# 5. Complex Numbers

It is never too late to be what you might have been. —George Eliot

## COMPLEX NUMBERS CLASS 11 MATHS

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 $$x = \frac{− b \pm \sqrt{4ac – b2} i}{2a}$$.