🔢 Complex Numbers and Quadratic Equations

1. Introduction to Complex Numbers

The square root of a negative number does not exist in the real number system. Thus, we introduce 'iota' (i).

A complex number is of the form z = a + ib, where 'a' is the real part Re(z) and 'b' is the imaginary part Im(z).

2. Algebra of Complex Numbers

• Equality: a + ib = c + id implies a = c and b = d.
• Addition: (a + ib) + (c + id) = (a + c) + i(b + d).
• Multiplication: (a + ib)(c + id) = (ac - bd) + i(ad + bc).

3. Modulus and Conjugate

4. Argand Plane and Polar Representation

A complex number z = x + iy can be represented as a point P(x,y) on a 2D plane called the Argand plane.

Polar Form: z = r(cos θ + i sin θ), where r = |z| and θ is the argument or amplitude of z.

The principal argument lies in the range -π < θ ≤ π.

5. Quadratic Equations

For ax² + bx + c = 0, if the discriminant D = b² - 4ac < 0, the roots are complex conjugates: