Complex Number Calculator
Add, subtract, multiply, and divide numbers of the form a + bi.
Complex Number Calculator
Perform operations on complex numbers with step-by-step solutions and visualizations
Complex Numbers
Polar Form
Complex Plane Visualization
About Complex Numbers
Complex numbers in short
A complex number is a + bi, where a is real, b is imaginary, and i² = -1. That form is rectangular (Cartesian). Polar form uses modulus r and argument θ: r(cos θ + i sin θ), or re^(iθ) via Euler's formula.
Rectangular form is easier for addition and subtraction. Polar form is easier for multiplication, division, powers, and roots. Convert with r = √(a² + b²) and θ = atan2(b, a).
Operations
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division: multiply by the conjugate of the denominator, then divide by c² + d².
Conjugate of a + bi is a - bi. Modulus |z| = √(a² + b²). De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
On the Argand plane, addition is vector addition. Multiplying by i rotates 90° counterclockwise. Conjugation mirrors across the real axis.
Where they show up
- AC circuits: phasors and impedance Z = R + jX
- Signal processing: Fourier transforms and filters
- Quantum mechanics: complex wave amplitudes
- Fluid flow: complex potentials and conformal maps
Frequently Asked Questions
Why call them "imaginary"?
Historical naming. Descartes used "imaginary" for roots of x² + 1 = 0. They are standard mathematical objects with concrete uses in physics and engineering.
Polar or rectangular?
Use rectangular for +/− and when you need a and b separately. Use polar for ×, ÷, powers, and roots.
How does multiplication look geometrically?
Multiply moduli and add angles. The product is a scale plus a rotation in the plane.
Link to trigonometry?
Euler's formula e^(iθ) = cos θ + i sin θ ties complex exponentials to sine and cosine on the unit circle.
How do they help with polynomials?
Every degree-n polynomial with complex coefficients has n complex roots (counting multiplicity). So equations without real roots still have solutions in ℂ.
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