Complex Number to Polar Form Calculator
Convert complex numbers from rectangular (a + bi) to polar form with step-by-step solutions
Example Complex Numbers
Complex Number Input
The real component of the complex number
The coefficient of the imaginary unit i
Complex Number:
Polar Form Results
Exponential Form
Trigonometric Form
Step-by-Step Conversion
Given complex number: z = 3 +4i
Calculate magnitude: r = √(a² + b²) = √(3² + 4²) = √25.0000 = 5.0000
Calculate argument: φ = arctan(b/a) = arctan(4/3) = arctan(1.3333)
Argument: φ = 0.9273 radians = 53.1301°
Complex number is in quadrant 1
Polar form: z = 5.0000 × (cos(0.9273) + i×sin(0.9273))
Exponential form: z = 5.0000 × exp(i×0.9273)
Conversion Formulas
Magnitude
Argument
Polar Form
Complex Plane Quadrants
Mathematical Symbols
Understanding Polar Form of Complex Numbers
What is Polar Form?
The polar form represents a complex number using its distance from the origin (magnitude) and the angle it makes with the positive real axis (argument). This is particularly useful for multiplication and division of complex numbers.
Key Components
- •Magnitude (r): Distance from origin to the complex number
- •Argument (φ): Angle from positive real axis
- •Exponential form: r × e^(iφ)
Conversion Process
Step 1: Calculate Magnitude
Use Pythagorean theorem: r = √(a² + b²)
Step 2: Find Argument
Use atan2 function for correct quadrant: φ = atan2(b, a)
Step 3: Write Polar Form
Express as r × e^(iφ) or r × [cos(φ) + i×sin(φ)]
Special Cases and Important Examples
Pure Real Numbers
z = 5 + 0i
Magnitude: r = 5
Argument: φ = 0
Polar: 5 × e^(i×0)
Pure Imaginary Numbers
z = 0 + 3i
Magnitude: r = 3
Argument: φ = π/2
Polar: 3 × e^(i×π/2)
Unit Circle
z = 1 + i
Magnitude: r = √2
Argument: φ = π/4
Polar: √2 × e^(i×π/4)