A complex number is an ordered pair of real numbers that can be manipulated through specific algebraic rules. The study of complex numbers fundamentally extends the real number system, enabling solutions to equations that are unsolvable in the reals alone, such as those involving the square roots of negative numbers. Quadratic equations with real or complex coefficients often yield complex roots, and therefore, their combined study is essential in higher mathematics, especially in the algebraic curriculum for competitive examinations.

Algebraic Representation and Equality of Complex Numbers

A complex number is defined as $z = a + ib$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit, satisfying $i^2 = -1$.

Given two complex numbers $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$, the numbers are equal if and only if $a_1 = a_2$ and $b_1 = b_2$.

Real Part: For $z = a + ib$, the real part is denoted by $\operatorname{Re}(z) = a$.

Imaginary Part: For $z = a + ib$, the imaginary part is denoted by $\operatorname{Im}(z) = b$.

Addition, Subtraction, Multiplication, and Division of Complex Numbers

The sum of $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ is computed as follows:

\[ z_1 + z_2 = (a_1 + ib_1) + (a_2 + ib_2) \]

\[ = (a_1 + a_2) + i(b_1 + b_2) \]

Subtraction is performed as: \[ z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2) \]

For multiplication, expand $(a_1 + ib_1)(a_2 + ib_2)$:

\[ (a_1 + ib_1)(a_2 + ib_2) = a_1a_2 + a_1(ib_2) + ib_1a_2 + (ib_1)(ib_2) \]

\[ = a_1a_2 + i a_1 b_2 + i b_1 a_2 + i^2 b_1 b_2 \]

\[ = a_1a_2 + i(a_1b_2 + b_1a_2) - b_1b_2 \]

\[ = (a_1a_2 - b_1b_2) + i(a_1b_2 + b_1a_2) \]

To divide $z_1$ by $z_2$ (with $z_2 \ne 0$), multiply numerator and denominator by the conjugate of the denominator:

\[ \frac{z_1}{z_2} = \frac{a_1 + ib_1}{a_2 + ib_2} \cdot \frac{a_2 - ib_2}{a_2 - ib_2} \]

\[ = \frac{(a_1a_2 + b_1b_2) + i(b_1a_2 - a_1b_2)}{a_2^2 + b_2^2} \]

\[ = \frac{a_1 a_2 + b_1 b_2}{a_2^2 + b_2^2} + i \frac{b_1 a_2 - a_1 b_2}{a_2^2 + b_2^2} \]

For further algebraic operations and theories, visit Understanding Complex Numbers.

Powers of the Imaginary Unit $i$

The imaginary unit $i$ satisfies $i^2 = -1$. The integral powers of $i$ are derived systematically as:

\[ i^1 = i \] \[ i^2 = -1 \] \[ i^3 = i^2 \cdot i = (-1) \cdot i = -i \] \[ i^4 = (i^2)^2 = (-1)^2 = 1 \]

For any integer $n$, the value of $i^n$ depends on the remainder when $n$ is divided by $4$:

\[ i^{4k} = 1, \quad i^{4k + 1} = i, \quad i^{4k + 2} = -1, \quad i^{4k + 3} = -i, \qquad (k \in \mathbb{Z}) \]

Modulus, Argument, and Conjugate of a Complex Number

The modulus (or absolute value) of $z = x + iy$ is $|z| = \sqrt{x^2 + y^2}$.

The conjugate of $z = x + iy$ is given by $\bar{z} = x - iy$.

The argument, denoted as $\arg(z)$, is the principal value of the angle $\theta$ such that $z = |z| (\cos \theta + i \sin \theta)$, with $\theta \in (-\pi, \pi]$.

\[ \arg(z) = \tan^{-1} \left( \frac{y}{x} \right) \]

The quadrant of $(x,y)$ must be taken into account to determine the correct value of $\theta$.

Geometric Representation: The Argand Plane and Locus Interpretations

Each complex number $z = x + iy$ corresponds to the point $(x, y)$ in the Argand plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

The modulus $|z|$ represents the distance from the origin to the point $(x, y)$. The argument $\arg(z)$ gives the angle formed with the positive real axis.

The set $\{z: |z - z_0| = r\}$ describes a circle in the Argand plane of radius $r$ centered at $z_0$.

The equation $\operatorname{Re}(z) = a$ is a vertical line, and $\operatorname{Im}(z) = b$ is a horizontal line in the Argand diagram.

For advanced geometric problems on the plane, refer to Matrices and Determinants Overview.

Powers, Roots, and Polar Representation

If $z = r (\cos \theta + i \sin \theta)$, then by De Moivre's Theorem, for any integer $n$:

\[ z^n = r^n [\cos(n\theta) + i\sin(n\theta)] \]

The $n^\text{th}$ roots of $z$ are given by

\[ z_k^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2\pi k}{n} + i \sin \frac{\theta + 2\pi k}{n} \right), \qquad k = 0, 1, ..., n-1 \]

The polar form of a complex number is $z = r \text{cis} \theta$, where $\text{cis} \theta = \cos \theta + i \sin \theta$.

For practical applications and further exploration, see Complex Numbers Practice Paper.

Cube Roots of Unity in Complex Numbers

The solutions to $z^3 = 1$ are called the cube roots of unity. They are:

\[ 1, \quad \omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \quad \omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]

These roots satisfy $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.

Solving Quadratic Equations with Complex Numbers

A quadratic equation in $x$ with real or complex coefficients is $ax^2 + bx + c = 0, \, a \ne 0$.

The discriminant is $D = b^2 - 4ac$.

The nature of the roots is as follows:

Case 1: If $D > 0$, the roots are real and distinct.

Case 2: If $D = 0$, the roots are real and equal.

Case 3: If $D < 0$, the roots are non-real and conjugate, given by:

\[ x = \frac{-b \pm i \sqrt{|D|}}{2a} \]

To see detailed properties of roots and formation of quadratic equations from roots, read Roots of Quadratic Equations.

Location and Nature of Roots in Quadratic Equations

Given $a x^2 + b x + c = 0$, the sum of roots $\alpha + \beta = -\dfrac{b}{a}$ and the product of roots $\alpha \beta = \dfrac{c}{a}$.

If both roots are positive, coefficients $a$ and $c$ must have the same sign.

If both roots are negative, $a$, $b$, and $c$ must all have the same sign.

If roots are equal in magnitude but opposite in sign, then $b = 0$.

If the roots are reciprocals, then $a = c$.

These criteria often arise in advanced JEE Main algebraic problems. For related inequalities, examine Quadratic Inequalities Explained.

Geometric Interpretations and Transformations in the Argand Plane

If a point $P$ divides the segment joining $z_1$ and $z_2$ in the ratio $m:n$, then the corresponding complex number is:

\[ z_P = \frac{m z_1 + n z_2}{m + n} \]

The centroid of the triangle with vertices at $z_1, z_2, z_3$ is:

\[ z_G = \frac{z_1 + z_2 + z_3}{3} \]

The equation $|z - z_0| = r$ represents a circle of radius $r$ centered at $z_0$ on the Argand plane.

A rotation about the origin by angle $\theta$ is equivalent to multiplication by $e^{i\theta}$. For loci and advanced geometric relations, the properties of transformations are indispensable.

For advanced connections with determinants, read Properties of Determinants.

Worked Example: Solving a Quadratic Equation with Complex Roots

Given: Solve $x^2 + 4x + 13 = 0$.

Step 1: Compute discriminant \[ D = b^2 - 4ac = 16 - 52 = -36 \]

Step 2: Apply the quadratic formula: \[ x = \frac{-4 \pm \sqrt{-36}}{2} \]

Step 3: Write $\sqrt{-36} = 6i$

Therefore, \[ x = \frac{-4 \pm 6i}{2} \]

\[ = -2 \pm 3i \]

Final result: The roots are $x = -2 + 3i,\, -2 - 3i$.

Summary of Fundamental Results

Every non-zero complex number $z = x + iy$ can be expressed in polar form as $z = r (\cos \theta + i \sin \theta)$, where $r = |z|$ and $\theta = \arg(z)$. The conjugate $\bar{z}$ reflects $z$ across the real axis. The set of all $z$ satisfying $|z| = r$ is a circle, and $z$-plane geometric interpretations streamline many algebraic and analytic results. All real-coefficient quadratic equations with negative discriminant have roots occurring in conjugate pairs, and identities such as $1 + \omega + \omega^2 = 0$ for cube roots of unity are foundational in JEE algebra.