Problem 93 - 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: \(\bar{z}\)

Step-by-Step Solution

Answer

The complex conjugate of \(z\) is \(3 + 2i\).

Step 1: Understanding Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If a complex number is given as \( a + bi \), its complex conjugate is \( a - bi \). In this exercise, \( z = 3 - 2i \), so we need to find \( \bar{z} \).

Step 2: Apply Conjugation to z

To find \( \bar{z} \), change the sign of the imaginary part of \( z \). Given that \( z = 3 - 2i \), its complex conjugate is \( \bar{z} = 3 + 2i \).

Key Concepts

Understanding the Complex Conjugate

Complex conjugates are a fundamental aspect of complex numbers, and they are especially useful in various mathematical operations, such as dividing complex numbers or finding magnitudes.

The complex conjugate of a complex number is formed by keeping the real part constant while switching the sign of its imaginary part. Consider any complex number represented as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The complex conjugate of this number is written as \( a - bi \).

In practical applications, complex conjugates are often used to simplify the process of rationalizing denominators when dividing complex numbers. They also appear in the calculation of the magnitude of a complex number, where the product of a complex number and its conjugate yields a real number. This concept is crucial for deeper analysis in fields like electrical engineering and quantum physics.

Identifying the Imaginary Part

The imaginary part of a complex number is tied to the component with the imaginary unit \( i \), which is defined as the square root of -1.

For a complex number expressed as \( a + bi \), "\( b \)" is the coefficient of \( i \), and it is designated as the imaginary part of the complex number. While the imaginary part alone cannot represent concrete values like its real counterpart, it is integral to complex solutions and modeling scenarios that involve oscillations or waveforms.

The concept of the imaginary part becomes pivotal in operations involving complex numbers and helps in visualizing complex number behaviors on a 2-dimensional plane, known as the Argand plane. Remember that when you compute the conjugate of a complex number, only the sign of the imaginary part changes, while the real part remains unchanged.

Performing Complex Number Operations

Complex number operations encompass addition, subtraction, multiplication, and division, very much like real numbers. However, these operations obey rules specific to complex numbers.

**Addition and Subtraction**
When adding or subtracting complex numbers, you treat them by separately handling their real and imaginary parts. For example, if \( z = a + bi \) and \( w = c + di \), then \( z + w = (a+c) + (b+d)i \) and \( z - w = (a-c) + (b-d)i \).

**Multiplication**
Multiplying complex numbers requires the use of the distributive property, where each term is muliplied individually. Remember, \( i^2 = -1\), which simplifies many expressions. Following this rule, the product \( (a + bi)(c + di) \) would expand to \( (ac - bd) + (ad + bc)i \).

**Division**
Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator, thereby eliminating the imaginary unit in the denominator. This process allows the division to result in a standard complex number form \( a + bi \).

By mastering these operations, you can effortlessly handle equations and expressions involving complex numbers in various fields of mathematics and engineering.