Problem 112 - Calculus for Biology and Medicine

Question

If \(z=a+b i\), find \(\bar{z}\). Use your answer to compute \(\overline{(\bar{z})}\), and compare your answer with \(z\).

Step-by-Step Solution

Answer

\(\overline{(\bar{z})} = z\) for any complex number \(z = a + bi\).

Step 1: Identify the Complex Conjugate

To find the complex conjugate of a complex number \(z = a + bi\), we change the sign of the imaginary part. The complex conjugate of \(z\) is denoted as \(\bar{z}\) and is given by \(\bar{z} = a - bi\).

Step 2: Compute the Double Conjugate

Next, we need to find \(\overline{(\bar{z})}\). The complex conjugate of \(\bar{z} = a - bi\) is found by changing the sign of its imaginary part, resulting in \(\overline{(\bar{z})} = a + bi\).

Step 3: Compare \(\overline{(\bar{z})}\) with \(z\)

Now we compare \(\overline{(\bar{z})}\) with the original complex number \(z\). We have \(z = a + bi\) and \(\overline{(\bar{z})} = a + bi\). Thus, \(\overline{(\bar{z})} = z\).

Key Concepts

Complex Conjugate

In the world of complex numbers, every number can be paired with a unique counterpart called its complex conjugate. A complex number is typically written as \( z = a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part. The complex conjugate of \( z \) is denoted as \( \bar{z} \). To find it, you simply reverse the sign in front of the imaginary part.

• Original complex number: \( z = a + bi \)
• Complex conjugate: \( \bar{z} = a - bi \)

This transformation results in a mirrored version of the original number along the real axis in the complex plane. Understanding the concept of complex conjugation is crucial as it plays a significant role in simplifying complex number equations and finding complex solutions.

Imaginary Part

The imaginary part of a complex number is a fundamental component that defines its place in the complex plane. For a number \( z = a + bi \), the term \( bi \) is the imaginary part, where \( b \) represents the coefficient and \( i \) is the square root of -1. The imaginary part can be thought of as the vertical distance from the real number line in the complex plane.

• Real part: \( a \)
• Imaginary part: \( bi \)

When working with complex numbers, altering the sign of the imaginary part is key for operations like finding the complex conjugate. Understanding and identifying the imaginary part helps in various applications, such as solving complex equations or computing signals.

Double Conjugate

The double conjugate is an interesting property of complex numbers that helps us confirm the original number after performing operations. The process starts by finding the complex conjugate of a given number. For \( z = a + bi \), we get \( \bar{z} = a - bi \). Performing the conjugation again (the double conjugate) on \( \bar{z} \) will revert it back to the original number.

• Original complex number: \( z = a + bi \)
• First conjugate: \( \bar{z} = a - bi \)
• Double conjugate: \( \overline{(\bar{z})} = a + bi \)

Thus, \( \overline{(\bar{z})} = z \). This property shows the symmetry of complex conjugation in the complex plane. Understanding the double conjugate aids in problem solving and verifying the consistency of solutions when dealing with complex numbers.