Complex Numbers and Quadratic Equations

The imaginary unit i is defined

The imaginary unit i is defined as a solution to x² = -1, so i² = -1. This extends the real number system to include solutions to equations that previ

A complex number is any number

A complex number is any number of the form z = a + ib, where a and b are real numbers. Here, a is the real part (Re z = a) and b is the imaginary part

For complex numbers z₁ =

For complex numbers z₁ = a + ib and z₂ = c + id, the sum is z₁ + z₂ = (a + c) + i(b + d). Addition combines real parts separately and imaginary parts

Subtraction is defined as z₁

Subtraction is defined as z₁ - z₂ = z₁ + (-z₂), where -z₂ is the additive inverse of z₂. For z₁ = a + ib and z₂ = c + id, we get z₁ - z₂ = (a - c) + i

For z₁ = a + ib

For z₁ = a + ib and z₂ = c + id, multiplication is defined as z₁z₂ = (ac - bd) + i(ad + bc). This follows from treating complex numbers algebraically