Calculate The Product Of Complex Numbers (2+i)(2-i)(5+4i)

Introduction to Complex Number Multiplication

In the realm of mathematics, complex numbers extend the familiar real number system by incorporating the imaginary unit, denoted as 'i', which is defined as the square root of -1. Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The multiplication of complex numbers involves applying the distributive property and simplifying terms using the fact that i² = -1. Understanding complex number multiplication is fundamental in various fields, including electrical engineering, quantum mechanics, and signal processing. In this article, we embark on a comprehensive exploration of multiplying complex numbers, focusing on the specific expression (2+i)(2-i)(5+4i). We will dissect the process step by step, providing clear explanations and insights to enhance your understanding of this essential mathematical operation.

Before we delve into the intricacies of multiplying the given complex numbers, let's establish a solid foundation by revisiting the fundamental principles of complex number multiplication. When multiplying two complex numbers, such as (a + bi) and (c + di), we apply the distributive property, similar to multiplying binomials in algebra. This yields: (a + bi)(c + di) = a(c + di) + bi(c + di). Expanding further, we get ac + adi + bci + bdi². Now, the crucial step is to remember that i² = -1. Substituting this into the expression, we have ac + adi + bci - bd. Grouping the real and imaginary terms, we arrive at the general formula for complex number multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. This formula provides a systematic way to multiply any two complex numbers. By applying this principle and meticulously tracking each term, we can confidently navigate the multiplication of more complex expressions, such as the one we'll be tackling in this article.

Step-by-Step Calculation of (2+i)(2-i)(5+4i)

Our mission is to unravel the product of three complex numbers: (2+i)(2-i)(5+4i). To conquer this challenge, we'll adopt a step-by-step approach, ensuring clarity and precision in our calculations. The journey begins by focusing on the first two complex numbers, (2+i) and (2-i). This pair holds a special significance as they are complex conjugates of each other. Complex conjugates have the form a + bi and a - bi, where the imaginary part has opposite signs. A remarkable property of complex conjugates is that their product always results in a real number. Let's witness this in action as we multiply (2+i) and (2-i). Applying the distributive property, we have (2+i)(2-i) = 2(2-i) + i(2-i). Expanding further, we get 4 - 2i + 2i - i². Now, recall the fundamental identity i² = -1. Substituting this into our expression, we have 4 - 2i + 2i + 1. Observe that the imaginary terms, -2i and +2i, gracefully cancel each other out. This leaves us with 4 + 1, which simplifies to 5. As anticipated, the product of the complex conjugates (2+i) and (2-i) is indeed a real number, specifically 5. This simplification significantly streamlines our calculations, paving the way for the next step in our quest.

With the product of the first two complex numbers in hand, we can now proceed to the next stage of our calculation. We have successfully simplified (2+i)(2-i) to 5. Our original expression, (2+i)(2-i)(5+4i), now transforms into 5(5+4i). This simplification makes the subsequent multiplication significantly easier. To complete the calculation, we need to multiply the real number 5 by the complex number (5+4i). This is a straightforward application of the distributive property. We multiply 5 by each term within the parentheses: 5 * 5 + 5 * 4i. This yields 25 + 20i. Therefore, the final product of the complex numbers (2+i)(2-i)(5+4i) is 25 + 20i. This completes our step-by-step journey, revealing the ultimate result of this complex number multiplication.

Significance of Complex Conjugates

The complex conjugates (2+i) and (2-i) played a pivotal role in simplifying our calculation. Their product, as we witnessed, resulted in a real number. This property of complex conjugates is not merely a coincidence; it's a fundamental characteristic that holds immense significance in various mathematical and scientific applications. Understanding why complex conjugates behave in this manner provides deeper insights into the structure of complex numbers. When we multiply complex conjugates (a + bi) and (a - bi), the middle terms, which involve the imaginary unit 'i', always cancel each other out. Let's see this in action: (a + bi)(a - bi) = a(a - bi) + bi(a - bi) = a² - abi + abi - b²i². The terms -abi and +abi cancel each other out. Substituting i² = -1, we get a² + b². Notice that the result is a real number, devoid of any imaginary component. This elegant cancellation is the essence of why complex conjugates are so powerful in simplifying expressions and solving problems involving complex numbers.

The real-number product of complex conjugates finds extensive use in various mathematical operations. One prominent application is in dividing complex numbers. To divide one complex number by another, we multiply both the numerator and the denominator by the conjugate of the denominator. This maneuver cleverly eliminates the imaginary part from the denominator, making the division process much more manageable. Complex conjugates also play a crucial role in finding the magnitude (or modulus) of a complex number. The magnitude of a complex number a + bi is defined as √(a² + b²), which is precisely the square root of the product of the complex number and its conjugate. Furthermore, complex conjugates are indispensable in solving quadratic equations with complex roots. When a quadratic equation has complex roots, they always occur in conjugate pairs. This property allows us to efficiently determine both roots if one of them is known. The significance of complex conjugates extends beyond pure mathematics, permeating fields such as electrical engineering, where they are used in circuit analysis, and quantum mechanics, where they appear in wave functions and probability calculations. Understanding and utilizing complex conjugates unlocks a powerful set of tools for tackling a wide range of problems in diverse domains.

Alternative Approaches to the Solution

While our step-by-step method provides a clear and structured path to the solution, it's often beneficial to explore alternative approaches. This not only reinforces our understanding but also equips us with a broader problem-solving toolkit. One alternative approach to calculating (2+i)(2-i)(5+4i) involves a slightly different order of operations. Instead of first multiplying (2+i) and (2-i), we could initially multiply (2-i) and (5+4i). Let's embark on this alternative route and witness how it unfolds. Applying the distributive property, we have (2-i)(5+4i) = 2(5+4i) - i(5+4i). Expanding further, we get 10 + 8i - 5i - 4i². Substituting i² = -1, we have 10 + 8i - 5i + 4. Combining the real and imaginary terms, we arrive at 14 + 3i. Now, we need to multiply this result by (2+i). So, we have (2+i)(14+3i) = 2(14+3i) + i(14+3i). Expanding, we get 28 + 6i + 14i + 3i². Substituting i² = -1, we have 28 + 6i + 14i - 3. Finally, combining the real and imaginary terms, we get 25 + 20i. As we can see, this alternative approach, while involving a different sequence of multiplications, leads us to the same final answer: 25 + 20i. This reaffirms the consistency and robustness of complex number multiplication.

Another insightful approach involves visualizing complex numbers on the complex plane. The complex plane is a two-dimensional coordinate system where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. Each complex number can be represented as a point on this plane. Multiplication of complex numbers has a beautiful geometric interpretation on the complex plane. When we multiply two complex numbers, their magnitudes are multiplied, and their arguments (angles with respect to the positive real axis) are added. This geometric perspective can provide a deeper understanding of complex number multiplication and offer an alternative way to approach problems. While this geometric approach might not be the most efficient method for direct calculation in this specific case, it provides valuable insights into the nature of complex number multiplication. Exploring different approaches like these not only enhances our problem-solving skills but also fosters a more profound appreciation for the interconnectedness of mathematical concepts.

Conclusion: Mastering Complex Number Multiplication

Our journey through the multiplication of complex numbers, specifically the expression (2+i)(2-i)(5+4i), has unveiled the power and elegance of this fundamental mathematical operation. We embarked on a step-by-step calculation, meticulously multiplying the complex numbers and simplifying the expression. The crucial role of complex conjugates in simplifying the calculation was highlighted, showcasing their ability to transform complex expressions into real numbers. We also ventured into alternative approaches, reinforcing our understanding and broadening our problem-solving horizons. Through this exploration, we have not only arrived at the solution, 25 + 20i, but also gained a deeper appreciation for the principles and techniques involved in complex number multiplication. Mastering complex number multiplication is not merely an academic exercise; it's a gateway to a wide range of applications in various fields, from electrical engineering to quantum mechanics.

The ability to confidently manipulate complex numbers is a valuable asset in any mathematical or scientific endeavor. The techniques we've explored in this article, such as the distributive property and the use of complex conjugates, are essential tools in your mathematical toolkit. By practicing these techniques and exploring different approaches, you can develop a strong command of complex number multiplication and unlock its potential in solving a diverse array of problems. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and developing the ability to apply them creatively. As you continue your mathematical journey, embrace the challenges, explore the connections, and relish the beauty of mathematical reasoning. The world of complex numbers is rich with fascinating patterns and applications, waiting to be discovered. So, continue your exploration, and let the power of mathematics guide you!