Complex Number to Polar Form Calculator

Convert a + bi to polar form r(cosθ + i·sinθ) or r·eiθ. Find magnitude and argument.

Enter Complex Number (a + bi)

Result

Polar Form
5(cos 53.13° + i sin 53.13°)
r(cosθ + i·sinθ)
Rectangular Form 3 + 4i
Magnitude (r) 5
Argument (degrees) 53.1301°
Argument (radians) 0.9273 rad
Exponential Form 5e^(0.9273i)
Quadrant I

Step-by-Step Solution

r = sqrt(a² + b²), theta = atan2(b, a)

Understanding Polar Form of Complex Numbers

Every complex number z = a + bi can be represented in polar form as z = r(cosθ + i·sinθ), where r is the magnitude (or modulus) and θ is the argument (or angle). This representation is especially useful for multiplication, division, and finding powers and roots of complex numbers.

The polar form leverages the geometric interpretation of complex numbers on the complex plane, where r is the distance from the origin to the point (a, b), and θ is the angle measured counterclockwise from the positive real axis.

Conversion Formulas

Magnitude (r)

The distance from the origin, computed using the Pythagorean theorem.

r = sqrt(a² + b²)

Argument (theta)

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

theta = atan2(b, a)

Trigonometric Form

Expresses the complex number using cosine and sine.

z = r(cos theta + i sin theta)

Exponential Form

Using Euler's formula, the most compact polar representation.

z = r * e^(i * theta)

Why Use Polar Form?

Polar form simplifies many operations with complex numbers:

  • Multiplication: Multiply magnitudes and add angles. z1 * z2 = r1*r2 * e^(i(theta1+theta2))
  • Division: Divide magnitudes and subtract angles. z1/z2 = (r1/r2) * e^(i(theta1-theta2))
  • Powers (De Moivre's Theorem): z^n = r^n * e^(i*n*theta)
  • Roots: The nth roots of a complex number are evenly spaced around a circle of radius r^(1/n).

Quadrant Considerations

When computing the argument, it is critical to consider which quadrant the complex number lies in. The basic arctan function only returns values between -90° and 90°, so adjustments are needed for quadrants II and III. The atan2(b, a) function handles all four quadrants correctly:

  • Quadrant I (a > 0, b > 0): theta = arctan(b/a)
  • Quadrant II (a < 0, b > 0): theta = arctan(b/a) + 180°
  • Quadrant III (a < 0, b < 0): theta = arctan(b/a) - 180°
  • Quadrant IV (a > 0, b < 0): theta = arctan(b/a) (negative result)

Special Cases

  • Real numbers (b = 0): Positive reals have theta = 0°, negative reals have theta = 180°.
  • Pure imaginary (a = 0): Positive imaginary has theta = 90°, negative imaginary has theta = -90° (or 270°).
  • Zero (a = 0, b = 0): r = 0 and the argument is undefined.

De Moivre's Theorem

One of the most powerful applications of polar form is De Moivre's Theorem, which states that for any integer n: [r(cosθ + i·sinθ)]^n = r^n(cos(nθ) + i·sin(nθ)). This makes computing powers of complex numbers straightforward and is essential for finding roots of complex numbers.