Complex Number to Polar Form Converter

Transform rectangular coordinates into polar representation instantly

Enter Complex Number (a + bi format)

Radians Degrees

Polar Form Results

Modulus (r): –

Argument (θ): –

Polar Form: –

Exponential Form: –

Quick Conversions

What is Polar Form?

The polar form represents a complex number using its distance from the origin (modulus) and the angle it makes with the positive real axis (argument). Instead of expressing a complex number as z = a + bi where a and b are rectangular coordinates, polar form expresses it as z = r(cos θ + i sin θ) or z = r e^iθ.

Polar Form: z = r(cos θ + i sin θ) = r cis θ

Exponential Form: z = r e^iθ

Where:

• r (modulus) = √(a² + b²)

• θ (argument) = atan2(b, a)

Conversion Formulas & Steps

Formulas

To convert from rectangular form a + bi to polar form, use these formulas:

Modulus: r = √(a² + b²)

Argument: θ = atan2(b, a)

The atan2 function handles all four quadrants correctly:

If a > 0: θ = arctan(b/a)
If a < 0 and b ≥ 0: θ = arctan(b/a) + π
If a < 0 and b < 0: θ = arctan(b/a) - π
If a = 0 and b > 0: θ = π/2
If a = 0 and b < 0: θ = -π/2

Conversion Steps

Identify the real part (a) and imaginary part (b) from your complex number a + bi
Calculate the modulus r using the formula r = √(a² + b²)
Calculate the argument θ using atan2(b, a) to get the correct quadrant
Write the polar form as r(cos θ + i sin θ) or r e^iθ
Convert angle to degrees if needed by multiplying by 180/π

Common Conversions

Rectangular Form	Modulus (r)	Argument (θ)	Polar Form
1 + 0i	1	0° (0 rad)	1(cos 0 + i sin 0)
0 + i	1	90° (π/2 rad)	1(cos π/2 + i sin π/2)
-1 + 0i	1	180° (π rad)	1(cos π + i sin π)
0 – i	1	-90° (-π/2 rad)	1(cos(-π/2) + i sin(-π/2))
1 + i	√2 ≈ 1.414	45° (π/4 rad)	√2(cos π/4 + i sin π/4)
√3 + i	2	30° (π/6 rad)	2(cos π/6 + i sin π/6)
3 + 4i	5	≈53.13° (≈0.927 rad)	5(cos 0.927 + i sin 0.927)
-1 + i	√2 ≈ 1.414	135° (3π/4 rad)	√2(cos 3π/4 + i sin 3π/4)

Real-World Applications

Electrical Engineering

AC circuit analysis uses polar form to represent impedance, making it easier to calculate phase angles and magnitudes in alternating current systems.

Signal Processing

Fourier transforms convert signals to frequency domain where polar form simplifies amplitude and phase calculations for audio and image processing.

Quantum Mechanics

Wave functions in quantum physics are represented as complex numbers, with polar form revealing probability amplitudes and phase relationships.

Control Systems

Transfer functions in control theory use polar coordinates for stability analysis and frequency response plots like Bode and Nyquist diagrams.

Computer Graphics

Rotations and transformations in 2D graphics are simplified using polar coordinates, making animation and geometric calculations more efficient.

Navigation Systems

GPS and radar systems use polar coordinates to represent direction and distance, making position tracking and course corrections more intuitive.

Advantages of Polar Form

Multiplication: When multiplying complex numbers in polar form, multiply the moduli and add the arguments:

z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

Division: When dividing, divide the moduli and subtract the arguments:

z₁/z₂ = (r₁/r₂)[cos(θ₁ – θ₂) + i sin(θ₁ – θ₂)]

Powers (De Moivre’s Theorem): Raising to a power becomes simple:

zⁿ = rⁿ[cos(nθ) + i sin(nθ)]

Roots: Finding nth roots is straightforward:

z^1/n = r^1/n[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], k = 0, 1, 2, …, n-1

Frequently Asked Questions

How do I convert a complex number to polar form by hand?

First, calculate the modulus using r = √(a² + b²). Then find the argument using θ = arctan(b/a), adjusting for the correct quadrant. Finally, write it as r(cos θ + i sin θ) or r e^iθ.

What is the difference between argument and principal argument?

The argument can have infinite values differing by 2π multiples. The principal argument is the unique value in the range (-π, π] or [0, 2π), depending on convention. Our calculator uses the range (-π, π].

Why use atan2 instead of arctan?

The atan2 function takes both x and y coordinates as separate arguments, automatically determining the correct quadrant. Regular arctan only gives values between -π/2 and π/2, requiring manual quadrant adjustments.

What is the polar form of zero?

Zero has a modulus of 0, but its argument is undefined since any angle works. By convention, it’s often written as 0(cos 0 + i sin 0), but technically the argument can be any value.

Can negative modulus values exist?

No, the modulus represents distance and is always non-negative. If you encounter a negative value, check your calculation. However, you can represent the same complex number with r > 0 and θ + π instead.

How do I convert back from polar to rectangular form?

Use a = r cos θ and b = r sin θ to get the rectangular form a + bi. Simply evaluate the trigonometric functions and multiply by the modulus.

Why is polar form useful for multiplication?

In polar form, multiplying complex numbers means multiplying their moduli and adding their arguments. This is much simpler than the FOIL method required in rectangular form, especially for multiple multiplications.

What are cis notation and Euler’s formula?

“cis θ” is shorthand for cos θ + i sin θ. Euler’s formula states that e^iθ = cos θ + i sin θ, allowing us to write polar form compactly as z = r e^iθ.