Simplifying Complex Expressions Into A + Bi Form A Step By Step Guide

In the realm of mathematics, particularly when dealing with complex numbers, it's often necessary to simplify intricate expressions into a standard form. This article will serve as a comprehensive guide on how to simplify a given expression into the form a + bi, where a and b are rational numbers. We'll break down the process step-by-step, using a specific example to illustrate the techniques involved. Our example expression is: 2(-36 - 3i) + (5 + 2i)(12 - 2i). We aim to transform this expression into its simplest form, making it easier to work with in subsequent calculations or analyses. Understanding how to manipulate complex numbers is crucial for various fields, including electrical engineering, quantum mechanics, and advanced mathematical modeling. Let's delve into the world of complex numbers and master the art of simplification. Before we dive into the specifics of our example, it's important to grasp the fundamental concepts surrounding complex numbers. A complex number is composed of two parts: a real part and an imaginary part. The real part is a rational number, while the imaginary part is a rational number multiplied by the imaginary unit, denoted as i. The imaginary unit i is defined as the square root of -1, meaning i² = -1. This seemingly simple definition opens up a whole new dimension in mathematics, allowing us to solve equations that were previously considered unsolvable within the realm of real numbers. When adding, subtracting, or multiplying complex numbers, we treat i as a variable, much like x in algebraic expressions. However, we must always remember the fundamental property i² = -1, which allows us to eliminate the imaginary unit squared and express our final answer in the standard a + bi form. This form makes it easy to identify the real and imaginary components of the complex number, facilitating further operations and comparisons. Now that we have a solid understanding of the basics, let's move on to the step-by-step process of simplifying complex expressions.

Step 1: Distribute and Expand

In this initial step, the primary goal is to eliminate parentheses by applying the distributive property and expanding any products of complex numbers. Taking our example expression, distribute the 2 in the first term and expand the product of the two binomials in the second term. This involves multiplying each term in the first set of parentheses by each term in the second set. The distributive property states that a(b + c) = ab + ac. Applying this to the first term, 2(-36 - 3i), we get 2 * -36 + 2 * -3i, which simplifies to -72 - 6i. For the second term, (5 + 2i)(12 - 2i), we use the FOIL method (First, Outer, Inner, Last) to expand the product. First, we multiply the first terms in each set of parentheses: 5 * 12 = 60. Outer, we multiply the outer terms: 5 * -2i = -10i. Inner, we multiply the inner terms: 2i * 12 = 24i. Last, we multiply the last terms: 2i * -2i = -4i². So, the expanded form of (5 + 2i)(12 - 2i) is 60 - 10i + 24i - 4i². This step is crucial because it lays the foundation for combining like terms and simplifying the expression further. It's important to be meticulous in this step, ensuring that every term is multiplied correctly and that no signs are missed. A small error in this initial expansion can propagate through the rest of the simplification process, leading to an incorrect final answer. After expanding, we have -72 - 6i + 60 - 10i + 24i - 4i². The next step involves simplifying the expression by substituting i² with -1 and combining like terms. Remember, the key to simplifying complex expressions lies in the systematic application of algebraic rules and a thorough understanding of the properties of imaginary numbers. The careful execution of this distributive and expansion step sets the stage for a smooth and accurate simplification process. We've successfully removed the parentheses and expanded the products, paving the way for the next stage of our simplification journey.

Step 2: Simplify Using i² = -1

The imaginary unit i is defined as the square root of -1, which means that i² equals -1. This seemingly simple relationship is the cornerstone of simplifying complex number expressions. In this step, we substitute every instance of i² with -1, effectively eliminating the squared imaginary unit and transforming it into a real number. This substitution is crucial because it allows us to combine the real and imaginary terms separately, ultimately leading to the standard a + bi form. Looking back at our expanded expression from Step 1, which is -72 - 6i + 60 - 10i + 24i - 4i², we identify the term -4i². We replace i² with -1, resulting in -4 * -1, which simplifies to +4. Our expression now becomes -72 - 6i + 60 - 10i + 24i + 4. By performing this substitution, we've successfully converted the imaginary unit squared into a real number, making it possible to combine it with the other real terms in the expression. This step is not just about algebraic manipulation; it's about understanding the fundamental nature of complex numbers and how they differ from real numbers. The relationship i² = -1 is what allows us to perform arithmetic operations on complex numbers and express them in a consistent and meaningful way. Without this substitution, we would be left with an expression that is not in the standard form and is more difficult to interpret and use. It's important to be mindful of the signs when substituting i² with -1, as a negative sign multiplied by another negative sign results in a positive sign. This is a common area for errors, so it's always a good idea to double-check your work. After this substitution, the expression is ready for the next step, which involves combining the real and imaginary terms separately. The goal is to group the real numbers together and the imaginary numbers together, ultimately expressing the complex number in the a + bi format. By carefully applying the i² = -1 rule, we've taken a significant step towards simplifying our complex expression.

Step 3: Combine Like Terms

After substituting i² with -1, the next logical step is to combine like terms. This involves grouping together the real numbers and the imaginary numbers separately. By doing so, we simplify the expression and bring it closer to the desired a + bi form. Remember that real terms are those without the imaginary unit i, while imaginary terms are those that include i. In our expression, -72 - 6i + 60 - 10i + 24i + 4, we identify the real terms as -72, 60, and 4. We add these together: -72 + 60 + 4 = -8. Next, we identify the imaginary terms as -6i, -10i, and 24i. We add their coefficients: -6 - 10 + 24 = 8. So, the combined imaginary term is 8i. By carefully grouping and adding the like terms, we have significantly simplified the expression. This step is essential for clarity and for ensuring that the final answer is in the standard complex number form. Combining like terms is a fundamental algebraic skill, but it's particularly important when working with complex numbers because it allows us to separate the real and imaginary components. This separation is what defines the a + bi form, where a represents the real part and b represents the imaginary part. It's crucial to pay attention to the signs when combining terms, as a simple sign error can lead to an incorrect result. Double-checking your work in this step can prevent errors from propagating to the final answer. Once the like terms are combined, the expression is in its simplest form, ready to be written in the standard a + bi format. This step brings us closer to our goal of expressing the given complex expression in its most concise and understandable form. The process of combining like terms not only simplifies the expression but also provides a clear understanding of the real and imaginary components, which is essential for further operations and analysis.

Step 4: Express in a + bi Form

The final step in simplifying a complex expression is to express it in the standard a + bi form, where a represents the real part and b represents the imaginary part of the complex number. This form is the universally accepted way of representing complex numbers, making it easy to compare and perform operations on them. After combining like terms in the previous step, we have a real part and an imaginary part. We simply need to write them in the correct order, with the real part first and the imaginary part second, separated by a plus sign. It's important to note that if the imaginary part is negative, the plus sign becomes a minus sign. Looking at our simplified expression from Step 3, we have -8 + 8i. Here, -8 is the real part (a) and 8 is the coefficient of the imaginary part (b). Therefore, the complex number is already in the a + bi form. We can simply rewrite it as -8 + 8i to emphasize that it is in the standard form. This final step is crucial because it presents the complex number in a clear and concise manner, making it readily usable for further calculations or analysis. The a + bi form allows for easy identification of the real and imaginary components, which is essential for operations such as addition, subtraction, multiplication, and division of complex numbers. It also facilitates the graphical representation of complex numbers on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. Expressing a complex number in the a + bi form is not just about adhering to a convention; it's about ensuring clarity and facilitating communication in the world of mathematics and its applications. By reaching this final step, we have successfully simplified the given complex expression and presented it in its most understandable and usable form. The journey from the initial expression to the a + bi form demonstrates the power of algebraic manipulation and the importance of understanding the properties of complex numbers. The final result, -8 + 8i, is a testament to the systematic approach we've taken and the careful execution of each step.

Conclusion

In conclusion, simplifying complex expressions into the standard a + bi form is a fundamental skill in mathematics. By following a systematic approach, we can transform intricate expressions into a clear and concise representation that is easy to work with. This process involves distributing and expanding, substituting i² with -1, combining like terms, and finally, expressing the result in the a + bi form. Throughout this article, we've used the example expression 2(-36 - 3i) + (5 + 2i)(12 - 2i) to illustrate each step of the simplification process. By carefully applying the distributive property, the definition of i², and the rules of combining like terms, we successfully simplified the expression to -8 + 8i. This final result is a complex number in the standard form, where the real part is -8 and the imaginary part is 8. The ability to simplify complex expressions is crucial for various applications in mathematics, physics, engineering, and computer science. Complex numbers are used to model phenomena in alternating current circuits, quantum mechanics, fluid dynamics, and signal processing, among other fields. Therefore, mastering the techniques presented in this article is an investment in your mathematical toolkit. The a + bi form is not just a convention; it's a powerful tool for understanding and manipulating complex numbers. It allows us to visualize complex numbers as points on a plane, perform arithmetic operations with ease, and solve equations that would be impossible to solve using only real numbers. The process of simplification may seem daunting at first, but with practice and a clear understanding of the steps involved, it becomes a straightforward and rewarding task. Remember to be meticulous in your calculations, pay attention to signs, and double-check your work to avoid errors. The journey through complex numbers is a fascinating one, and the ability to simplify expressions is a key to unlocking their power and potential. This article has provided a comprehensive guide to this essential skill, equipping you with the knowledge and confidence to tackle complex number problems effectively. The world of complex numbers is vast and intriguing, and the ability to simplify expressions is just the first step in exploring its depths. With a solid foundation in the basics, you can delve into more advanced topics and discover the beauty and utility of complex numbers in various fields of study and application.