Section 1: Mindmap

Section 2: Main Content with Inline Diagrams

Introduction: The modulus of a complex number is a fundamental concept in algebra and complex number theory, crucial for understanding the magnitude or length of complex numbers in the Argand plane. This section covers the definition, properties, and applications of the modulus of complex numbers, with clear explanations and diagrams to build a strong conceptual foundation for NDA entrance exam preparation.

1. Definition and Geometric Interpretation

A complex number is generally expressed in the form:

\( z = x + iy \), where \( x, y \in \mathbb{R} \) and \( i = \sqrt{-1} \).

The modulus (or absolute value) of \( z \), denoted by \( |z| \), is defined as the distance of the point \( (x, y) \) from the origin \( (0,0) \) in the Cartesian plane (also called the Argand plane).

Mathematically,

\( |z| = \sqrt{x^2 + y^2} \).

Geometric Diagram:

In this diagram, the blue line represents the modulus \( |z| \), which is the hypotenuse of the right triangle formed by the real part \( x \) (horizontal green dashed line) and the imaginary part \( y \) (vertical green dashed line).

2. Properties of Modulus

• Non-negativity: \( |z| \geq 0 \) and \( |z| = 0 \iff z = 0 \).
• Multiplicative Property: For any two complex numbers \( z_1 \) and \( z_2 \), \[ |z_1 z_2| = |z_1| \cdot |z_2|. \]
• Division Property: If \( z_2 \neq 0 \), then \[ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}. \]
• Triangle Inequality: For any two complex numbers \( z_1 \) and \( z_2 \), \[ |z_1 + z_2| \leq |z_1| + |z_2|. \] This inequality is fundamental in complex analysis and geometry.
• Reverse Triangle Inequality: \[ \big|,|z_1| - |z_2|,\big| \leq |z_1 - z_2|. \]

3. Calculation Methods

a) Using Cartesian Form

Given \( z = x + iy \), calculate modulus as:

\( |z| = \sqrt{x^2 + y^2} \).

b) Using Polar Form

A complex number can also be expressed in polar form:

\( z = r(\cos \theta + i \sin \theta) \), where \( r = |z| \) and \( \theta = \arg(z) \).

Here, the modulus \( |z| = r \) is the radius or distance from the origin.

4. Applications

• Distance Between Complex Numbers: The distance between \( z_1 \) and \( z_2 \) is given by \[ |z_1 - z_2|. \]
• Solving Equations: Modulus helps in solving equations involving complex numbers, especially when magnitude constraints are given.
• Inequalities: Triangle inequality is used to estimate bounds and prove inequalities involving complex numbers.

5. Common Mistakes & Tips

• Do not confuse modulus \( |z| \) with argument \( \arg(z) \).
• Remember modulus is always non-negative.
• Apply triangle inequality carefully; equality holds only when \( z_1 \) and \( z_2 \) are collinear in the Argand plane.

Section 3: Worked Examples

Section 4: Formula Bank

• Modulus of \( z = x + iy \):
  \( |z| = \sqrt{x^2 + y^2} \)
• Modulus in Polar Form:
  \( |z| = r \) where \( z = r(\cos \theta + i \sin \theta) \)
• Multiplicative Property:
  \( |z_1 z_2| = |z_1| \cdot |z_2| \)
• Division Property:
  \( \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \), \( z_2 \neq 0 \)
• Triangle Inequality:
  \( |z_1 + z_2| \leq |z_1| + |z_2| \)
• Reverse Triangle Inequality:
  \( \big|,|z_1| - |z_2|,\big| \leq |z_1 - z_2| \)
• Distance Between Two Complex Numbers:
  \( d = |z_1 - z_2| \)