Multiply And Simplify Complex Numbers (3-5i)(-2+4i)
Introduction: Mastering Complex Number Multiplication
In the realm of mathematics, complex numbers play a crucial role, extending the familiar number system to include the imaginary unit, denoted as 'i', where i² = -1. Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. Understanding complex number arithmetic, particularly multiplication, is essential for various fields, including electrical engineering, quantum mechanics, and signal processing. This article delves into the process of multiplying and simplifying complex number products, using the specific example of (3 - 5i)(-2 + 4i) to illustrate the steps involved. By mastering these techniques, you'll gain a solid foundation for working with complex numbers and their applications. Understanding the fundamentals of complex number arithmetic, especially multiplication, is crucial for success in various technical fields. This guide will walk you through a step-by-step process to master complex number multiplication, ensuring you have a solid foundation. We will explore the core principles and provide a detailed solution to the given problem, ensuring clarity and comprehension.
Understanding the Fundamentals of Complex Numbers
Before diving into the multiplication process, it's crucial to grasp the basic structure and properties of complex numbers. A complex number is composed of two parts: a real part and an imaginary part. The real part is a standard real number, while the imaginary part is a real number multiplied by the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1, meaning i² = -1. This property is fundamental to simplifying complex number expressions. When multiplying complex numbers, we use the distributive property, similar to multiplying binomials in algebra. We then simplify the result by combining like terms and substituting i² with -1. Complex numbers, with their real and imaginary components, extend the real number system and enable solutions to equations that are unsolvable within real numbers alone. The imaginary unit, denoted by 'i', is the cornerstone of complex numbers, defined as the square root of -1. This seemingly simple definition unlocks a wealth of mathematical possibilities, allowing us to perform operations and solve problems that were previously beyond our reach. The complex number system is not just a mathematical curiosity; it's a powerful tool with applications in diverse fields such as physics, engineering, and computer science. Understanding how to manipulate and simplify complex numbers is crucial for anyone working in these areas.
Step-by-Step Multiplication of Complex Numbers
To multiply complex numbers, we apply the distributive property, often referred to as the FOIL (First, Outer, Inner, Last) method, which is a mnemonic for the standard method of multiplying two binomials. This involves multiplying each term in the first complex number by each term in the second complex number. Let's apply this to our example: (3 - 5i)(-2 + 4i). First, multiply the first terms: 3 * -2 = -6. Next, multiply the outer terms: 3 * 4i = 12i. Then, multiply the inner terms: -5i * -2 = 10i. Finally, multiply the last terms: -5i * 4i = -20i². Now, combine the terms: -6 + 12i + 10i - 20i². The next step is crucial: substitute i² with -1. This gives us -6 + 12i + 10i - 20(-1). Simplify further: -6 + 12i + 10i + 20. Now, combine the real and imaginary parts: (-6 + 20) + (12i + 10i). This results in 14 + 22i. Therefore, the product of (3 - 5i) and (-2 + 4i) is 14 + 22i. The distributive property is a fundamental concept in algebra and is equally crucial when dealing with complex numbers. It allows us to break down the multiplication of binomials into manageable steps. The FOIL method, a popular mnemonic, provides a structured approach to ensure that all terms are multiplied correctly. By systematically applying the distributive property and the FOIL method, we can confidently tackle complex number multiplication problems. This method not only simplifies the process but also minimizes the chances of errors, leading to accurate solutions.
Simplifying the Product: Combining Real and Imaginary Terms
After applying the distributive property, the resulting expression typically contains both real and imaginary terms. To simplify the product, we need to combine the like terms. This involves grouping the real terms together and the imaginary terms together. In our example, after substituting i² with -1, we have -6 + 12i + 10i + 20. The real terms are -6 and 20, which combine to give 14. The imaginary terms are 12i and 10i, which combine to give 22i. Therefore, the simplified product is 14 + 22i. This is the standard form of a complex number, where the real part is 14 and the imaginary part is 22. Simplifying the product of complex numbers involves meticulous organization and application of fundamental arithmetic principles. By carefully combining like terms – real with real and imaginary with imaginary – we arrive at the standard form of a complex number. This process not only streamlines the expression but also provides clarity and facilitates further calculations. Mastering the art of simplification is essential for anyone working with complex numbers, ensuring accuracy and efficiency in problem-solving. The ability to confidently simplify expressions is a cornerstone of complex number arithmetic.
Identifying the Correct Product: A Detailed Comparison
Now that we have multiplied and simplified the product (3 - 5i)(-2 + 4i) to 14 + 22i, we can compare our result with the given options. The options are: A. 14 + 22i, B. 14 + 2i, C. 15 + 22i, and D. 26 + 2i. By comparing our result with these options, we can see that option A, 14 + 22i, matches our simplified product. Therefore, option A is the correct answer. The other options are incorrect due to errors in either the multiplication or simplification process. Identifying the correct product requires careful comparison and attention to detail. Each step in the multiplication and simplification process must be meticulously checked to ensure accuracy. By systematically comparing our result with the given options, we can confidently identify the correct answer. This process not only reinforces our understanding of complex number arithmetic but also cultivates critical thinking and problem-solving skills. The ability to accurately identify the correct product is a testament to a thorough understanding of the underlying principles.
Conclusion: Solidifying Your Understanding of Complex Number Multiplication
In this article, we have explored the process of multiplying and simplifying complex numbers, using the example of (3 - 5i)(-2 + 4i). We covered the fundamental properties of complex numbers, the application of the distributive property (FOIL method), and the simplification process of combining like terms and substituting i² with -1. By following these steps, we arrived at the correct product, 14 + 22i. This comprehensive guide provides a solid foundation for understanding complex number arithmetic. By practicing these techniques and applying them to various problems, you can master complex number multiplication and confidently tackle more advanced mathematical concepts. Solidifying your understanding of complex number multiplication requires consistent practice and application of the principles discussed. By working through various examples and challenging problems, you can reinforce your knowledge and develop fluency in this area. Complex numbers are a powerful tool in mathematics and its applications, and mastering their arithmetic is a valuable asset. This guide serves as a stepping stone to further exploration and deeper understanding of the fascinating world of complex numbers.
Final Answer: The final answer is
