Complex Number Simplification: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of complex numbers. We'll be simplifying a complex expression and rewriting it in the neat form of a + bi, where a and b are rational numbers. This is a fundamental skill in complex number arithmetic, so let's get started! We'll be breaking down the expression step-by-step, making sure everyone can follow along. Don't worry if you're new to this; we'll explain everything clearly. Our goal is to make complex number simplification a breeze. By the end of this, you'll be able to confidently tackle similar problems.

Understanding the Basics of Complex Numbers

Before we jump into the simplification, let's quickly recap what complex numbers are all about. A complex number is a number that can be expressed in the form a + bi, where:

  • a is the real part, and it's a real number.
  • b is the imaginary part, and it's also a real number.
  • i is the imaginary unit, defined as the square root of -1 (√-1). This is the key element that distinguishes complex numbers from real numbers.

So, basically, complex numbers extend the concept of real numbers by introducing the imaginary unit. The imaginary unit i allows us to solve equations that don't have real number solutions, such as the square root of a negative number. Complex numbers are used widely in various fields like electrical engineering, quantum mechanics, and signal processing. Understanding their basic form is crucial for any further operations. Think of a as the 'real' component and bi as the 'imaginary' component of the number. When we simplify complex expressions, we're essentially trying to isolate these two parts to get our final a + bi form. Keep in mind that i² = -1. This is a crucial rule you'll use frequently during our simplification process. Now that we've gone over the basics, let's get our hands dirty with the main task.

Simplifying the Complex Expression: The Main Task

Alright, guys, let's tackle the main task: Simplifying the expression 2(-36 - 3i) + (5 + 2i)(12 - 2i) into the a + bi form. Here's how we'll break it down:

  1. Distribute the multiplication in the first term: Multiply 2 by both terms inside the first set of parentheses.
  2. Multiply the two complex numbers: Use the FOIL method (First, Outer, Inner, Last) to multiply the two complex numbers.
  3. Combine real and imaginary terms: Group the real parts together and the imaginary parts together.
  4. Simplify: Perform the necessary arithmetic operations.

Let's start with the first part of the expression: 2(-36 - 3i). We distribute the 2 to both terms inside the parentheses:

  • 2 * -36 = -72
  • 2 * -3i = -6i

So, 2(-36 - 3i) simplifies to -72 - 6i. Now, let's move on to the second part: (5 + 2i)(12 - 2i). We'll use the FOIL method:

  • First: 5 * 12 = 60
  • Outer: 5 * -2i = -10i
  • Inner: 2i * 12 = 24i
  • Last: 2i * -2i = -4i²

Remember, i² = -1, so -4i² becomes -4 * (-1) = 4. Thus, the expression becomes 60 - 10i + 24i + 4. Next, we'll combine the real and imaginary terms. Adding the real parts (60 + 4), which is 64. Then, combine the imaginary parts (-10i + 24i) which is 14i. The expression simplifies to 64 + 14i. Finally, combine everything together: we had -72 - 6i and 64 + 14i. Add those together: -72 + 64 = -8 and -6i + 14i = 8i.

Therefore, the final simplified form of the expression is -8 + 8i.

Step-by-Step Breakdown and Explanation

Let's walk through the simplification process in more detail, making sure we haven't missed any steps. We are aiming for absolute clarity, so no one will be lost. We started with the expression: 2(-36 - 3i) + (5 + 2i)(12 - 2i). Here's a step-by-step breakdown:

  1. Distribute the 2: This means multiplying 2 by each term inside the first set of parentheses. This step eliminates the parentheses and simplifies the initial part of the expression. 2 * -36 gives us -72, and 2 * -3i results in -6i. So far, our expression looks like: -72 - 6i + (5 + 2i)(12 - 2i).
  2. Apply the FOIL method: FOIL stands for First, Outer, Inner, Last. It's a systematic way to multiply two binomials. This is where we multiply the two complex numbers (5 + 2i) and (12 - 2i). The FOIL method is fundamental when you have to deal with multiplying two binomials. By using FOIL, we ensure that we multiply every term in the first binomial with every term in the second. Applying the FOIL method, 5 times 12 is 60, then 5 times -2i is -10i, then 2i times 12 is 24i, and finally 2i times -2i is -4i².
  3. Simplify i²: Remember that i² = -1. The expression contains -4i², and substituting i² with -1, -4i² becomes -4 * (-1) = 4. This simplifies the expression by removing the imaginary unit squared.
  4. Combine like terms: Now, gather the real parts (the numbers without i) and the imaginary parts (the terms with i). Our expression from step 2 was 60 - 10i + 24i + 4. The real parts are 60 and 4, and the imaginary parts are -10i and 24i. Combining them gives us 64 + 14i.
  5. Combine the results: At this point, we had -72 - 6i and 64 + 14i. Add the real parts together: -72 + 64 = -8. Add the imaginary parts together: -6i + 14i = 8i. Therefore, the simplified form is -8 + 8i. This is the final simplified form, and it adheres to the a + bi format, where a is -8 and b is 8.

Tips for Mastering Complex Number Simplification

Here are some tips and tricks to help you become a pro at simplifying complex expressions:

  • Practice regularly: The more you practice, the more familiar you'll become with the steps involved. Do as many exercises as possible. Consistent practice helps build muscle memory and reduces errors.
  • Memorize i² = -1: This is the most important rule. Remembering this will save you a lot of time and effort. Make sure you always remember the fundamental rule that i² equals -1. Many problems are simply about applying this simple fact.
  • Pay attention to signs: Be careful with positive and negative signs, especially when distributing or multiplying. A small mistake in sign can lead to a completely different answer. Always double-check the signs before you proceed to the next step. This is one of the most common sources of errors, so keep an eye out!
  • Use the FOIL method systematically: When multiplying two complex numbers, use the FOIL method to ensure you don't miss any terms. This is a reliable method to avoid errors when multiplying.
  • Break down complex problems into smaller steps: Don't try to do everything at once. Break down the problem into smaller, more manageable steps, as we did in this guide. This approach reduces the likelihood of making mistakes and makes the problem less intimidating.
  • Check your work: Always double-check your work, especially when dealing with signs and calculations. It’s always a good practice to re-evaluate your calculations. It is always useful to verify and re-calculate your steps. If possible, solve the problem again and compare your answers.

Conclusion: Simplifying Complex Numbers

So there you have it, guys! We've successfully simplified a complex expression, transforming it into the a + bi form. We have learned to identify the real and imaginary parts, perform the necessary operations, and combine the results. Remember that understanding the basics and practicing regularly are key to mastering this skill. You're now well-equipped to handle similar problems. Keep practicing, and you'll be a complex number whiz in no time. If you have any further questions, feel free to ask. Keep up the great work, and happy calculating!