Understanding the Modulus of a Complex Number

The modulus of a complex number represents a foundational concept in the study of complex numbers for the JEE Main mathematics syllabus. Its understanding is crucial for mastering advanced results and for applications in geometrical and algebraic contexts. This discussion provides a detailed, precise explanation of the modulus, including derivations, properties, formulas, examples, and its analytical significance within the complex plane.

Complex Number and Its Geometric Interpretation

A complex number $z$ is an ordered pair $(x,\,y)$, commonly written as $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part. The number $i$ satisfies $i^2 = -1$.

Geometrically, $z$ is represented as the point $(x,\,y)$ in the complex plane, also called the Argand plane. The horizontal axis represents real numbers, and the vertical axis represents imaginary numbers.

The position of $z$ in the Argand plane motivates the definition of its modulus as the Euclidean distance from the origin $(0,\,0)$ to the point $(x,\,y)$. This geometric interpretation is central to many advanced results in complex analysis.

Definition and Formula for Modulus

The modulus of a complex number $z = x + iy$ is denoted by $|z|$ and is defined mathematically as \[ |z| = \sqrt{x^2 + y^2} \] This formula expresses the distance from the origin to the point $(x,\,y)$ in the complex plane.

The modulus always yields a real, non-negative value. It provides a measure of the size, or magnitude, of the complex number, independent of direction in the Argand plane.

Derivation of the Modulus Formula

For $z = x + iy$, the distance from the origin to $(x,\,y)$ is calculated using the distance formula in coordinate geometry: \[ |z| = \sqrt{(x-0)^2 + (y-0)^2} = \sqrt{x^2 + y^2} \]

Alternatively, in terms of $z$ and its complex conjugate $\bar{z} = x - iy$, observe that \[ z \cdot \bar{z} = (x + iy)(x - iy) = x^2 + y^2 \] Therefore, the modulus can also be written as $|z| = \sqrt{z \cdot \bar{z}}$.

Properties of Modulus

The modulus function $|z|$ satisfies several important algebraic and geometric properties. These facilitate simplification and solution of equations involving complex numbers. Some of the core properties are as follows.

• Non-negativity: $|z| \geq 0$ for all $z$
• $|z| = 0$ if and only if $z = 0$
• Multiplicative: $|z_1 z_2| = |z_1|\,|z_2|$
• Quotient: $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}, \, z_2 \neq 0$
• Triangle Inequality: $|z_1 + z_2| \leq |z_1| + |z_2|$
• Reverse Triangle: $||z_1| - |z_2|| \leq |z_1 - z_2|$

Each property follows directly from vector and Euclidean geometry, or from algebraic manipulations using the definition of modulus and conjugation.

Proofs of Crucial Modulus Properties

To establish the multiplicative property, let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. Compute their product:

\[ z_1 z_2 = (x_1 + iy_1)(x_2 + iy_2) = (x_1 x_2 - y_1 y_2) + i(x_1 y_2 + y_1 x_2) \]

Its modulus is then: \[ |z_1 z_2| = \sqrt{(x_1 x_2 - y_1 y_2)^2 + (x_1 y_2 + y_1 x_2)^2} \] Upon expanding and simplifying, this equals \[ \sqrt{(x_1^2 + y_1^2)(x_2^2 + y_2^2)} = |z_1|\,|z_2| \]

For the quotient property, observe: \[ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \] since division by a complex number scales the modulus by the reciprocal of the divisor's modulus. The formal derivation parallels that for the multiplicative property, using conjugation to rationalize denominators where needed.

The geometric triangle inequality for moduli reflects the standard triangle inequality for vectors: \[ |z_1 + z_2| \leq |z_1| + |z_2| \] It arises from the Euclidean norm applied to vector addition in $\mathbb{R}^2$.

Special Cases and Notable Results

If $z$ is a real number, i.e., $y = 0$, then $|z|$ reduces to the absolute value $|x|$. If $y \neq 0$, the modulus always exceeds the real part unless $x = 0$.

For unit modulus, $|z| = 1$ characterizes all complex numbers lying on the unit circle in the Argand plane, a central set in trigonometric and geometric constructions.

Examples of Modulus Calculation and Properties

Consider $z = -4 + 3i$. The modulus is:

\[ |z| = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Suppose $v = (1 + 3i)(-2 + 4i)$. Compute individual moduli:

\[ |1 + 3i| = \sqrt{1^2 + 3^2} = \sqrt{10} \] \[ |-2 + 4i| = \sqrt{(-2)^2 + 4^2} = \sqrt{20} \] \[ |v| = |1 + 3i| \cdot |-2 + 4i| = \sqrt{10} \times \sqrt{20} = \sqrt{200} = 10\sqrt{2} \]

Thus, the modulus of $v$ is $10\sqrt{2}$, exemplifying the multiplicative property.

For more solved questions, refer to Complex Numbers Overview.

Applications and Analytical Significance

The modulus is significant in expressing complex numbers in polar form as $z = r\,(\cos\theta + i\sin\theta)$, where $r = |z|$. It is essential in interpreting the distance formula and in bounding the values of expressions involving complex numbers.

Many problems in JEE Main require direct use of modulus properties to simplify expressions and deduce geometric loci in the complex plane. Mastery of modulus ensures advanced command over both algebraic operations and geometric reasoning within the topic.

Detailed relationships between modulus and conjugate can be studied further in Modulus And Conjugate Discussion.

In advanced applications, modulus properties underpin analytic geometry of circles, lines, and transformations in the complex domain, and are integral to understanding loci and trigonometric identities.

Summary of Principal Modulus Properties

A comprehensive understanding of modulus properties is a prerequisite for efficient problem-solving and deeper study of loci, transformations, and analytic geometry involving complex numbers. For an expanded review of advanced results, consult Complex Numbers And Quadratic Equations.