Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{6-2 i}{3} $$

Step-by-Step Solution

Answer

The simplified complex number is \(2 - \frac{2}{3}i\).

Step 1: Write Down the Complex Expression

The given complex expression to simplify is \( \frac{6-2i}{3} \). Our task is to express this as a simplified complex number.

Step 2: Separate the Real and Imaginary Parts

To simplify, separate the real and imaginary components by splitting the fraction: \( \frac{6}{3} - \frac{2i}{3} \).

Step 3: Simplify Each Part

Calculate each component: \( \frac{6}{3} = 2 \) is the real part, and \( \frac{2i}{3} = \frac{2}{3}i \) is the imaginary part.

Step 4: Combine the Components

Combine the simplified real and imaginary parts: \( 2 - \frac{2}{3}i \). This is the simplified form of the complex number.

Key Concepts

Understanding Imaginary Numbers

Imaginary numbers might seem odd at first glance, but they're a fundamental aspect of mathematics, especially when dealing with complex numbers. At the heart of imaginary numbers is the unit "i" which is defined as the square root of -1. This unique attribute allows us to explore numbers beyond the real number system.

When working with imaginary numbers, remember:

• The imaginary unit "i" satisfies the equation: \(i^2 = -1\).
• Any imaginary number can be represented as a multiple of "i". For example, \(2i\) and \(-3.5i\) are imaginary numbers.
• In calculations, treat "i" like an algebraic variable but remember its defining property \(i^2 = -1\).

Imaginary numbers become particularly useful in engineering, physics, and complex systems where equations might not initially have real solutions. The concept can be fascinating, allowing us to extend the mathematical landscape beyond what is traditionally possible with just real numbers.

Diving into Real Numbers

Real numbers are the numbers we most commonly use in everyday life, including:

• Whole numbers like 0, 1, 2, 3...
• Fractions such as \(\frac{1}{2}\), \(\frac{3}{4}\)
• Decimals like 3.14
• Negative numbers such as -5

The set of real numbers encompasses all these kinds of numbers and more, forming a part of the number line from negative infinity to positive infinity.

Real numbers can be further divided into rational numbers, which can be expressed as a ratio of two integers, and irrational numbers, which cannot be neatly written as fractions. Examples of irrational numbers include \(\sqrt{2}\) and \(\pi\). These numbers play essential roles in mathematical operations and various fields of science and engineering. Understanding real numbers helps provide a solid foundation for exploring more complex mathematical concepts.

Simplifying Expressions with Complex Numbers

Simplifying expressions involving complex numbers is a crucial skill in mathematics. Complex numbers have a real part and an imaginary part, and to simplify them, we often need to handle each part separately. Let's take an example given in the problem: \( \frac{6-2i}{3} \).

To simplify:

• Split the fraction into "real" and "imaginary" parts: \(\frac{6}{3} - \frac{2i}{3}\).
• Simplify each part separately. Here, \(\frac{6}{3} = 2\), and \(\frac{2i}{3} = \frac{2}{3}i\).
• Combine the simplified parts to form a new, simplified complex number: \(2 - \frac{2}{3}i\).

This approach lets you neatly sort through complex number expressions, ensuring clear and concise results. Remember to always simplify the coefficients and the imaginary portions individually before recombining them into the final expression.