Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-62: Verify the property. $$ \overline{z \cdot w}=\bar{z} \cdot \bar{w} $$

Step-by-Step Solution

Answer

The property \( \overline{z \cdot w} = \overline{z} \cdot \overline{w}\) is verified.

Step 1: Understand Complex Conjugates

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. For a complex number \(z = a + bi\), its conjugate is \(\overline{z} = a - bi\). Similarly, for \(w = c + di\), its conjugate is \(\overline{w} = c - di\).

Step 2: Multiply Two Complex Numbers

To find the product \(z \cdot w\), multiply the complex numbers \((a + bi)\) and \((c + di)\). By expanding using distribution: \[(a + bi)(c + di) = ac + adi + bci + bdi^2\].Since \(i^2 = -1\), this becomes:\[ac + adi + bci - bd\]. Rewriting gives: \((ac - bd) + (ad + bc)i\).

Step 3: Find Conjugate of the Product

Take the conjugate of the result from Step 2, i.e., the conjugate of \((ac - bd) + (ad + bc)i\): \[\overline{z \cdot w} = (ac - bd) - (ad + bc)i\].

Step 4: Find Product of Conjugates

Now, find the product of the conjugates \(\overline{z} \cdot \overline{w}\). Using \(\overline{z} = a - bi\) and \(\overline{w} = c - di\),multiply:\[(a - bi)(c - di) = ac - adi - bci + bdi^2\].Since \(i^2 = -1\), this simplifies to: \[ac - adi - bci - bd\]. Rewriting gives: \[(ac - bd) - (ad + bc)i\].

Step 5: Compare Results

Compare the results from Step 3 and Step 4. Both expressions obtained are \((ac - bd) - (ad + bc)i\). This confirms that:\[\overline{z \cdot w} = \overline{z} \cdot \overline{w}\].

Key Concepts

Complex Conjugates

Complex conjugates are a fundamental concept when working with complex numbers. The complex conjugate of a complex number changes the sign of its imaginary part while retaining the real part. For instance, if you have a complex number represented as \( z = a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, its conjugate is \( \overline{z} = a - bi \).

Understanding this flip in the imaginary component is essential because it allows various operations and properties to simplify to real numbers when used in certain mathematical contexts. Similarly, for another complex number \( w = c + di \), the conjugate would be \( \overline{w} = c - di \). This concept of conjugation helps in solving equations, analyzing functions, and plays a key role in complex analysis.

By transforming the imaginary component's sign, complex conjugates also facilitate easier calculations of the modulus and other expressions that require a real result. Always remember that conjugating twice returns you to the original complex number: \( \overline{\overline{z}} = z \).

Multiplication of Complex Numbers

Multiplying two complex numbers requires distributing each part of the numbers carefully. Consider two complex numbers, \( z = a + bi \) and \( w = c + di \). The product \( z \cdot w \) is calculated by expanding the terms as you would in normal algebra:

• First, multiply the real parts: \( ac \).
• Then, multiply the outer terms: \( adi \).
• Next, multiply the inner terms: \( bci \).
• Finally, multiply the imaginary components and bear in mind that \( i^2 = -1 \): the result of \( bdi^2 \) is \( -bd \).

This results in: \((ac - bd) + (ad + bc)i\).

During complex multiplication, recognizing the influence of \( i^2 = -1 \) is crucial, as it affects the outcome, converting part of the imaginary product back into a real number. This technique of multiplying ensures you uphold the fundamental structure of complex numbers, keeping parts distinctly real and imaginary. Always double-check your arithmetic to ensure that the imaginary terms are simplified correctly.

Properties of Complex Numbers

Complex numbers possess intriguing properties that make operations elegant and straightforward. One key property involves understanding the addition, multiplication, and behavior under conjugation and complex division.

• Closure: The set of complex numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means any operation within these rules results in another complex number.
• Associative and Commutative: Complex number addition and multiplication are both associative and commutative, which means the order of operations does not affect the result.
• Distributive Property: Similar to real numbers, complex numbers follow this property, \( z(x + y) = zx + zy \).
• Conjugate Symmetry: The conjugate of a sum is the sum of the conjugates, and the conjugate of a product is the product of the conjugates, which underpins the principle we need to verify.
  The modulus of a complex number \( |z| \) remains the same as that of its conjugate. This holds true as \( |z| = |\overline{z}| \).

These properties empower you to solve problems with confidence using a solid understanding of how operations affect the structure and outcome of complex expressions.

Conjugate of a Product

The beauty of complex numbers lies partially in their conjugation properties, especially when multiplied. To understand the conjugate of a product, consider the product of two complex numbers \( z = a + bi \) and \( w = c + di \).

When you multiply these numbers to get \( z \cdot w = (ac - bd) + (ad + bc)i \), the conjugate of the product, \( \overline{z \cdot w} \), naturally switches the sign of its imaginary part, yielding \( (ac - bd) - (ad + bc)i \). This is derived by applying the conjugation rule to the whole product.

Similarly, when you consider the individual conjugates, \( \overline{z} = a - bi \) and \( \overline{w} = c - di \), and calculate their product, you will find it leads to the same expression: \( (ac - bd) - (ad + bc)i \), affirming the property \( \overline{z \cdot w} = \overline{z} \cdot \overline{w} \).

This symmetry under conjugation is not just a curiosity but a vital characteristic that supports the complex field's integrity and allows accurate algebraic manipulations to stay consistent across expressions involving complex numbers.