Problem 95 - 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: \(\overline{z+v}\)

Step-by-Step Solution

Answer

The value of \(\overline{z+v}\) is \(6 - 3i\).

Step 1: Add the Complex Numbers

First, compute the sum of the complex numbers \(z\) and \(v\). The complex number \(z = 3 - 2i\) and \(v = 3 + 5i\). Adding these together, we work component-wise: add the real parts and the imaginary parts separately.\[(3 + 3) + (-2i + 5i) = 6 + 3i\]

Step 2: Compute the Conjugate

Now, compute the conjugate of the resulting complex number from Step 1. The conjugate of a complex number \(a + bi\) is \(a - bi\). Thus, for \(6 + 3i\), its conjugate is:\[6 - 3i\]

Key Concepts

Understanding Complex Conjugate

A complex conjugate is a concept that often puzzles students initially, but it's quite straightforward once you get the hang of it! Think of a complex number in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). The complex conjugate is simply a reflection across the real axis. To find it, you keep the real part \(a\) unchanged, and flip the sign of the imaginary part \(b\).

• Original complex number: \(a + bi\)
• Complex conjugate: \(a - bi\)

Why is the complex conjugate important? It helps simplify the division of complex numbers and is used to find the modulus or magnitude of complex numbers. For example, the magnitude of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). The conjugate \(\overline{z} = a - bi\) plays a crucial role in this computation.

Addition of Complex Numbers

Adding complex numbers is very much like adding traditional numbers, but with a twist! Each complex number consists of a real part and an imaginary part. During addition, you combine these components separately, which is very straightforward.

Here's how:

• Add the real parts of the numbers together.
• Add the imaginary parts of the numbers together.

For instance, given two complex numbers \(z = 3 - 2i\) and \(v = 3 + 5i\), the addition process looks like this:
- The real parts: \(3 + 3 = 6\)- The imaginary parts: \(-2i + 5i = 3i\)
So, when you add them, you get the resultant complex number \(6 + 3i\). Addition in complex numbers is commutative and associative, meaning you can add them in any order and still get the same result!

Real and Imaginary Components

Real and imaginary components are fundamental to understanding complex numbers. Each complex number comprises two parts:

• The real part is the "regular" number you're used to, like \(3\) in the number \(3 - 2i\).
• The imaginary part is the component multiplied by \(i\), like \(-2i\) in \(3 - 2i\).

The real part represents a position on the horizontal axis of the complex plane, while the imaginary part represents a position on the vertical axis. Together, they determine the position of the complex number in the plane.

Having these two components allows complex numbers to represent phenomena in multiple dimensions, such as waves and oscillations in electrical engineering. It's crucial to handle the real and imaginary parts properly in operations like addition, subtraction, multiplication, and division to accurately work with complex numbers.