Problem 24 - Algebra 2 and Trigonometry

Question

In \(18-25,\) write the complex conjugate of each number. $$ \frac{5}{3}-\frac{2}{3} i $$

Step-by-Step Solution

Answer

The complex conjugate is \(\frac{5}{3} + \frac{2}{3} i\).

Step 1: Identify the Complex Number

The given complex number is \(\frac{5}{3} - \frac{2}{3} i\). It is in the form \(a + bi\), where \(a = \frac{5}{3}\) and \(b = -\frac{2}{3}\).

Step 2: Understand the Concept of Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). It involves changing the sign of the imaginary part.

Step 3: Write the Complex Conjugate

To find the complex conjugate of \(\frac{5}{3} - \frac{2}{3} i\), change the sign of the imaginary part. Thus, the complex conjugate is \(\frac{5}{3} + \frac{2}{3} i\).

Key Concepts

Understanding Complex Numbers

Complex numbers are an extension of the real numbers and include both a real and an imaginary part. They are used to solve equations that do not have solutions within the realm of real numbers alone, like the square root of negative numbers. A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) representing the square root of \(-1\).
The combination of these parts allows us to work with problems in algebra and calculus that go beyond basic real number solutions. Complex numbers are significant in many fields of science and engineering. For example, they are essential in signal processing and control systems.

• The real part, \(a\), lies along the horizontal axis, while the imaginary part, \(bi\), lies along the vertical axis, allowing for a straightforward graphical representation.

Understanding the layout and utility of complex numbers is a critical aspect of various advanced mathematics courses, as they help bridge concepts from algebra into more complex theories.

Identifying the Imaginary Part

The imaginary part of a complex number is the component that includes the imaginary unit \(i\). In a complex number expressed as \(a + bi\), \(b\) is the coefficient of \(i\), and it represents the imaginary part.
The imaginary unit \(i\) is defined such that \(i^2 = -1\). This unique quality of \(i\) allows it to facilitate the extension of real numbers into the complex plane. When studying or working with imaginary numbers:

• Make sure you properly identify and understand where the imaginary part lies in the complex number, which will be crucial when performing operations like addition, subtraction, or finding complex conjugates.
• The imaginary part is what sets complex numbers apart from real numbers and allows them to solve equations that real numbers cannot.

This part doesn't need to be "imaginary" in the sense of being unreal or fictional, just distinct from the real axis.

Integrating Complex Numbers into Algebra

Algebra incorporates complex numbers to expand the types of problems you can solve. Understanding the operations involving complex numbers, such as addition, subtraction, and finding the complex conjugate, is crucial.
In the given exercise, the complex number is \(\frac{5}{3} - \frac{2}{3} i\), where algebraic operations are applied directly to find the complex conjugate. To determine the complex conjugate, you simply invert the sign of the imaginary part, resulting in \(\frac{5}{3} + \frac{2}{3} i\).
Some key algebraic manipulations with complex numbers include:

• For addition or subtraction, combine like parts: add real parts with real parts, and imaginary parts with imaginary parts.
• For multiplication, use the distributive property (FOIL method) and remember that \(i^2 = -1\).
• Conjugates are especially useful in division, as multiplying by the conjugate eliminates the imaginary part in the denominator, simplifying expressions.

Incorporating complex numbers into algebra not only enables solving a wider range of equations but also illuminates a deeper understanding of algebraic structures.