Notes from Toppers
Detailed Notes for Complex Numbers (JEE Preparation)
1. Basic Concepts:
 Complex numbers are ordered pairs of real numbers, represented as (a + bi), where (a) and (b) are real numbers and (i) is the imaginary unit defined by (i^2 = 1).
Reference:
 NCERT Mathematics Class 11, Chapter 5, Complex Numbers and Quadratic Equations
2. Graphical Representation:
 Complex numbers can be plotted on the complex plane, a twodimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Reference:
 NCERT Mathematics Class 11, Chapter 5, Complex Numbers and Quadratic Equations
3. Complex Conjugates:
 The complex conjugate of a complex number (a + bi) is (a  bi).
 Complex conjugates have the same real part but differ in the sign of the imaginary part.
Reference:
 NCERT Mathematics Class 11, Chapter 5, Complex Numbers and Quadratic Equations
4. Modulus and Argument:
 The modulus (magnitude) of a complex number (a + bi) is given by (\sqrt{a^2 + b^2}).
 The argument (angle) of a complex number (a + bi) is the angle formed by the positive real axis and the line connecting the origin to the point ((a, b)) on the complex plane. It is denoted as (\theta).
Reference:
 NCERT Mathematics Class 11, Chapter 5, Complex Numbers and Quadratic Equations
5. De Moivre’s Theorem:
 De Moivre’s Theorem states that for any complex number (z = r(\cos\theta + i\sin\theta)) and any positive integer (n), we have (z^n = r^n(\cos n\theta + i\sin n\theta)).
 This theorem is used to simplify powers and roots of complex numbers.
Reference:
 NCERT Mathematics Class 12, Chapter 5, Complex Numbers and Quadratic Equations
6. Exponential and Logarithmic Forms:
 A complex number (z = a + bi) can be expressed in exponential form as (z = r(\cos\theta + i\sin\theta)), where (r = \sqrt{a^2 + b^2}) and (\theta) is the argument of (z).
 The natural logarithm of a complex number (z = r(\cos\theta + i\sin\theta)) is given by (\ln z = \ln r + i\theta), where (r) is the modulus and (\theta) is the argument of (z).
Reference:
 NCERT Mathematics Class 12, Chapter 5, Complex Numbers and Quadratic Equations
7. Applications:

Complex numbers have wide applications in various fields such as electrical engineering, mechanical engineering, signal processing, and quantum mechanics.

In electrical engineering, complex numbers are used to represent and analyze alternating current (AC) circuits, impedance, and power factor.

In mechanical engineering, complex numbers are used to analyze vibrations and the behavior of mechanical systems.
Important Notes:
 To excel in complex numbers, practice solving a variety of problems involving graphical representation, complex conjugates, modulus and argument, De Moivre’s Theorem, and exponential and logarithmic forms.
 Understand the geometric interpretations of complex numbers and their operations.
 Develop your problemsolving skills by attempting challenging questions and problems from previous years’ JEE examinations and practice tests.
 Regularly revise and reinforce your understanding of complex numbers to ensure longterm retention.