Simplifying (2 + 4i)(1 + 3i) A Step-by-Step Guide
In the realm of mathematics, complex numbers play a crucial role, extending the real number system to include the imaginary unit, denoted by i, where i² = -1. Operations with complex numbers, such as multiplication, often require careful attention to detail to ensure accuracy. This article delves into simplifying the expression (2 + 4i)(1 + 3i), providing a comprehensive, step-by-step guide suitable for students and enthusiasts alike. Understanding how to manipulate complex numbers is fundamental in various fields, including electrical engineering, quantum mechanics, and applied mathematics. This detailed exploration aims to clarify the process and reinforce the underlying principles, ensuring a solid grasp of complex number arithmetic.
Understanding Complex Numbers
Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The real part of the complex number is a, and the imaginary part is b. Operations with complex numbers involve treating i as a variable while remembering that i² = -1. This unique property is essential when performing arithmetic operations, particularly multiplication. Complex numbers are not just abstract mathematical constructs; they have practical applications in various scientific and engineering disciplines. For example, in electrical engineering, complex numbers are used to represent alternating currents and impedances. In quantum mechanics, they are crucial for describing wave functions. Therefore, mastering the manipulation of complex numbers is vital for anyone pursuing studies or careers in these fields. The ability to simplify expressions involving complex numbers accurately is a fundamental skill that underpins more advanced concepts and applications. By understanding the basic principles and practicing the techniques, one can confidently tackle complex number problems and appreciate their significance in diverse areas of science and technology. This foundational knowledge serves as a building block for further exploration and application of complex numbers in more complex scenarios.
The Distributive Property and Complex Number Multiplication
The core technique for multiplying complex numbers is the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. This method ensures that each term in the first complex number is multiplied by each term in the second complex number. Applying the distributive property to (2 + 4i)(1 + 3i) involves multiplying the first terms (2 * 1), the outer terms (2 * 3i), the inner terms (4i * 1), and the last terms (4i * 3i). This yields: 2 + 6i + 4i + 12i². The next step is to simplify the expression by combining like terms and substituting i² with -1. This substitution is crucial because it transforms the imaginary unit squared into a real number, which is a fundamental step in simplifying complex number expressions. The distributive property is a cornerstone of algebraic manipulation, and its application in complex number multiplication highlights its versatility. Understanding and mastering this property is essential for simplifying not just complex numbers but also a wide range of algebraic expressions. The FOIL method provides a structured approach to ensure that no terms are missed during the multiplication process, leading to accurate results. This methodical approach is particularly useful when dealing with more complex expressions involving multiple terms or higher powers of i. By consistently applying the distributive property and remembering the key substitution i² = -1, one can confidently and accurately multiply complex numbers.
Step-by-Step Simplification of (2 + 4i)(1 + 3i)
Let's break down the simplification of (2 + 4i)(1 + 3i) step by step. First, apply the distributive property: (2 + 4i)(1 + 3i) = 2 * 1 + 2 * 3i + 4i * 1 + 4i * 3i. This expands to 2 + 6i + 4i + 12i². Next, combine the like terms, which are the terms involving i: 6i + 4i = 10i. So, the expression becomes 2 + 10i + 12i². Now, substitute i² with -1: 12i² = 12 * (-1) = -12. The expression further simplifies to 2 + 10i - 12. Finally, combine the real parts: 2 - 12 = -10. The simplified form is -10 + 10i. This step-by-step approach ensures clarity and minimizes the chance of errors. Each step builds upon the previous one, making the process manageable and easy to follow. The initial expansion using the distributive property is crucial, followed by the careful combination of like terms. The substitution of i² with -1 is a key step that transforms the imaginary part of the expression. Finally, combining the real parts leads to the simplified complex number. This methodical approach not only yields the correct answer but also enhances understanding of the underlying principles of complex number arithmetic.
Combining Like Terms and Substituting i²
After applying the distributive property, the next crucial step is combining like terms and substituting i² with -1. In the expression 2 + 6i + 4i + 12i², the like terms are 6i and 4i. Combining them gives 10i. Then, we focus on the term 12i². Since i² = -1, we substitute i² with -1, which gives 12 * (-1) = -12. This substitution is fundamental because it converts the imaginary unit squared into a real number, allowing us to further simplify the expression. The expression now becomes 2 + 10i - 12. The real parts of the expression are 2 and -12. Combining these gives -10. Therefore, the simplified expression is -10 + 10i. The process of combining like terms is a standard algebraic technique that simplifies expressions by grouping terms with the same variable or constant. In the context of complex numbers, this involves combining the terms with i and the real number terms separately. The substitution of i² with -1 is a unique aspect of complex number arithmetic and is essential for reducing the expression to its simplest form. This step highlights the importance of understanding the properties of the imaginary unit and how it interacts within algebraic expressions. By carefully combining like terms and making the correct substitution, we can accurately simplify complex number expressions and arrive at the final result.
The Final Simplified Form: -10 + 10i
After performing all the necessary steps, the simplified form of the expression (2 + 4i)(1 + 3i) is -10 + 10i. This result is a complex number in the standard form a + bi, where a = -10 and b = 10. The real part of the complex number is -10, and the imaginary part is 10. This final form represents the most concise and simplified version of the original expression. The process of simplifying complex number expressions involves a series of algebraic manipulations, including the application of the distributive property, combining like terms, and substituting i² with -1. Each step is crucial for arriving at the correct simplified form. The final simplified form -10 + 10i clearly presents the complex number in its standard representation, making it easy to identify the real and imaginary components. This is particularly important in various applications where the real and imaginary parts have distinct meanings. For example, in electrical engineering, the real part might represent resistance, while the imaginary part represents reactance. Understanding how to simplify complex number expressions to their standard form is therefore essential for interpreting and applying them in practical contexts. The ability to arrive at the final simplified form with confidence demonstrates a solid grasp of complex number arithmetic and its underlying principles.
Common Mistakes to Avoid
When simplifying expressions involving complex numbers, several common mistakes can lead to incorrect results. One frequent error is forgetting to apply the distributive property correctly, which can result in missing terms. Another common mistake is failing to substitute i² with -1, leaving the expression in an unsimplified form. Additionally, errors can occur when combining like terms, particularly if the signs are not handled carefully. To avoid these mistakes, it is essential to follow a systematic approach, breaking down the simplification process into clear, manageable steps. Double-checking each step can help identify and correct errors before they propagate through the calculation. Paying close attention to the signs and ensuring that all terms are accounted for during the distributive property application are crucial. Remembering the fundamental property i² = -1 and applying it consistently is also vital for accurate simplification. By being mindful of these common mistakes and adopting a methodical approach, one can significantly reduce the likelihood of errors and confidently simplify complex number expressions. Practice and careful attention to detail are key to mastering complex number arithmetic and avoiding these pitfalls.
Conclusion
Simplifying complex number expressions, such as (2 + 4i)(1 + 3i), requires a systematic approach that includes applying the distributive property, combining like terms, and substituting i² with -1. By following these steps carefully, we can transform the expression into its simplified form, -10 + 10i. Understanding the properties of complex numbers and avoiding common mistakes are essential for accurate simplification. This detailed guide provides a solid foundation for further exploration of complex number arithmetic and its applications in various fields. The ability to manipulate complex numbers is a valuable skill in mathematics, science, and engineering, and mastering these techniques opens doors to more advanced concepts and applications. This comprehensive explanation aims to empower readers with the knowledge and confidence to tackle complex number problems effectively. The process of simplification not only yields the final result but also enhances understanding of the underlying principles, making it a crucial aspect of learning and applying complex number arithmetic. With practice and attention to detail, one can confidently navigate the world of complex numbers and their diverse applications.
