Complex Number Polar Form Calculator

Convert complex numbers between rectangular (a + bi) and polar form r(cos theta + i sin theta).

Enter Complex Number

3 + 4i

Results

Polar Form (Trigonometric)
--
Exponential Form
--
Modulus (r) --
Argument theta (radians) --
Argument theta (degrees) --
Quadrant --
Conjugate --
|z|^2 --

Step-by-Step Solution

Understanding Complex Number Polar Form

Every complex number z = a + bi can be represented in polar form as z = r(cos theta + i sin theta), where r is the modulus (distance from the origin) and theta is the argument (angle from the positive real axis). This representation is also written using Euler's formula as z = re^(i*theta).

Key Formulas

Modulus (r)

The distance from the origin in the complex plane.

r = sqrt(a^2 + b^2)

Argument (theta)

The angle from the positive real axis, using atan2 for correct quadrant.

theta = atan2(b, a)

Trigonometric Form

The polar representation using cosine and sine.

z = r(cos theta + i sin theta)

Exponential Form

Using Euler's formula e^(i*theta) = cos theta + i sin theta.

z = r * e^(i * theta)

De Moivre's Theorem

For computing powers of complex numbers.

z^n = r^n(cos(n*theta) + i*sin(n*theta))

Multiplication in Polar

Multiply moduli and add arguments.

z1*z2 = r1*r2 * e^(i(t1+t2))

Converting Between Forms

Rectangular to Polar

  1. Calculate the modulus: r = sqrt(a^2 + b^2)
  2. Calculate the argument: theta = atan2(b, a)
  3. Write in polar form: z = r(cos theta + i sin theta)

Polar to Rectangular

  1. Calculate the real part: a = r cos theta
  2. Calculate the imaginary part: b = r sin theta
  3. Write in rectangular form: z = a + bi

Applications

Polar form is essential in electrical engineering (AC circuit analysis, phasors), signal processing (Fourier transforms), quantum mechanics (wave functions), and any field dealing with oscillations and rotations. The exponential form makes multiplication, division, and exponentiation of complex numbers much simpler.