Problem 32 - Advanced Engineering Mathematics

Question

In Problems 27-32, let \(z=x+i y\). Find the indicated expression. $$ |z+5 \bar{z}| $$

Step-by-Step Solution

Answer

The modulus is \(\sqrt{36x^2 + 16y^2}\).

Step 1: Understand the Formulas

Before we start solving, recall the formulas for complex numbers: For any complex number \(z = x + iy\), its complex conjugate \(\bar{z} = x - iy\). The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\).

Step 2: Substitute and Simplify the Expression

Express \(z + 5\bar{z}\) in terms of \(x\) and \(y\), where \(z = x + iy\) and \(\bar{z} = x - iy\). Thus, \(z + 5\bar{z} = (x + iy) + 5(x - iy) = 6x - 4iy\).

Step 3: Calculate the Modulus

Find the modulus of \(6x - 4iy\). Using the modulus formula \(|a + bi| = \sqrt{a^2 + b^2}\), we get \(|6x - 4iy| = \sqrt{(6x)^2 + (-4y)^2} = \sqrt{36x^2 + 16y^2}\).

Step 4: Express the Final Answer

The expression simplifies to \(|z + 5\bar{z}| = \sqrt{36x^2 + 16y^2}\). This is the magnitude of the combined complex expression.

Key Concepts

Complex Conjugate

The concept of a complex conjugate is crucial in understanding complex numbers. When you have a complex number, like \( z = x + iy \), its complex conjugate is denoted as \( \bar{z} \), and it is simply \( x - iy \). This involves changing the sign of the imaginary part.

Complex conjugates have interesting properties. For instance:

• The product of a complex number and its conjugate is a real number. This is because \( z \cdot \bar{z} = (x + iy)(x - iy) = x^2 + y^2 \), which is purely real.
• Conjugates are used to rationalize denominators in complex fractions. By multiplying the numerator and denominator by the conjugate of the denominator, we eliminate the imaginary component.

Understanding these properties can help make complex number calculations simpler and more intuitive. The problem we faced involves using conjugates to manipulate the expression into a more manageable form, making it easier to find its modulus.

Modulus of a Complex Number

The modulus of a complex number is a measure of its size or magnitude. For a complex number \( z = a + bi \), the modulus \(|z|\) is calculated by \( \sqrt{a^2 + b^2} \). Essentially, the modulus gives us the length of the vector representing the complex number in the complex plane.

Why is the modulus useful? Here are a few important points:

• It helps in understanding the distance of a point from the origin in the complex plane.
• Modulus is always a non-negative real number, giving us a clear sense of the magnitude without direction.
• It is particularly helpful in finding distances and in rotating complex numbers.

In the given problem, after simplifying \(z + 5\bar{z} = 6x - 4iy\), the modulus is \(\sqrt{36x^2 + 16y^2}\). This represents the magnitude of the resultant complex number.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions using algebraic rules to simplify or solve them. To tackle complex number problems, knowing how to add, subtract, multiply, and divide complex numbers using algebra is essential.

Here are some key steps used in algebraic manipulation of complex numbers:

• Substitution: Replacing a complex number with its components; for example, writing \(z = x + iy\) into a new form to suit the problem's need.
• Combining like terms: In our problem, \( z + 5\bar{z} = (x + iy) + 5(x - iy) \) simplifies to \(6x - 4iy\).
• Simplification: Breaking expressions to their simplest form, which helps in easily calculating their modulus or other properties.

Using proper algebraic manipulation, we turned a complex expression into a simpler one, making it easier to compute its modulus and understand its nature. This step-by-step simplification is crucial for solving complex number problems effectively.