Complex Numbers Operations
Add, subtract, multiply and divide two complex numbers; includes modulus and argument.
Complex numbers: addition and Argand plane
A complex number has the form z = a + bi, where a is the real part, b the imaginary part, and i² = −1. Addition is componentwise: (a + bi) + (c + di) = (a + c) + (b + d)i — geometrically, this is vector addition in the Argand plane. Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The modulus is |z| = √(a² + b²) and the argument is arg(z) = atan2(b, a). Polar form: z = r·e^(iθ) (Euler's formula). The identity e^(iπ) + 1 = 0 — Euler's identity — is often voted the most beautiful equation in mathematics, tying together five fundamental constants.
Applications: engineering, physics and fractals
Complex numbers are essential in electrical engineering (impedance Z = R + jX), signal processing (FFT, Fourier transforms), quantum physics (wave functions are complex-valued), fractal geometry (Mandelbrot and Julia sets are defined by iterations on ℂ) and aerodynamics (conformal mappings for airfoil design).
FAQ
Why "imaginary"? Historical naming by Descartes (17th century), who viewed √−1 with suspicion. Modern mathematics treats complex numbers as fully legitimate.
Why use j instead of i in engineering? In electrical engineering i denotes current; j avoids the clash.
Is addition commutative? Yes — z₁ + z₂ = z₂ + z₁, just like real numbers.
How to subtract? (a + bi) − (c + di) = (a − c) + (b − d)i — componentwise, same as addition.
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