Complex Number Calculator – Guide & Formulas
Perform arithmetic operations (addition, subtraction, multiplication, division) on complex numbers. Converts to polar coordinates instantly.
Add, subtract, multiply, and divide complex numbers in rectangular form (a + bi). Convert expressions to polar form (magnitude r and phase angle θ) with complete step breakdowns.
Key Takeaway
Use the free Complex Number Calculator to perform arithmetic operations (addition, subtraction, multiplication, division) on complex numbers. converts to polar coordinates instantly. Get instant results with step-by-step explanations.
How to Use the Complex Number Calculator
- Enter the real and imaginary parts of the first complex number (A).
- Enter the real and imaginary parts of the second complex number (B).
- Choose the math operation: Addition, Subtraction, Multiplication, or Division.
- Review the rectangular result, the calculated absolute value (magnitude), and the phase angle.
The Formula
Variable Definitions
- a, c: Real components
- b, d: Imaginary components
- i: Imaginary unit where i² = -1
Multiplying Complex Impedances
An electrical engineer needs to multiply impedance A (3 + 4i) by impedance B (2 - 1i).
- Step 1: Set A = 3 + 4i, B = 2 - 1i.
- Step 2: Choose multiplication.
- Step 3: Expand: (3*2 - 4*-1) + (3*-1 + 4*2)i = (6 + 4) + (-3 + 8)i.
- Step 4: The rectangular result is 10 + 5i.
Frequently Asked Questions
How do you divide complex numbers?
Multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary component in the denominator.
What is the complex conjugate?
The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate always yields a real number (a² + b²).
What is the polar form of a complex number?
Polar form represents a complex number as r(cos θ + i sin θ) or re^(iθ), where r is the distance from the origin (magnitude) and θ is the angle from the positive real axis.