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Chapter 1: Problem 78

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) show that each statement is true. $$\overline{z w}=\bar{z} \cdot \bar{w}$$

Short Answer

Expert verified
The property \( \overline{zw} = \bar{z} \cdot \bar{w} \) holds for complex numbers.

Step by step solution

01

Understand the Statement

We need to prove that the conjugate of the product of two complex numbers is equal to the product of their conjugates, i.e., \( \overline{z w} = \bar{z} \cdot \bar{w} \). Given \( z = a + bi \) and \( w = c + di \).
02

Calculate the Product of Complex Numbers

First, calculate the product \( z w \) where \( z = a + bi \) and \( w = c + di \). The product is given by \((a + bi)(c + di) = ac + adi + bci + bdi^2.\) Since \(i^2 = -1\), we have \( z w = (ac - bd) + (ad + bc)i \).
03

Find the Conjugate of the Product

The conjugate of \( z w = (ac - bd) + (ad + bc)i \) is \( \overline{z w} = (ac - bd) - (ad + bc)i \).
04

Calculate the Conjugates of z and w Separately

The conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \). The conjugate of \( w = c + di \) is \( \bar{w} = c - di \).
05

Multiply the Conjugates

Multiply \( \bar{z} \) and \( \bar{w} \) to get \((a - bi)(c - di) = ac - adi - bci + bdi^2.\) Simplifying using \(i^2 = -1\), this becomes \( (ac - bd) - (ad + bc)i \).
06

Compare Results

Compare the result of \( \overline{z w} = (ac - bd) - (ad + bc)i \) with the result of \( \bar{z} \cdot \bar{w} = (ac - bd) - (ad + bc)i \). They are equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Conjugate
The complex conjugate of a complex number is a reflection of the number across the real axis in the complex plane. Any complex number can be expressed in the form \(z = a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Its complex conjugate is denoted by \(\bar{z}\) and is given by \(\bar{z} = a - bi\). This operation essentially changes the sign of the imaginary component.

Understanding the role of the complex conjugate is central to many mathematical applications, especially in simplifying expressions and solving equations that include complex numbers.
Product of Complex Numbers
When multiplying two complex numbers, \(z = a + bi\) and \(w = c + di\), the distributive property of multiplication is used. The formula can be expanded as \((a+bi)(c+di) = ac + adi + bci + bdi^2\).

Remember, \(i^2 = -1\), therefore \(bdi^2 = -bd\). Simplifying gives \((ac - bd) + (ad + bc)i\) as the result. This result, like other complex numbers, has a real and an imaginary part. This technique of multiplication forms the basis for many operations involving complex numbers.
Conjugate of a Product
To understand the conjugate of a product, consider two complex numbers \(z = a + bi\) and \(w = c + di\). Their conjugates are \(\bar{z} = a - bi\) and \(\bar{w} = c - di\), respectively. According to the problem, we need to show that \(\overline{z w}=\bar{z} \cdot \bar{w}\).

Following the process: Calculate \(zw = (a+bi)(c+di)\) giving \((ac-bd) + (ad+bc)i\) and find its conjugate \((ac-bd)-(ad+bc)i\). On the other hand, multiplying the conjugates gives \((a - bi)(c - di) = ac - adi - bci + bdi^2 = (ac-bd)-(ad+bc)i\). The results are identical, proving the property that the conjugate of the product is the product of the conjugates.
Complex Number Properties
Complex numbers exhibit several interesting properties that simplify many calculations.
  • Closure: The sum or product of complex numbers is also a complex number.
  • Associativity: The order of addition and multiplication does not affect the outcome, e.g., \((z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)\).
  • Commutativity: The order of adding or multiplying two complex numbers does not matter, i.e., \(z_1 + z_2 = z_2 + z_1\) and \(z_1 \cdot z_2 = z_2 \cdot z_1\).
  • Distributive Property: Multiplication distributes over addition as in real numbers. If we have a sum of two numbers multiplied by a third, \((z_1 + z_2)z_3 = z_1z_3 + z_2z_3\).


The conjugate and modulus are frequently used tools, offering a way to solve equations and perform operations within the realm of complex numbers.

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