How do you convert a complex number to polar form?

Given a complex number z = a + bi, the polar form is z = r∠θ (or r(cos θ + i sin θ), also written as re^(iθ)). The modulus r equals the square root of (a² + b²). The argument θ equals atan2(b, a), which is the angle from the positive real axis. For example, z = 3 + 4i gives r = sqrt(9 + 16) = 5, and θ = atan2(4, 3) ≈ 53.13°. So the polar form is 5∠53.13°.

What is the difference between the modulus and the argument of a complex number?

The modulus (r) is the distance from the origin to the point (a, b) on the complex plane. It is always a non-negative real number and is calculated as r = sqrt(a² + b²). The argument (θ) is the angle the line from the origin to the point makes with the positive real axis, measured anticlockwise. It is usually expressed in radians in the range (-π, π] or in degrees (-180°, 180°]. Together, r and θ fully describe the position of a complex number.

Why does converting to polar form use atan2 rather than plain arctan?

The plain arctan function only returns values in the range (-90°, 90°), covering only the first and fourth quadrants. The atan2(b, a) function takes both the real part (a) and the imaginary part (b) as separate arguments and returns the correct angle in all four quadrants, ranging from -180° to 180°. For example, atan(4/(-3)) gives about -53.13°, but atan2(4, -3) gives the correct 126.87° for a complex number in the second quadrant.

Complex Number to Polar Form Calculator

Convert a complex number from rectangular form (a + bi) to polar form (r∠θ). Enter the real part and imaginary part to instantly get the modulus (r), argument (θ) in degrees or radians, and the equivalent exponential form.

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Standard formula r = √(a² + b²), θ = atan2(b, a). Works for all quadrants.

1. Enter the Complex Number

Real part (a) the number without i

Please enter a valid number.

Imaginary part (b) the coefficient of i

Please enter a valid number.

Angle unit for output

2. Input Summary

Complex number 3 + 4i

Real part (a) 3

Imaginary part (b) 4

Quadrant I (a > 0, b > 0)

Polar Form

5∠53.13°

5e^i0.9273

Polar Form Results

Modulus (r)
5
r = √(a² + b²)

Argument (θ) in Degrees

53.13°

atan2(b, a)

Argument (θ) in Radians

0.9273 rad

θ × π / 180

Argument as π fraction

0.2952π

θ / π radians

Step-by-Step Working

Step 1: a²9

Step 2: b²16

Step 3: a² + b²25

Step 4: r = √(a² + b²)5

Step 5: θ = atan2(b, a)53.1301°

Polar form5∠53.1301°

Alternative Notations

Phasor notation5∠53.13°

Trigonometric form5(cos 53.13° + i sin 53.13°)

Euler form (re^iθ)5e^i0.9273

Modulus squared (|z|²)25

Conjugate3 - 4i

|z|5

Result: The complex number 3 + 4i has modulus 5 and argument 53.13°. In polar form: 5∠53.13°.

What Is Polar Form?

Every complex number z = a + bi can be represented as a point on the complex plane, with the real part a on the horizontal axis and the imaginary part b on the vertical axis. Polar form describes this same point using two values: the distance from the origin (the modulus r) and the angle from the positive real axis (the argument θ).

Polar form is written as z = r∠θ, or equivalently as z = r(cos θ + i sin θ), or in Euler notation as z = re^(iθ). All three forms are equivalent and interchangeable.

The Conversion Formulas

To convert from rectangular form (a + bi) to polar form (r∠θ):

Quantity	Formula	Description
Modulus (r)	r = √(a² + b²)	Distance from the origin; always ≥ 0
Argument (θ) in radians	θ = atan2(b, a)	Angle from positive real axis, range (−π, π]
Argument (θ) in degrees	θ = atan2(b, a) × (180/π)	Range (−180°, 180°]

The atan2(b, a) function is used rather than plain arctan(b/a) because it correctly handles all four quadrants, including when a is zero or negative.

Worked Example

Convert z = 3 + 4i to polar form.

Calculate a² = 3² = 9
Calculate b² = 4² = 16
Sum: a² + b² = 9 + 16 = 25
Modulus: r = √25 = 5
Argument: θ = atan2(4, 3) ≈ 0.9273 radians ≈ 53.13°
Polar form: z = 5∠53.13° (or 5e^i0.9273)

With the default inputs (a = 3, b = 4), this calculator gives exactly these results: modulus 5 and argument 53.13°.

Quadrant Reference

Quadrant	Signs of a, b	Angle range (degrees)
I	a > 0, b > 0	0° to 90°
II	a < 0, b > 0	90° to 180°
III	a < 0, b < 0	-180° to -90°
IV	a > 0, b < 0	-90° to 0°

Why Use Polar Form?

Polar form makes certain operations with complex numbers much simpler. Multiplication of two complex numbers in polar form is carried out by multiplying the moduli and adding the arguments: (r1∠θ1) × (r2∠θ2) = r1r2∠(θ1 + θ2). Division is equally straightforward: divide the moduli and subtract the arguments. Raising a complex number to a power uses De Moivre's theorem: (r∠θ)^n = r^n ∠ nθ. These operations would require expanding brackets in rectangular form, making polar form far more efficient for engineering, signal processing, and physics applications.

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Method: Modulus calculated as r = sqrt(a² + b²) using the Pythagorean theorem. Argument calculated using JavaScript Math.atan2(b, a) which correctly handles all quadrants and returns values in the range (-π, π]. Degrees converted from radians by multiplying by 180/π.

This calculator computes the exact modulus and argument for any complex number with real and imaginary parts you enter. Results are displayed to 4 decimal places. For very large or very small numbers, the result is computed using standard IEEE 754 double-precision floating point arithmetic.

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