Complex Numbers

The imaginary unit i is defined

The imaginary unit i is defined as i = √(-1), so i² = -1. An imaginary number has the form ki where k ∈ R, k ≠ 0. For example, √(-25) = 5i. Powers of

A complex number z =

A complex number z = a + ib where a, b ∈ R and i = √(-1). Here 'a' is the real part Re(z) and 'b' is the imaginary part Im(z). The set of complex numb

The conjugate of z =

The conjugate of z = a + ib is z̄ = a - ib. Properties: (z̄) = z, if z = z̄ then z is real, if z = -z̄ then z is purely imaginary. Important result: z

Addition

Addition: (a+ib) + (c+id) = (a+c) + (b+d)i. Subtraction: (a+ib) - (c+id) = (a-c) + (b-d)i. Multiplication: (a+ib)(c+id) = (ac-bd) + (ad+bc)i. Division

To find √(x+iy)

To find √(x+iy), let √(x+iy) = a+ib. Squaring both sides: x+iy = (a+ib)² = (a²-b²) + 2abi. Equating real and imaginary parts: x = a²-b², y = 2ab. Solv