Problem 96 - Calculus for Biology and Medicine

Question

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\overline{v-w}\)

Step-by-Step Solution

Answer

The conjugate of the expression is \(2 - 6i\).

Step 1: Find the Result of v-w

First, we need to find the difference between the complex numbers \(v\) and \(w\). We have \(v = 3 + 5i\) and \(w = 1 - i\). By subtracting \(w\) from \(v\), we get: \[ v-w = (3 + 5i) - (1 - i) = (3 - 1) + (5i + i) = 2 + 6i. \]

Step 2: Compute the Conjugate of the Result

Next, we need to find the conjugate of the complex number obtained from Step 1. The conjugate of a complex number \(a + bi\) is \(a - bi\). For the complex number \(2 + 6i\), the conjugate is: \[ \overline{v-w} = \overline{2 + 6i} = 2 - 6i. \]

Key Concepts

Conjugate of a Complex Number

In the world of complex numbers, each complex number has a sibling known as its conjugate. Conjugation is the mathematical operation where we swap the sign of the imaginary part of the complex number while the real part remains the same.
For example, for a complex number of the form \(a + bi\), its conjugate is \(a - bi\). This operation is crucial when it comes to simplifying expressions involving complex numbers. You can think of the conjugate as a reflection across the real axis in the complex plane.
The conjugate is especially helpful in division operations and in rationalizing denominators, but it also plays a significant role in various other operations, such as taking differences or powers of complex numbers in their conjugate pair forms.

Complex Number Subtraction

Understanding complex number subtraction requires us to consider both the real and imaginary parts of the complex numbers involved. Subtraction of two complex numbers \(a + bi\) and \(c + di\) is performed by separately subtracting the real parts and the imaginary parts.
Mathematically, this looks like \((a - c) + (b - d)i\). By applying this method, we simultaneously manage the real and imaginary components, ensuring a smooth and logical operation.
For the given problem, to find \(v - w\), with \(v = 3 + 5i\) and \(w = 1 - i\), we calculated that:

• The real parts: \(3 - 1 = 2\)
• The imaginary parts: \(5i - (-i) = 5i + i = 6i\)

Hence, the result of \(v-w\) is \(2 + 6i\). This step is crucial as it sets up the next operation of finding the conjugate.

Step-by-step Solutions

The need for step-by-step solutions arises from the complexity not only of the problem but often of the operations involved. Breaking down a problem allows for a better understanding of each component before combining it for the final answer.
Let's revisit our given problem where the operations were dissected in clear steps:

• Step 1: Subtraction - We found the difference between two complex numbers \(v\) and \(w\) resulting in \(2 + 6i\).
• Step 2: Conjugation - Next, we identified the conjugate of \(2 + 6i\), which is \(2 - 6i\).

By following these steps, learners can observe how operations are applied sequentially.
This approach does not merely aim to give the answer but equips you with the understanding necessary for tackling similar problems on your own.