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