Polar Form Calculator
Convert complex numbers between rectangular (a + bi) and polar (r × e^(iφ)) forms
Complex Number Conversion
The real component of the complex number
The imaginary component (coefficient of i)
Conversion Results
Rectangular Form
Real part (a): 3.0000
Imaginary part (b): 4.0000
Polar Form
Magnitude (r): 5.0000
Phase (φ): 0.9273 radians
Phase (φ): 53.1301°
Trigonometric Form
cos(φ): 0.6000
sin(φ): 0.8000
Conversion Formulas
Rectangular to Polar:
• r = √(a² + b²)
• φ = atan2(b, a)
Polar to Rectangular:
• a = r × cos(φ)
• b = r × sin(φ)
Step-by-Step Calculation
r = √(a² + b²) = √((3)² + (4)²)
r = √(9.0000 + 16.0000) = 5.0000
φ = atan2(b, a) = atan2(4, 3)
φ = 0.9273 radians = 53.1301°
Example Calculations
Rectangular to Polar
Given: z = 3 + 4i
Magnitude: r = √(3² + 4²) = √25 = 5
Phase: φ = atan2(4, 3) ≈ 0.927 radians
Result: z = 5 × exp(0.927i)
Special Cases
• Real numbers: z = a (φ = 0 or π)
• Imaginary numbers: z = bi (φ = π/2 or -π/2)
• Unity: z = 1 (r = 1, φ = 0)
• Negative unity: z = -1 (r = 1, φ = π)
Polar to Rectangular
Given: z = 2 × exp(π/4 i)
Real: a = 2 × cos(π/4) = 2 × √2/2 ≈ 1.414
Imaginary: b = 2 × sin(π/4) = 2 × √2/2 ≈ 1.414
Result: z = 1.414 + 1.414i
Applications
• Electrical engineering (AC circuits)
• Signal processing (Fourier analysis)
• Quantum mechanics (wave functions)
• Computer graphics (rotations)
Complex Number Forms
Rectangular Form
Format: z = a + bi
a: Real part
b: Imaginary part
Polar Form
Format: z = r × e^(iφ)
r: Magnitude
φ: Phase (argument)
Key Formulas
Magnitude
r = √(a² + b²)
Phase
φ = atan2(b, a)
Real Part
a = r × cos(φ)
Imaginary Part
b = r × sin(φ)
Calculator Tips
Magnitude r is always non-negative
Phase φ is typically in [-π, π] range
atan2 function handles all quadrants correctly
Both radians and degrees are supported
Understanding Complex Number Forms
What is Rectangular Form?
The rectangular form (also called Cartesian form) represents a complex number as z = a + bi, where 'a' is the real part and 'b' is the imaginary part. This form directly shows the horizontal and vertical components on the complex plane.
What is Polar Form?
The polar form represents a complex number using its distance from the origin (magnitude r) and the angle it makes with the positive real axis (phase φ). It's written as z = r × e^(iφ).
Why Use Different Forms?
- •Rectangular form is easier for addition and subtraction
- •Polar form is easier for multiplication and division
- •Polar form reveals magnitude and rotation clearly
- •Different forms suit different applications
Remember: Both forms represent the same complex number, just in different ways. Choose the form that makes your calculations easier!
The atan2 Function
The atan2(b, a) function is crucial for correct phase calculation. Unlike the regular arctan function, atan2 considers the signs of both arguments to determine the correct quadrant:
Quadrant I (a > 0, b > 0):
φ = arctan(b/a)
Quadrant II (a < 0, b > 0):
φ = arctan(b/a) + π
Quadrant III (a < 0, b < 0):
φ = arctan(b/a) - π
Quadrant IV (a > 0, b < 0):
φ = arctan(b/a)