Complex Number Converter
Switch between Rectangular (a + bi) and Polar (r ∠ θ) forms.
Enter Complex Number z
Polar Form
Rectangular Form
Argand Diagram Visualization
Visual: The Green arrow is the complex number. Its shadow on X is the Real part, shadow on Y is the Imaginary part.
Understanding Complex Numbers
A Complex Number is a number that has two parts: a Real part and an Imaginary part. We can write it in two main ways, and being able to switch between them is crucial for engineering and physics.
Rectangular Form (a + bi)
Think of this like coordinates on a map. You go a steps East (Real axis) and b steps North (Imaginary axis).
z = 3 + 4i
Polar Form (r ∠ θ)
Think of this like radar. You face direction θ and walk distance r.
z = 5 (cos 53° + i sin 53°)
The Formulas
To convert from Rectangular to Polar, we use the Pythagorean theorem and trigonometry:
• r = √(a² + b²) (Modulus)
• θ = tan⁻¹(b/a) (Argument/Phase)
To convert back, we resolve the vector components:
• a = r cos(θ)
• b = r sin(θ)
Frequently Asked Questions (FAQ)
Q: What is ‘i’?
A: ‘i’ is the imaginary unit, defined as the square root of -1. It allows us to solve equations like x² + 1 = 0 which have no real solution.
Q: Why use Polar form?
A: Polar form makes multiplication and division incredibly easy. To multiply two complex numbers, you simply multiply their lengths (r) and add their angles (θ). This is vital in AC circuit analysis.
Q: What is Euler’s Formula?
A: It is the famous equation eiθ = cos(θ) + i sin(θ). It connects exponential growth to rotation, showing that Polar form is just a compact exponential notation: z = reiθ.