Mathematics Complex Numbers and Quadratic Equations JEE Syllabus

Complex Numbers and Quadratic Equations is an important chapter in JEE Maths because it introduces a new number system beyond the real numbers and helps you understand solutions to quadratic equations that cannot be solved using real numbers alone. The chapter explains the concept of complex numbers, their algebraic operations, modulus, conjugate, and graphical representation on the Argand plane. 

These concepts strengthen algebraic problem-solving skills and form the foundation for advanced mathematics topics. Questions from complex numbers frequently appear in competitive examinations, making this chapter an essential part of JEE preparation.

Complex Numbers

The concept of complex numbers emerged from the need to solve equations that have no real solutions. Mathematicians gradually accepted numbers involving √−1 and developed a complete system around them.

W. R. Hamilton later provided a rigorous mathematical foundation for complex numbers.

The Symbol i

The symbol i, called iota, is defined as:

i = √−1

Therefore:

i² = −1

This number is a direct solution to the equation:

x² + 1 = 0

which has no real solution.

Definition of a Complex Number

Any number of the form:

z = a + ib

where a and b are real numbers, is called a complex number.

Real and Imaginary Parts

For a complex number:

z = a + ib

a is called the real part and is written as Re(z).

b is called the imaginary part and is written as Im(z).

Equality of Complex Numbers

Two complex numbers are equal only when their corresponding real parts and imaginary parts are equal.

If:

a + ib = c + id

then:

a = c

and

b = d

Algebra of Complex Numbers

Complex numbers follow arithmetic operations similar to real numbers.

Addition of Complex Numbers

If:

z₁ = a + ib

z₂ = c + id

then:

z₁ + z₂ = (a + c) + i(b + d)

Properties of Addition

Closure Property

The sum of two complex numbers is always a complex number.

Commutative Property

z₁ + z₂ = z₂ + z₁

Associative Property

(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

Additive Identity

0 + 0i acts as the additive identity.

z + 0 = z

Additive Inverse

For every complex number z = a + ib, there exists:

−z = −a − ib

such that:

z + (−z) = 0

Subtraction of Complex Numbers

Subtraction is defined as:

z₁ − z₂ = z₁ + (−z₂)

Multiplication of Complex Numbers

If:

z₁ = a + ib

z₂ = c + id

then:

z₁z₂ = (ac − bd) + i(ad + bc)

Properties of Multiplication

Multiplicative Inverse

For a non-zero complex number:

z = a + ib

The multiplicative inverse is:

1/z = a/(a²+b²) − ib/(a²+b²)

This inverse satisfies:

z × (1/z) = 1

Division of Complex Numbers

Division is defined as:

z₁/z₂ = z₁ × (1/z₂)

where z₂ ≠ 0.

To simplify division, the conjugate of the denominator is usually used.

Powers of i

This topic covers how powers of the imaginary unit i repeat in a fixed cycle of four values.

i¹ = i

i² = −1

i³ = −i

i⁴ = 1

i⁵ = i

i⁶ = −1

The pattern repeats after every four powers.

Square Roots of Negative Real Numbers

In the real number system, the square of any real number is always non-negative. For example:

• 22=42^2 = 422=4

• (−2)2=4(-2)^2 = 4(−2)2=4

Since:

i² = −1

and

(−i)² = −1

Both i and −i are square roots of −1.

For a positive real number a:

√−a = i√a

Modulus and Conjugate

The modulus of a complex number represents its distance from the origin in the complex plane.

Let:

z = a + ib

The modulus of z is:

|z| = √(a² + b²)

The modulus represents the distance of the point from the origin in the complex plane.

Conjugate of a Complex Number

The conjugate of:

z = a + ib

is:

z̄ = a − ib

The sign of the imaginary part changes.

Important Identities

z × z̄ = |z|²

1/z = z̄/|z|²

|z₁z₂| = |z₁||z₂|

(z₁z₂)̄ = z̄₁z̄₂

(z₁ ± z₂)̄ = z̄₁ ± z̄₂

These identities are frequently used in problem-solving.

Argand Plane

Complex numbers can be represented geometrically on a coordinate plane known as the Argand Plane.

A complex number:

z = x + iy

is represented by the point:

P(x, y)

Components of the Argand Plane

• Real Axis: The horizontal axis represents real numbers.

• Imaginary Axis: The vertical axis represents imaginary numbers.

Representation of Complex Numbers

The complex number:

2 + 3i

is represented by the point (2, 3).

Similarly:

−1 + 4i

is represented by the point (−1, 4).

The Argand plane helps visualise modulus, conjugates, and geometric properties of complex numbers.

Complex Numbers and Quadratic Equations

One of the most important applications of complex numbers is solving quadratic equations.

Complex Numbers and Quadratic Equations is a mathematics chapter that introduces a new number system called complex numbers and explains how these numbers help solve certain quadratic equations that cannot be solved using real numbers alone.

A quadratic equation has the form:

ax² + bx + c = 0

where a ≠ 0.

The roots are given by:

x = [−b ± √(b² − 4ac)] / 2a

The quantity:

D = b² − 4ac

is called the discriminant.

Nature of Roots

If D > 0

The equation has two distinct real roots.

If D = 0

The equation has equal real roots.

If D < 0

The equation has complex roots.

Important Notes

• The development of complex numbers involved contributions from several mathematicians.

• Mahavira (850 AD) discussed the impossibility of ordinary square roots for negative numbers.

• Bhaskara (1150 AD) also stated that negative numbers cannot have real square roots.

• Euler introduced the symbol i for √−1.

• W. R. Hamilton later defined a complex number as an ordered pair of real numbers and established a strong mathematical foundation for the subject.