Math

Polar to Rectangular Converter

Converting Between Polar and Rectangular Form

Complex numbers can be represented in rectangular form (a + bi) or polar form (r∠θ). Converting between the two is essential for solving problems in trigonometry, engineering, and physics. Use r = √(a²+b²) and θ = atan2(b,a) for rectangular to polar, and a = r·cosθ, b = r·sinθ for polar to rectangular.

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Signal Addition

Add two phasor signals in complex form to find the resultant.

Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition

Circuit Impedance

Divide two impedances to compute the transfer ratio.

Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division

De Moivre Power

Raise a complex number to the 10th power using De Moivre's theorem.

Key values: z₁ = 1 + i · n = 10 · Power

Cube Roots of −8

Find all three cube roots of −8, visualized as equidistant points on a circle.

Key values: z₁ = −8 + 0i · n = 3 · nth Root

Documentation

This calculator is also known as Polar to Rectangular Converter.

Read the complete guide

Rectangular vs. Polar Form

Every complex number z = a + bi can be written in polar form as z = r(cosθ + i·sinθ) = r·cis(θ), where r = |z| is the modulus and θ = arg(z) is the argument. Polar form makes multiplication, division, powers, and roots much simpler.

Conversion Formulas

The two-way conversion uses these formulas:

Rectangular to polar: r = √(a² + b²), θ = atan2(b, a)
Polar to rectangular: a = r·cosθ, b = r·sinθ

Examples

Convert 3 + 4i to Polar Form

Find the modulus and argument of z = 3 + 4i.

r = √(9 + 16) = 5, θ = atan2(4, 3) ≈ 53.13°. So z = 5∠53.13°.

Key takeaway: The 3-4-5 Pythagorean triple makes this a clean conversion. Not all conversions yield simple values.

Efficient Form Conversion

Tips for converting between forms:

Use atan2(b, a) instead of atan(b/a) to handle all four quadrants correctly
Remember that argument is measured from the positive real axis, counterclockwise
Euler form r·e^(iθ) is equivalent to r·cis(θ) and is the most compact notation

Frequently Asked Questions about Polar to Rectangular Converter

When should I use polar form?

Polar form is best for multiplication, division, powers, and roots because moduli multiply/divide and arguments add/subtract. Rectangular form is better for addition and subtraction.

What is the argument of a negative real number?

A negative real number like −5 has argument π radians (180°). It lies on the negative real axis.

Specialized Calculators

Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.

Operation

11 Calculators

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Complex Number Addition Calculator | Add Complex Numbers Online

Add two complex numbers with step-by-step results in rectangular and polar form.