Complex Number to Polar Form Calculator

Category: Algebra II
- May 14, 2025
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Convert complex numbers from rectangular form (a + bi) to polar form (r∠θ) and vice versa. Calculate magnitude, phase angle, and visualize the complex number on the complex plane.

Enter Complex Number

Rectangular (a + bi)
Polar (r∠θ)

Display Options

About Complex Numbers

Complex numbers extend the real number system by including the imaginary unit i, where i² = -1. They are typically written in the form a + bi, where a is the real part and b is the imaginary part.

Different Forms of Complex Numbers
  • Rectangular Form: z = a + bi
  • Polar Form: z = r∠θ, where r is magnitude and θ is the angle
  • Exponential Form: z = r·e^(iθ)
  • Trigonometric Form: z = r(cos(θ) + i·sin(θ))
Applications of Complex Numbers
  • Electrical Engineering: Analysis of alternating current circuits
  • Control Theory: System stability analysis
  • Signal Processing: Fourier transforms and frequency analysis
  • Quantum Mechanics: Wave functions and operators
  • Computer Graphics: Rotations and transformations
Formulas Used:

Magnitude (r): √(a² + b²)
Angle (θ): tan⁻¹(b/a)
Real Part (a): r·cos(θ)
Imaginary Part (b): r·sin(θ)

What is the Complex Number to Polar Form Calculator?

The Complex Number to Polar Form Calculator is a tool that helps you easily convert complex numbers between rectangular form (a + bi) and polar form (r∠θ). It also provides magnitude, phase angle, and different forms like exponential and trigonometric versions.

This calculator is useful for students, engineers, and anyone working with complex number operations or needing to understand polar and rectangular forms. It is part of a family of useful tools like the Complex Number Calculator, Inverse Function Calculator, and Polynomial Roots Calculator that help solve mathematical problems quickly and accurately.

Key Features

  • Convert between rectangular (a + bi) and polar (r∠θ) forms
  • Calculate magnitude and angle automatically
  • View results in exponential and trigonometric forms
  • Optional step-by-step solution display
  • Graphical visualization on the complex plane
  • Choose between degrees and radians for angle display
  • Select the number of decimal places for results

How to Use the Calculator

  • Select the input form: Rectangular or Polar
  • Enter the real and imaginary parts (for rectangular) or magnitude and angle (for polar)
  • Adjust display settings like angle unit and decimal places if needed
  • Click the Calculate button to see the results
  • Review the conversion results, calculation steps, and graph if enabled
  • Use the Reset button to start a new calculation

Why This Calculator is Useful

Working with complex numbers can often feel confusing, especially when you need to switch between forms or perform calculations. This calculator simplifies the process by doing the heavy lifting for you. It also connects to broader topics like using the De Moivre's Theorem Calculator for powers of complex numbers or the Complex Roots Calculator for solving equations.

Additionally, if you're involved in fields like electrical engineering, control theory, or signal processing, converting complex numbers to polar form is an essential task. This tool ensures accuracy and saves time when preparing complex arithmetic or transformations.

Frequently Asked Questions (FAQ)

What is a complex number?

A complex number is a combination of a real part and an imaginary part. It is written in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1.

What is polar form?

Polar form expresses a complex number using its magnitude (r) and an angle (θ). Instead of writing it as a + bi, you represent it as r∠θ, where r = √(a² + b²) and θ = tan⁻¹(b/a).

Can I convert from polar back to rectangular form?

Yes! You can use the calculator to find the rectangular form by using the formulas:
Real part (a) = r·cos(θ)
Imaginary part (b) = r·sin(θ)

What units are used for the angle?

You can choose between degrees and radians for the angle measurement, depending on your preference or the requirements of your work.

Where else are complex numbers used?

Complex numbers show up in many areas such as:

  • Electrical engineering for AC circuit analysis
  • Quantum mechanics for describing wave functions
  • Control theory for system stability
  • Signal processing for frequency domain analysis
  • Computer graphics for image transformations

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Final Thoughts

The Complex Number to Polar Form Calculator makes working with complex numbers fast, clear, and accurate. It is an essential tool when you need to convert complex to polar or explore complex number transformations. Whether you're solving for magnitudes, plotting points, or preparing for more advanced calculations like those done in the De Moivre's Theorem Calculator, this tool helps you complete the task with confidence.