Polar Form to Complex Number Calculator

Category: Algebra II

- April 26, 2025

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Convert between polar form (r∠θ) and rectangular form (a + bi) of complex numbers, and perform operations in either representation.

Input Complex Number

Input Form:

Magnitude (r):

Angle (θ):

Operations

⟷ Convert

z×n Multiply by n

zⁿ Raise to Power

ⁿ√z Find Roots

Display Options

Decimal Places:

Show steps

Show visualization

About Complex Numbers and Polar Form

Complex numbers can be represented in two main forms: rectangular form (a + bi) and polar form (r∠θ or re^iθ). These different representations are useful for different types of operations.

Conversion Between Forms

Rectangular to Polar:
- Magnitude (r): r = √(a² + b²)
- Angle (θ): θ = tan^-1(b/a) (with appropriate quadrant adjustments)
Polar to Rectangular:
- Real part (a): a = r × cos(θ)
- Imaginary part (b): b = r × sin(θ)

Advantages of Polar Form

Polar form makes certain operations particularly simple:

Multiplication: To multiply complex numbers, multiply their magnitudes and add their angles.
Division: Divide the magnitudes and subtract the angles.
Powers: Using De Moivre's formula: (r∠θ)ⁿ = rⁿ∠(nθ)
Roots: The n^th roots of r∠θ are given by: r^1/n∠((θ + 2πk)/n) for k = 0, 1, 2, ..., n-1

Applications

Polar representation of complex numbers is particularly useful in:

Electrical engineering and AC circuit analysis
Signal processing and Fourier transforms
Control systems and stability analysis
Fluid dynamics and aerodynamics
Computer graphics and geometric transformations

Formulas Used

Rectangular to Polar:
- Magnitude: \( r = \sqrt{a^2 + b^2} \)
- Angle: \( \theta = \tan^{-1}(b/a) \)
Polar to Rectangular:
- Real Part: \( a = r \cos(\theta) \)
- Imaginary Part: \( b = r \sin(\theta) \)
De Moivre's Theorem:
- \( (r\angle\theta)^n = r^n\angle(n\theta) \)
Roots of Complex Numbers:
- \( r^{1/n}\angle\left(\frac{\theta + 2\pi k}{n}\right), \, k=0,1,...,n-1 \)

About the Polar Form to Complex Number Calculator

The Polar Form to Complex Number Calculator is a simple tool that helps you convert numbers between polar coordinates (r∠θ) and rectangular form (a + bi). It also enables you to multiply complex numbers, raise them to a power, and find their roots. This calculator is a useful addition if you often work with imaginary number calculations or need a quick and accurate way to solve complex number operations.

Key Features

Convert between polar and rectangular form quickly.
Multiply complex numbers by a scalar.
Raise a complex number to any power using De Moivre's formula.
Find all n-th roots of a complex number, neatly visualized.
Display step-by-step solutions for better understanding.
Visualize results on the complex plane for clarity.

How to Use the Calculator

Choose the input form: Select either "Polar Form" or "Rectangular Form."
Enter the values: Input the magnitude and angle (for polar) or real and imaginary parts (for rectangular).
Select an operation:
- Convert
- Multiply by n
- Raise to Power
- Find Roots
Optional: Adjust decimal precision, and choose whether to show steps and a visual graph.
Click "Calculate": Instantly see results along with a clear explanation and optional complex plane visualization.

Why This Calculator is Helpful

Understanding and manipulating complex numbers is essential in fields such as engineering, Physics, and computer graphics. This calculator simplifies tasks that would otherwise involve multiple steps and reduces the chances of manual errors. It complements Other helpful Math tools such as the Complex Number Calculator, De Moivre's Theorem Calculator, and Complex Roots Calculator.

Related Tools You Might Like

If you find this calculator useful, you might also benefit from:

Complex Number to Polar Form Calculator: Convert complex numbers to polar form easily.
Polynomial Roots Calculator: Find roots of polynomial equations including complex roots.
Inverse Function Calculator: Find and solve for function inverses effortlessly.
Logarithm Calculator: Solve logarithmic functions and base conversions quickly.
Midpoint Calculator: Determine the midpoint between two points efficiently.

Frequently Asked Questions (FAQ)

What is the difference between polar and rectangular form?

Rectangular form expresses a complex number with real and imaginary parts (a + bi), while polar form uses a magnitude and an angle (r∠θ).

Can I use degrees or radians for the angle?

Yes, you can choose between degrees and radians when entering the angle for polar coordinates.

What happens when I multiply or raise a complex number to a power?

Multiplying a complex number changes its magnitude and keeps or adjusts the angle. Raising a complex number to a power uses De Moivre's formula to calculate the result accurately.

How are roots of complex numbers displayed?

The calculator shows all possible roots, spaced evenly around the circle, with clear values for each root in both rectangular and polar forms.

Which fields often use complex numbers?

Fields like electrical engineering, signal processing, fluid dynamics, and computer graphics frequently work with complex numbers.

Summary

The Polar Form to Complex Number Calculator is a practical and user-friendly tool for anyone needing to convert polar to complex or perform complex number operations with ease. It fits perfectly alongside other useful tools like the Inverse Hyperbolic Sine Calculator, Logarithm Calculator, Operations on Functions Calculator, and many more. Use it today to streamline your complex number calculations!