📐 Complex Number Calculator
Perform operations on complex numbers
(a₁ + b₁i) + (a₂ + b₂i)
First Complex Number
Second Complex Number
How to Use This Calculator
Select Operation
Choose addition, subtraction, multiplication, or division.
Enter First Number
Type real part (a₁) and imaginary part (b₁) of the first complex number.
Enter Second Number
Type real part (a₂) and imaginary part (b₂) of the second complex number.
Click Calculate
View the result in rectangular form, plus magnitude and angle.
Formulas
Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction:
(a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division:
(a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²)
Example 1: Add (3 + 4i) + (1 + 2i)
Real part: 3 + 1 = 4
Imaginary part: 4 + 2 = 6
Result: 4 + 6i
Example 2: Multiply (2 + 3i) × (1 + 4i)
ac - bd = 2×1 - 3×4 = 2 - 12 = -10
ad + bc = 2×4 + 3×1 = 8 + 3 = 11
Result: -10 + 11i
About Complex Number Calculator
The Complex Number Calculator performs basic arithmetic operations on complex numbers. Complex numbers extend real numbers by adding an imaginary unit i where i² = -1. They're essential in many fields including electrical engineering, signal processing, and quantum physics.
When to Use This Calculator
- AC Circuit Analysis: Calculate impedance and phase relationships
- Signal Processing: Work with Fourier transforms and filters
- Polynomial Roots: Handle complex solutions to equations
- Physics: Quantum mechanics and wave analysis
- Mathematics: Complete arithmetic operations on complex numbers
Why Use Our Calculator?
- ✅ Four Operations: Add, subtract, multiply, and divide
- ✅ Multiple Formats: Rectangular and polar forms
- ✅ Instant Results: Fast and accurate calculations
- ✅ Shows Magnitude & Angle: Complete representation
- ✅ Easy to Use: Simple interface for complex math
- ✅ Free Tool: No registration required
Understanding Complex Numbers
- Rectangular form: a + bi (real + imaginary parts)
- Polar form: r(cos θ + i sin θ) or r∠θ (magnitude and angle)
- Euler's form: re^(iθ) (exponential notation)
- Conjugate: a - bi (changing sign of imaginary part)
- Magnitude: |z| = √(a² + b²)
Real-World Applications
Electrical Engineering: AC circuits use complex numbers to represent impedance, with real part as resistance and imaginary part as reactance.
Signal Processing: Fourier transforms convert signals to frequency domain using complex exponentials.
Control Systems: Analyze system stability using complex poles and zeros.
Tips for Using This Calculator
- Division by zero is not allowed (denominator must be non-zero)
- For division, use the formula involving conjugate
- Magnitude is always non-negative
- Angle is measured in degrees from positive real axis
- Rectangular form is best for addition/subtraction
- Polar form is better for multiplication/division
Frequently Asked Questions
What is a complex number?
A complex number has the form a + bi where a and b are real numbers and i² = -1. The number a is the real part, bi is the imaginary part.
How do I multiply complex numbers?
Multiply like binomials: (a+bi)(c+di) = ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i.
How do I divide complex numbers?
Multiply numerator and denominator by the conjugate of the denominator, then simplify. The result is [(ac+bd)+(bc-ad)i]/(c²+d²).
What is the magnitude of a complex number?
The magnitude (or modulus) |z| = √(a² + b²). It represents the distance from origin in the complex plane.
What is polar form?
Polar form represents a complex number using magnitude r and angle θ: z = r(cos θ + i sin θ) or z = r∠θ. Useful for multiplication and division.
Where are complex numbers used?
Complex numbers are essential in electrical engineering (AC circuits), signal processing (Fourier analysis), quantum physics (wave functions), and control systems (stability analysis).