Complex Numbers

The imaginary unit i is defined

The imaginary unit i is defined as i = √(-1), so i² = -1. A complex number z = a + ib where a, b ∈ R. Here 'a' is the real part Re(z) and 'b' is the i

Addition

Addition: (a+ib) + (c+id) = (a+c) + (b+d)i. Multiplication: (a+ib)(c+id) = (ac-bd) + (ad+bc)i. Division: (a+ib)/(c+id) = [(ac+bd) + (bc-ad)i]/(c²+d²).

Conjugate of z = a +

Conjugate of z = a + ib is z̄ = a - ib. Modulus |z| = √(a² + b²) represents the distance from origin. Properties: z·z̄ = |z|², |z₁z₂| = |z₁||z₂|, |z₁/

Complex number z = a +

Complex number z = a + ib is represented as point (a,b) in Argand diagram. Polar form: z = r(cosθ + isinθ) where r = |z| and θ = arg(z) = tan⁻¹(b/a).

[r(cosθ + isinθ)]ⁿ = rⁿ(cosnθ +

[r(cosθ + isinθ)]ⁿ = rⁿ(cosnθ + isinnθ) for any integer n. This theorem simplifies finding powers and roots of complex numbers. Example: (1+i)⁴ = (√2)