# 5. Complex Numbers

A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.

Let z_1= a + ib \text{ and } z_2 = c + id . Then
(i) z_1 + z_2 = (a + c) + i (b + d)
(ii) z_1 z_2 = (ac – bd) + i (ad + bc)

For any non-zero complex number z = a + ib (a \ne 0, b \ne 0), there exists the complex number \frac{a}{a^2+ b^2} + i \frac{-b}{a^2 + b^2} , denoted by \frac{1}{z} or z^{-1}, called the multiplicative inverse of z such that (a + ib)\left ( \frac{a^2}{a^2+b^2}+i \frac{-b}{a^2+b^2} \right ) = 1 + i 0 = 1

For any integer k , i^{4k}=1, i{4k + 1}=i, i^{4k + 2}= – 1, i{4k + 3} = – i

The conjugate of the complex number z = a + ib, denoted by z , is given by z = a – ib.

The polar form of the complex number z = x + iy is r (\cos θ + i \sin θ), where r = \sqrt{x^2 + y^2} (the modulus of z) and \cos θ = \frac{x}{r} , \sin θ = \frac{y}{r}. (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z.

A polynomial equation of n degree has n roots.

The solutions of the quadratic equation ax^2 + bx + c = 0, where a, b, c ∈ R, a ≠ 0, b^2 – 4ac < 0 , are given by x = \frac{− b \pm \sqrt{4ac – b2} i}{2a} .