Find The Modulus Of The Following.1. Es002-1.jpg

Finding the Modulus: A Comprehensive Guide with Worked Examples

Finding the modulus, often represented as |z|, is a fundamental concept in complex numbers. It represents the distance of a complex number from the origin on the complex plane. Understanding how to find the modulus is crucial for various applications in mathematics, engineering, and physics. This article will provide a detailed explanation of how to find the modulus of complex numbers, covering various forms and scenarios with numerous worked examples.

What is a Complex Number?

Before diving into the modulus, let's briefly review complex numbers. A complex number, denoted as z, is a number that can be expressed in the form a + bi, where:

• a is the real part (Re(z))
• b is the imaginary part (Im(z))
• i is the imaginary unit, defined as the square root of -1 (√-1)

Finding the Modulus: The Formula

The modulus of a complex number z = a + bi is calculated using the following formula:

|z| = √(a² + b²)

This formula is derived from the Pythagorean theorem, considering the complex number as a point (a, b) in the Cartesian coordinate system. The modulus represents the distance from the origin (0, 0) to this point.

Methods for Finding the Modulus

Let's explore different methods for calculating the modulus, with examples for each:

1. Rectangular Form (a + bi):

This is the most common form of a complex number. To find the modulus, simply apply the formula directly.

Example 1: Find the modulus of z = 3 + 4i

|z| = √(3² + 4²) = √(9 + 16) = √25 = 5

Example 2: Find the modulus of z = -2 - 2i

|z| = √((-2)² + (-2)²) = √(4 + 4) = √8 = 2√2

Example 3: Find the modulus of z = 5i

Since the real part is 0, we have:

|z| = √(0² + 5²) = √25 = 5

2. Polar Form (r(cos θ + i sin θ)):

The polar form offers an alternative way to represent a complex number. Here, r represents the modulus, and θ represents the argument (angle).

• r = |z| = √(a² + b²)
• θ = arctan(b/a) (Note: Care must be taken with the quadrant when using arctan; consider the signs of a and b)

In polar form, finding the modulus is straightforward because r is already the modulus.

Example 4: Convert z = 1 + i to polar form and find its modulus.

• a = 1, b = 1
• r = √(1² + 1²) = √2
• θ = arctan(1/1) = π/4
• Polar form: z = √2(cos(π/4) + i sin(π/4))
• Modulus: |z| = √2

Example 5: Given z = 3(cos(π/3) + i sin(π/3)), find the modulus.

The modulus is directly given as r = 3.

3. Exponential Form (re^(iθ)):

Euler's formula connects the exponential form to the polar form: e^(iθ) = cos θ + i sin θ. Therefore, the modulus is again represented by r.

Example 6: Given z = 2e^(iπ/6), find the modulus.

The modulus is directly given as r = 2.

Properties of Modulus

Understanding the properties of the modulus is essential for simplifying calculations and solving problems involving complex numbers. Some key properties include:

• |z₁z₂| = |z₁||z₂|: The modulus of the product of two complex numbers is the product of their moduli.
• |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0): The modulus of the quotient of two complex numbers is the quotient of their moduli.
• |zⁿ| = |z|ⁿ: The modulus of a complex number raised to a power is the modulus raised to that power.
• |z| = |-z|: The modulus of a complex number and its negative are equal.
• |z̄| = |z|: The modulus of a complex number and its conjugate are equal (where z̄ is the conjugate of z, obtained by changing the sign of the imaginary part).

Applications of Modulus

The concept of modulus has wide-ranging applications, including:

• Complex Plane Geometry: Determining distances and lengths related to complex numbers.
• Signal Processing: Representing the amplitude of signals.
• Quantum Mechanics: Describing the probability amplitude of quantum states.
• Electrical Engineering: Analyzing alternating current (AC) circuits.
• Fluid Dynamics: Solving problems involving complex potential flow.

Advanced Examples

Let's tackle some more complex examples involving the properties of modulus:

Example 7: Find the modulus of z = (2 + 3i)(1 - i)

First, find the product: (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i

Then, find the modulus: |5 + i| = √(5² + 1²) = √26

Alternatively, using the property |z₁z₂| = |z₁||z₂|:

|2 + 3i| = √(2² + 3²) = √13 |1 - i| = √(1² + (-1)²) = √2 |(2 + 3i)(1 - i)| = √13 * √2 = √26

Example 8: Find the modulus of z = (3 + 4i) / (1 + 2i)

First, find the quotient: (3 + 4i) / (1 + 2i) = (3 + 4i)(1 - 2i) / (1 + 2i)(1 - 2i) = (3 - 6i + 4i - 8i²) / (1 - 4i²) = (11 - 2i) / 5 = 11/5 - (2/5)i

Then, find the modulus: |11/5 - (2/5)i| = √((11/5)² + (-2/5)²) = √(121/25 + 4/25) = √(125/25) = √5

Alternatively, using the property |z₁/z₂| = |z₁|/|z₂|:

|3 + 4i| = √(3² + 4²) = 5 |1 + 2i| = √(1² + 2²) = √5 |(3 + 4i) / (1 + 2i)| = 5 / √5 = √5

Conclusion:

Finding the modulus of a complex number is a fundamental skill in complex analysis. Understanding the formula and its application across different forms (rectangular, polar, exponential) along with the properties of modulus allows for efficient problem-solving in various fields. By mastering these concepts and practicing with diverse examples, you will solidify your understanding of complex numbers and their applications. Remember to always double-check your calculations and consider the use of alternative methods to verify your answers. This comprehensive guide provides a strong foundation for further exploration of complex number theory.