Complex Numbers and De Moivre’s Theorem

A complex number is expressed as

A complex number is expressed as z = a + bi, where a and b are real numbers and i = √(-1) is the imaginary unit. Here, 'a' is the real part (Re(z)) an

i¹ = i

i¹ = i, i² = -1, i³ = -i, i⁴ = 1. The pattern repeats every 4 powers. For any positive integer n: iⁿ = i^(n mod 4). Example: i¹⁷ = i¹ = i since 17 ≡ 1

The conjugate of z =

The conjugate of z = a + bi is z̄ = a - bi. Properties: (z̄) = z, z + z̄ = 2a, z - z̄ = 2bi, z·z̄ = |z|². Example: If z = 3 + 4i, then z̄ = 3 - 4i.

For z = a + bi

For z = a + bi, |z| = √(a² + b²). Properties: |z| = |z̄| = |-z|, |z₁z₂| = |z₁||z₂|, |z₁/z₂| = |z₁|/|z₂|, |z₁ + z₂| ≤ |z₁| + |z₂|. Example: |3 + 4i| =

Complex number z = a +

Complex number z = a + bi is represented as point (a, b) in the coordinate plane, where x-axis represents real part and y-axis represents imaginary pa