Problem 60 - Precalculus: Functions and Graphs

Question

Verify the property. $$\overline{z / w}=\bar{z} / \bar{w}$$

Step-by-Step Solution

Answer

The property \( \overline{z / w} = \bar{z} / \bar{w} \) is true.

Step 1: Understand the Problem

We need to verify that the conjugate of the division of two complex numbers is the same as the division of their respective conjugates. In mathematical terms, we need to show that \(\overline{z/w} = \bar{z}/\bar{w}\) holds true for any complex numbers \(z\) and \(w\).

Step 2: Express Complex Numbers

Write the complex numbers in their standard form. Let \(z = a + bi\) and \(w = c + di\) where \(a, b, c,\) and \(d\) are real numbers, and \(i\) is the imaginary unit.

Step 3: Find the Division \(z/w\)

To divide two complex numbers \(z = a + bi\) and \(w = c + di\), multiply the numerator and the denominator by the conjugate of the denominator: \[\frac{z}{w} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{ac + bd + i(bc - ad)}{c^2 + d^2}\] Here, the expression in the denominator is obtained using \((c+di)(c-di) = c^2 + d^2\).

Step 4: Conjugate of \(z/w\)

Take the conjugate of the expression obtained in Step 3: \[\overline{\frac{z}{w}} = \frac{ac + bd}{c^2 + d^2} - i\frac{bc - ad}{c^2 + d^2}\] This results from the fact that the conjugate of a complex number \(x + yi\) is \(x - yi\).

Step 5: Conjugate of \(z\) and \(w\)

Find the conjugates of \(z\) and \(w\): \(\bar{z} = a - bi\) and \(\bar{w} = c - di\).

Step 6: Division of Conjugates

Calculate \(\bar{z}/\bar{w}\):\[\frac{\bar{z}}{\bar{w}} = \frac{(a-bi)(c+di)}{(c-di)(c+di)} = \frac{ac + bd - i(bc - ad)}{c^2 + d^2}\] Thus, the result is \(\frac{ac + bd}{c^2 + d^2} - i\frac{bc - ad}{c^2 + d^2}\).

Step 7: Compare Results

Compare the results from Step 4 and Step 6. Both expressions match:\[\overline{\frac{z}{w}} = \frac{ac + bd}{c^2 + d^2} - i\frac{bc - ad}{c^2 + d^2}\]Therefore, \(\overline{z / w} = \bar{z} / \bar{w}\) is verified.

Key Concepts

Complex Conjugate

The concept of a complex conjugate is central when dealing with complex numbers. Each complex number is made up of a real part and an imaginary part. For example, if you have a complex number \( z = a + bi \), the complex conjugate of \( z \) is \( \bar{z} = a - bi \). Here, \( a \) and \( b \) are real numbers, while \( i \) is the imaginary unit, defined as \( i^2 = -1 \). The complex conjugate is simply obtained by changing the sign of the imaginary part.

• Used to simplify division of complex numbers.
• Helps cancel out the imaginary part during multiplication with another complex number.
• The product of a complex number and its conjugate is always a real number.

Understanding the complex conjugate is crucial because it allows operations like division to be carried out easily. Essentially, multiplying by the conjugate helps you eliminate the imaginary component from the denominator, making the division manageable.

Real and Imaginary Parts

When talking about complex numbers, it's important to understand their structure. A complex number \( z = a + bi \) consists of a real part and an imaginary part. In this form:

• The real part is \( a \).
• The imaginary part is \( bi \).

The real part \( a \) is on the horizontal axis, while \( bi \) is on the vertical axis when graphed on the complex plane. These parts are essential for calculations like addition, subtraction, and especially multiplication and division.To find the sum or difference of complex numbers, handle the real and imaginary parts separately. For multiplication and division, these parts are crucial in justifying how complex number operations resemble those with real numbers, with additional considerations for the imaginary unit \( i \). Understanding these parts helps clarify why the conjugate works for division simplification.

Complex Number Multiplication

Multiplication is fundamental when dealing with complex numbers, especially in the context of division where multiplying by the complex conjugate plays a critical role. Given two complex numbers \( z = a + bi \) and \( w = c + di \), their product is calculated as follows:Firstly, apply the distributive property:\[zw = (a+bi)(c+di) = ac + adi + bci + bdi^2\]Because \( i^2 = -1 \), this simplifies to:\[zw = (ac - bd) + (ad + bc)i\]When dividing complex numbers, the key is to eliminate the imaginary part in the denominator. This involves multiplying the numerator and the denominator by the conjugate of the denominator. This is why understanding and applying multiplication is a core concept when working with complex numbers. By converting division into multiplication using the conjugate, calculations become more straightforward, allowing clear handling of the real and imaginary parts separately.