z/w| = |z|/|w|; and |z|² = z · z̄.

What is the argument?

The argument of z ≠ 0 is an angle θ with cos θ = a/r and sin θ = b/r. It is determined only up to a multiple of 2π: if θ is an argument, so is θ + 2kπ. The principal argument, Arg z, is the one in the interval (−π, π] (some books use [0, 2π)). The argument of 0 is not defined.

The principal argument for the main cases:

• a > 0: Arg z = arctan(b/a), between −π/2 and π/2.
• a < 0 and b ≥ 0: Arg z = arctan(b/a) + π.
• a < 0 and b < 0: Arg z = arctan(b/a) − π.
• a = 0: Arg z = π/2 for b > 0 and −π/2 for b < 0.

Using the arctan alone gives a wrong angle when a < 0, because the calculator returns values between −90° and 90°.

How do you convert to polar form step by step?

1. Compute r = √(a² + b²).
2. Determine the quadrant of (a, b) from the signs.
3. Find the reference angle α = arctan(|b|/|a|).
4. Compute θ from the quadrant: I: α; II: π − α; III: −(π − α) or π + α; IV: −α.
5. Write z = r(cos θ + i sin θ). Check by converting back.

Example 1: quadrant I

z = 1 + i: r = √2, α = arctan 1 = π/4, so z = √2(cos π/4 + i sin π/4).

Example 2: quadrant II

z = −1 + √3 i: r = √(1 + 3) = 2, α = arctan(√3) = π/3, quadrant II so θ = π − π/3 = 2π/3. So z = 2(cos 2π/3 + i sin 2π/3). Check: 2(−1/2 + i√3/2) = −1 + √3 i ✓.

Example 3: quadrant III

z = −√3 − i: r = √(3 + 1) = 2, α = arctan(1/√3) = π/6, quadrant III so θ = −(π − π/6) = −5π/6 (or 7π/6). So z = 2(cos(−5π/6) + i sin(−5π/6)). Check: cos(−5π/6) = −√3/2 and sin(−5π/6) = −1/2, so 2(−√3/2 − i/2) = −√3 − i ✓.

Example 4: quadrant IV

z = 2 − 2i: r = 2√2, α = π/4, quadrant IV so θ = −π/4. So z = 2√2(cos(−π/4) + i sin(−π/4)).

Example 5: on the axes

z = −3: r = 3, θ = π, so z = 3(cos π + i sin π). z = −2i: r = 2, θ = −π/2. z = 5: r = 5, θ = 0.

Example 6: non-standard angle

z = 3 + 4i: r = 5, θ = arctan(4/3) ≈ 0.9273 rad ≈ 53.13°. So z ≈ 5(cos 53.13° + i sin 53.13°).

How do you convert back to algebraic form?

Evaluate the cosine and sine and multiply by r. For 4(cos 150° + i sin 150°): 4(−√3/2 + i/2) = −2√3 + 2i ≈ −3.46 + 2i. For 6(cos 270° + i sin 270°): 6(0 − i) = −6i.

The notation r cis θ is a shorthand for r(cos θ + i sin θ), and in exponential form z = re^iθ.

Table of standard conversions

Where does polar form come from?

The use of trigonometric form for complex numbers developed in the eighteenth century, in the work of Abraham de Moivre and Leonhard Euler, who linked complex numbers to trigonometry. The notation with the modulus and the argument became standard after Gauss’s geometric interpretation. The term “modulus” was introduced by Argand in 1814, and “argument” was used by Cauchy.

Where is it used today?

In electrical engineering, a sinusoidal voltage with amplitude V and phase φ is the phasor Ve^iφ, whose modulus is the amplitude and whose argument is the phase. A circuit with the impedance Z = 3 + 4 i ohms has the magnitude 5 ohms and the phase 53.13°, which means that the voltage leads the current by that angle. In navigation and robotics, polar form describes direction and distance.

Which mistakes are common?

Taking arctan(b/a) without checking the quadrant

For −1 + i, arctan(−1) = −π/4, but the correct argument is 3π/4.

Forgetting that the argument has multiple values

θ and θ + 2π describe the same number. State the principal argument, unless the context needs another one.

Mixing up r and θ in the formulas

The real part is r cos θ and the imaginary part is r sin θ.

Using degrees and radians together

Keep one unit in a computation. A mixture such as r(cos π/4 + i sin 45°) is correct only if the two are equal, but it invites errors.