Addition
\((3 + 4i) + (1 + 2i) = 4 + 6i\)
Complex Numbers Calculator
Perform operations with complex numbers and find polar forms
Results
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Theory & Formula
Complex numbers extend real numbers to include imaginary numbers (multiples of i, where i² = -1). They are essential in engineering, physics, and advanced mathematics.
Key concepts:
- Rectangular form: z = a + bi
- Polar form: z = r(cos θ + i sin θ) = r∠θ
- Magnitude: |z| = √(a² + b²)
- Argument: θ = arctan(b/a)
- Conjugate: z̄ = a - bi
- Euler's formula: e^(iθ) = cos θ + i sin θ
\(z = a + bi\), where \(i^2 = -1\)
Worked Examples
Multiplication
\((3 + 4i)(1 + 2i) = -5 + 10i\)
Polar Form
\(5 + 0i = 5\angle 0^\circ\), \(0 + 5i = 5\angle 90^\circ\)