Complex Numbers

The imaginary unit i is defined

The imaginary unit i is defined such that i² = -1. This allows us to find square roots of negative numbers. The powers of i follow a cyclic pattern: i

A complex number z is expressed

A complex number z is expressed as z = x + iy, where x is the real part (Re(z)) and y is the imaginary part (Im(z)). Two complex numbers are equal if

The Argand plane (or complex plane)

The Argand plane (or complex plane) is a 2D coordinate system where the horizontal axis represents the real part and the vertical axis represents the

Addition

Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂). Subtraction: z₁ - z₂ = (x₁ - x₂) + i(y₁ - y₂). Multiplication: z₁z₂ = (x₁x₂ - y₁y₂) + i(x₁

The conjugate of z = x

The conjugate of z = x + iy is z̄ = x - iy. Geometrically, the conjugate is the reflection of z across the real axis. Key properties: z · z̄ = |z|², a