Put Complex Number in Polar Form Calculator

A complex number in polar form represents a complex number using a magnitude (or modulus) and an angle (or argument). This form is particularly useful in applications involving rotation, waves, and periodic phenomena. Our calculator converts standard complex numbers to their polar form representation.

What is Polar Form?

Polar form expresses a complex number as the product of its magnitude and a complex exponential involving its angle. The general form is:

z = r(cosθ + i sinθ) = r e^iθ

Where:

z is the complex number
r is the magnitude (distance from origin)
θ is the angle (argument) in radians
i is the imaginary unit (√-1)

This representation is particularly valuable in physics, engineering, and signal processing where rotational symmetry is important.

How to Convert to Polar Form

To convert a complex number from standard form (a + bi) to polar form, follow these steps:

Identify the real part (a) and imaginary part (b) of the complex number
Calculate the magnitude (r) using the Pythagorean theorem: r = √(a² + b²)
Determine the angle (θ) using the arctangent function: θ = arctan(b/a)
Adjust the angle based on the quadrant of the complex number
Express the number in polar form using r and θ

Note: The angle θ is typically measured in radians. For degrees, convert using π/180.

Formula

The conversion formulas are:

r = √(a² + b²)
θ = arctan(b/a) (with quadrant adjustment)

Where:

a is the real part
b is the imaginary part
r is the magnitude
θ is the angle in radians

Worked Example

Let's convert the complex number 3 + 4i to polar form:

Identify a = 3 and b = 4
Calculate magnitude: r = √(3² + 4²) = √(9 + 16) = √25 = 5
Calculate angle: θ = arctan(4/3) ≈ 0.927 radians (53.13°)
The complex number is in the first quadrant, so no adjustment is needed
Polar form: 5(cos(0.927) + i sin(0.927)) or 5 e^i0.927

Result: The complex number 3 + 4i in polar form is approximately 5(cos(0.927) + i sin(0.927)).

FAQ

What is the difference between polar and rectangular form?

Rectangular form (a + bi) represents complex numbers using real and imaginary components, while polar form (r e^iθ) uses magnitude and angle. Polar form is often more intuitive for operations involving rotation.

How do I convert polar form back to rectangular form?

Use the formulas: a = r cosθ and b = r sinθ. For the example above, 3 = 5 cos(0.927) and 4 = 5 sin(0.927).

What are the units for the angle in polar form?

The angle is typically measured in radians, though degrees can be used with appropriate conversion. Our calculator uses radians by default.

When is polar form most useful?

Polar form is particularly useful in physics for wave equations, engineering for rotational systems, and signal processing for phase analysis.