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
Complex Number Calculator
Enter complex numbers and choose an operation to see the result.
This calculator is also known as Polar to Rectangular Converter.
Read the complete guideRectangular 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.
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