Evaluating Complex Expressions: A Step-by-Step Guide

In the world of mathematics, complex numbers open up fascinating possibilities. This article dives deep into the process of evaluating complex expressions, providing a clear, step-by-step approach. We'll tackle the expression $(2-5i)(p+q)(i)$ when $p=2$ and $q=5i$, revealing the underlying principles and techniques involved. Whether you're a student grappling with complex numbers for the first time or a seasoned mathematician looking for a refresher, this guide will equip you with the knowledge and skills to confidently navigate the realm of complex number evaluation.

Understanding Complex Numbers

Before we jump into the evaluation, it’s crucial to grasp the fundamentals of complex numbers. A complex number is essentially a combination of a real number and an imaginary number. It's expressed in the form a + bi, where a represents the real part, b represents the imaginary part, and i is the imaginary unit, defined as the square root of -1 (i = √-1). This seemingly simple concept unlocks a whole new dimension in mathematics, allowing us to solve equations that have no solutions within the realm of real numbers alone.

Why are complex numbers so important? They play a vital role in various fields, from electrical engineering and quantum mechanics to fluid dynamics and signal processing. Understanding complex numbers is not just an academic exercise; it's a gateway to comprehending and contributing to these advanced scientific and technological domains.

Think of the real part (a) as a point on the familiar number line, and the imaginary part (b) as a point on a perpendicular number line. Together, they define a point on a two-dimensional plane called the complex plane. This geometric representation provides a powerful visual tool for understanding and manipulating complex numbers. Operations like addition, subtraction, multiplication, and division can be visualized as geometric transformations on this plane, offering a deeper intuitive understanding of their behavior.

Now, let's delve into the arithmetic of complex numbers. Adding and subtracting complex numbers is straightforward: simply combine the real parts and the imaginary parts separately. For example, (2 + 3i) + (1 - i) = (2 + 1) + (3 - 1)i = 3 + 2i. Multiplication, however, requires a bit more care. We use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) and the crucial fact that i² = -1. This property is the key to simplifying complex expressions and obtaining results in the standard a + bi form.

Evaluating the Expression: A Step-by-Step Breakdown

Now, let's tackle the expression at hand: $(2-5i)(p+q)(i)$ when $p=2$ and $q=5i$. This expression might look intimidating at first glance, but by breaking it down into manageable steps, we can conquer it with confidence. The key is to follow the order of operations (PEMDAS/BODMAS) and carefully apply the rules of complex number arithmetic.

Step 1: Substitute the values of p and q.

Our first task is to substitute the given values of p and q into the expression. We're told that p = 2 and q = 5i. Plugging these values in, we get:

(2 - 5i)(2 + 5i)(i)

This substitution is a crucial first step, transforming the abstract expression into a concrete one that we can work with. It's like laying the foundation for a building – without it, we can't proceed further.

Step 2: Simplify the expression (2 + 5i).

Next, we focus on simplifying the term (2 + 5i). Notice that this term, along with (2 - 5i), forms a special pair called complex conjugates. Complex conjugates have the same real part but opposite imaginary parts. Their product is always a real number, a property that's incredibly useful in simplifying expressions. In this case, (2 - 5i) and (2 + 5i) are complex conjugates.

Step 3: Multiply the complex conjugates (2 - 5i) and (2 + 5i).

Now, let's multiply the complex conjugates (2 - 5i) and (2 + 5i). We'll use the distributive property (FOIL method) to expand the product:

(2 - 5i)(2 + 5i) = 2 * 2 + 2 * 5i - 5i * 2 - 5i * 5i

= 4 + 10i - 10i - 25i²

Notice that the imaginary terms (+10i and -10i) cancel each other out, leaving us with:

= 4 - 25i²

Now, we use the fundamental property of imaginary numbers: i² = -1. Substituting this in, we get:

= 4 - 25(-1)

= 4 + 25

= 29

As expected, the product of the complex conjugates is a real number, 29. This simplification significantly reduces the complexity of our expression.

Step 4: Multiply the result by i.

We're now left with 29(i). This is a simple multiplication: we just multiply the real number 29 by the imaginary unit i.

29(i) = 29i

This is our final result, expressed in the standard form of a complex number (0 + 29i). The real part is 0, and the imaginary part is 29.

The Final Result

Therefore, the value of the expression $(2-5i)(p+q)(i)$ when $p=2$ and $q=5i$ is 29i. This corresponds to option A in the multiple-choice options provided.

Common Mistakes to Avoid

When working with complex numbers, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting that i² = -1: This is the most crucial property to remember. Failing to substitute -1 for i² will lead to incorrect results.
• Incorrectly applying the distributive property: Be meticulous when expanding products of complex numbers. Ensure you multiply each term in the first complex number by each term in the second complex number.
• Mixing up real and imaginary parts: Keep track of which terms are real and which are imaginary. Don't accidentally combine them when adding or subtracting complex numbers.
• Skipping steps: It's tempting to rush through the calculations, but skipping steps increases the risk of making errors. Write out each step clearly to minimize mistakes.

By being aware of these common pitfalls and taking the time to work through the problems carefully, you can significantly improve your accuracy and confidence in handling complex number expressions.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. Evaluate $(3 + 2*i*)(1 - *i*)$.
2. Simplify (4 - 3i)².
3. Find the value of (2 + i)/(1 - i).

Working through these problems will help you internalize the concepts and techniques discussed in this article. Remember, practice makes perfect! The more you work with complex numbers, the more comfortable and proficient you'll become.

Conclusion

Evaluating complex expressions might seem daunting at first, but with a clear understanding of the fundamentals and a systematic approach, it becomes a manageable and even enjoyable task. By breaking down the expression into steps, carefully applying the rules of complex number arithmetic, and avoiding common mistakes, you can confidently navigate the world of complex numbers. This article has provided a comprehensive guide to evaluating the specific expression $(2-5i)(p+q)(i)$ when $p=2$ and $q=5i$. However, the principles and techniques discussed here are applicable to a wide range of complex number expressions.

Continue to explore the fascinating world of mathematics, and remember that consistent practice is the key to mastery. Don't hesitate to seek out additional resources and examples to deepen your understanding. The journey of mathematical discovery is a rewarding one, filled with challenges and triumphs. Embrace the challenges, celebrate the triumphs, and never stop learning.

For further exploration of complex numbers and their applications, consider visiting resources like Khan Academy's Complex Numbers section.