Problem 21 - Algebra 2 and Trigonometry

Question

In \(18-25,\) write the complex conjugate of each number. $$ -6-9 i $$

Step-by-Step Solution

Answer

The conjugate of \(-6 - 9i\) is \(-6 + 9i\).

Step 1: Understand the Concept of Conjugate

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. So for any complex number in the form \( a + bi \), the conjugate is \( a - bi \).

Step 2: Identify the Original Complex Number

Identify the real and imaginary components of the given complex number \(-6 - 9i\). Here, the real part is \(-6\) and the imaginary part is \(-9i\).

Step 3: Apply the Concept of Conjugation

Change the sign of the imaginary part while keeping the real part the same. The original number is \(-6 - 9i\), so the conjugate will be \(-6 + 9i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually expressed in the form \( a + bi \), where \( a \) represents the real component, and \( bi \) represents the imaginary component. In this format, \( b \) is a real number, and \( i \) is the imaginary unit. Complex numbers extend our number system beyond just real numbers, allowing us to solve equations that do not have real solutions. For example, the equation \( x^2 + 1 = 0 \) has solutions in the form of complex numbers. By incorporating these numbers into mathematics, we can solve a wider range of problems.

Real and Imaginary Parts

In a complex number like \( a + bi \), understanding the real and imaginary parts is crucial. The real part is \( a \), and it behaves just like any other real number you are familiar with. The imaginary part, \( bi \), includes the imaginary unit \( i \), which is defined as the square root of \(-1\). This setup allows us to handle the square roots of negative numbers and to perform mathematical operations that involve them. Identifying these parts helps in operations like finding conjugates, performing addition or subtraction with complex numbers, or even multiplying them.

Algebra 2

In Algebra 2, concepts such as complex numbers and their properties become more apparent and applicable. You learn how to manipulate and work with complex numbers, perform operations like addition, subtraction, multiplication, division, and conjugation. For example, in conjugation, if you take the complex number \( a + bi \), its conjugate would be \( a - bi \). This process helps in simplifying expressions and solving equations involving complex numbers. Algebra 2 expands the horizons of what can be solved by including these skills in its curriculum.

Imaginary Unit

The imaginary unit \( i \) is essential in the realm of complex numbers. Defined as \( i = \sqrt{-1} \), it allows us to extend the regular number line into a complex plane. This concept might initially seem abstract, but it holds significant importance in mathematics and engineering. With \( i \), we can represent and solve problems involving square roots of negative numbers. When combined with a real number, it creates the imaginary part of a complex number. It helps in calculations, especially when squaring \( i \), because \( i^2 = -1 \), which frequently simplifies expressions in complex number operations.