NCERT Book Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

76 MATHEMATICS

Chapter 4

COMPLEX NUMBERS AND
QUADRATIC EQUATIONS

vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v

4.1 Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x2 + 1 = 0 has no real
solution as x2 + 1 = 0 gives x2 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x2 = – 1. In fact, the main
objective is to solve the equation ax2 + bx + c = 0, where
D = b2 – 4ac < 0, which is not possible in the system of
real numbers.
W. R. Hamilton
4.2 Complex Numbers (1805-1865)

Let us denote −1 by the symbol i. Then, we have i = −1 . This means that i is a
2

solution of the equation x2 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
 −1 
complex number. For example, 2 + i3, (– 1) + i 3 , 4 + i   are complex numbers.
 11 
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.