Multiplying Complex Numbers (6 + 3i)(4 - I) A Step-by-Step Guide

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When delving into the realm of complex numbers, mastering the art of multiplication is paramount. Complex numbers, those enigmatic entities composed of a real and an imaginary part, often appear daunting at first glance. However, with a structured approach and a dash of algebraic finesse, even the most intricate multiplications can be unraveled. In this comprehensive guide, we'll meticulously dissect the multiplication of two complex numbers, (6 + 3i) and (4 - i), and express the final result as a simplified complex number. This exploration will not only illuminate the mechanics of the process but also enhance your overall understanding of complex number arithmetic.

Before we embark on the multiplication itself, let's lay a solid foundation by revisiting the fundamental definition of a complex number. A complex number is generally represented in the form a + bi, where 'a' denotes the real part and 'b' represents the imaginary part. The imaginary unit, denoted by 'i', is defined as the square root of -1. This seemingly simple definition unlocks a whole new dimension in mathematics, allowing us to tackle equations and problems that were previously unsolvable within the realm of real numbers. The ability to manipulate complex numbers is crucial in various fields, including electrical engineering, quantum mechanics, and signal processing. Therefore, a firm grasp of their operations, especially multiplication, is an invaluable asset for anyone venturing into these domains.

In our specific case, we are presented with two complex numbers: (6 + 3i) and (4 - i). The first complex number, (6 + 3i), has a real part of 6 and an imaginary part of 3. The second complex number, (4 - i), has a real part of 4 and an imaginary part of -1. Our mission is to multiply these two complex numbers together and express the result in the standard form of a + bi. This process involves applying the distributive property, a cornerstone of algebra, and carefully handling the imaginary unit 'i'. As we progress, we will emphasize the importance of simplifying the result by combining like terms and utilizing the fundamental property of i, which is i² = -1. This property is the key to eliminating the imaginary unit from the real part of the complex number, ultimately leading to a simplified expression.

Now, let's embark on the step-by-step journey of multiplying the complex numbers (6 + 3i) and (4 - i). The cornerstone of this process is the distributive property, a fundamental principle in algebra that dictates how expressions involving sums and products are expanded. The distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), provides a systematic way to ensure that each term in the first complex number is multiplied by each term in the second complex number. This meticulous approach guarantees that we don't miss any crucial terms and maintains the accuracy of our calculation.

Applying the distributive property to our problem, (6 + 3i)(4 - i), we begin by multiplying the 'First' terms: 6 multiplied by 4, which yields 24. Next, we move to the 'Outer' terms: 6 multiplied by -i, resulting in -6i. Then, we multiply the 'Inner' terms: 3i multiplied by 4, which gives us 12i. Finally, we multiply the 'Last' terms: 3i multiplied by -i, leading to -3i². This meticulous application of the distributive property transforms our initial expression into 24 - 6i + 12i - 3i². It's crucial to remember that each of these four terms plays a vital role in determining the final result.

The next step in our journey is to simplify the expression we've obtained. This involves two key operations: combining like terms and utilizing the fundamental property of the imaginary unit, i² = -1. Like terms, in this context, are the terms that have the same imaginary unit. In our expression, -6i and 12i are like terms. Combining them, we get -6i + 12i = 6i. This simplifies our expression to 24 + 6i - 3i². Now, we turn our attention to the term -3i². Remembering that i² = -1, we can substitute -1 for i² in our expression. This gives us -3(-1) = 3. Our expression now transforms into 24 + 6i + 3. The substitution of i² with -1 is a pivotal step, as it eliminates the imaginary unit from this term and allows us to combine it with the real part of the complex number.

Finally, we arrive at the last stage of simplification: combining the real terms. In our expression, 24 and 3 are both real numbers. Adding them together, we get 24 + 3 = 27. This leaves us with 27 + 6i. This is the simplified form of the product of the two complex numbers, expressed in the standard form of a + bi, where 27 is the real part and 6 is the imaginary part. This meticulous step-by-step process, from the application of the distributive property to the simplification using i² = -1, showcases the elegance and precision of complex number multiplication.

Having meticulously navigated the multiplication process, we arrive at the crucial step of expressing our result as a simplified complex number. This final touch ensures that our answer is presented in the standard and universally recognized form for complex numbers, which is a + bi. As we've seen, 'a' represents the real part of the complex number, and 'b' represents the imaginary part, with 'i' being the imaginary unit (√-1). Presenting the answer in this format not only adheres to mathematical conventions but also facilitates clear communication and comparison with other complex numbers.

From our previous calculations, we arrived at the expression 27 + 6i. This expression is already in the form of a complex number, with 27 clearly identified as the real part and 6 as the coefficient of the imaginary part. Therefore, our final answer is indeed a simplified complex number. There are no further simplifications possible, as the real and imaginary parts are already combined, and there are no higher powers of 'i' remaining. The expression 27 + 6i elegantly encapsulates the result of multiplying (6 + 3i) and (4 - i).

It's worth emphasizing the significance of this simplified form. Expressing complex numbers in the a + bi format allows us to easily visualize them on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis. This geometric representation provides a powerful tool for understanding the properties and relationships of complex numbers. Moreover, this standard form is essential for performing further operations with complex numbers, such as addition, subtraction, division, and finding magnitudes and arguments. The ability to express complex numbers in this simplified form is a cornerstone of complex number arithmetic and unlocks a wide range of applications in mathematics, physics, and engineering.

In conclusion, the simplified complex number resulting from the multiplication of (6 + 3i) and (4 - i) is 27 + 6i. This result embodies the culmination of our step-by-step multiplication process, which involved applying the distributive property, combining like terms, and utilizing the fundamental property i² = -1. This journey through complex number multiplication not only provides the answer to our specific problem but also strengthens our understanding of complex number arithmetic in general. The ability to confidently manipulate complex numbers is an invaluable skill for anyone venturing into the mathematical sciences.

While the process of multiplying complex numbers, as we've demonstrated, is relatively straightforward, there are common pitfalls that students often encounter. Being aware of these potential errors can significantly enhance your accuracy and understanding of complex number operations. Let's explore some of these common mistakes and how to avoid them.

One of the most frequent errors occurs during the application of the distributive property. As we've emphasized, the distributive property requires that every term in the first complex number be multiplied by every term in the second complex number. However, it's easy to overlook a term or misapply the multiplication, especially when dealing with negative signs or the imaginary unit 'i'. For example, in the multiplication (6 + 3i)(4 - i), a student might forget to multiply the '3i' by the '-i', or they might incorrectly calculate the product. To avoid this, it's crucial to use a systematic approach, such as the FOIL method (First, Outer, Inner, Last), and to double-check each multiplication to ensure accuracy. Practice and attention to detail are key in mastering the distributive property.

Another common mistake arises when simplifying expressions involving the imaginary unit 'i'. The fundamental property i² = -1 is the cornerstone of simplification, but it's often forgotten or misapplied. Students might incorrectly assume that i² = 1 or simply ignore the i² term altogether. To avoid this error, it's essential to explicitly substitute -1 for i² whenever it appears in the expression. This substitution transforms the imaginary term into a real term, allowing us to combine it with the other real numbers. Forgetting this crucial step can lead to an incorrect final answer. Therefore, memorizing and consistently applying the property i² = -1 is paramount.

A third common mistake involves combining like terms. As we've discussed, like terms are those that have the same imaginary unit. Real terms can be combined with other real terms, and imaginary terms (those with 'i') can be combined with other imaginary terms. However, students sometimes incorrectly combine real and imaginary terms, treating them as like terms. For example, in the expression 27 + 6i, it's incorrect to add 27 and 6 together, as one is a real number and the other is the coefficient of an imaginary number. To avoid this error, always remember that real and imaginary terms are distinct and should be kept separate until the final answer is expressed in the form a + bi.

Finally, carelessness with signs can also lead to errors. Complex number multiplication often involves negative signs, and a small mistake in handling these signs can propagate through the entire calculation, leading to an incorrect result. For instance, when multiplying 3i by -i, the correct result is -3i², but a student might mistakenly write 3i². To avoid sign errors, it's crucial to pay close attention to the signs of each term and to use parentheses when necessary to maintain clarity. Double-checking each step and carefully reviewing the signs can significantly reduce the likelihood of errors.

By being aware of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and confidence in multiplying complex numbers. Practice, attention to detail, and a solid understanding of the underlying principles are the keys to success in this area.

To solidify your understanding of complex number multiplication, engaging with practice problems is essential. These problems provide an opportunity to apply the concepts we've discussed and to identify any areas where further clarification may be needed. Let's delve into a few practice problems, complete with step-by-step solutions, to hone your skills in this area.

Problem 1: Multiply the complex numbers (2 - i) and (3 + 2i) and express the result as a simplified complex number.

Solution:

  1. Apply the distributive property (FOIL method):
    • (2 - i)(3 + 2i) = 2(3) + 2(2i) - i(3) - i(2i)
    • = 6 + 4i - 3i - 2i²
  2. Simplify by combining like terms and substituting i² = -1:
    • = 6 + i - 2(-1)
    • = 6 + i + 2
  3. Combine the real terms:
    • = 8 + i

Therefore, the simplified complex number is 8 + i.

Problem 2: Multiply the complex numbers (-1 + 4i) and (5 - 3i) and express the result as a simplified complex number.

Solution:

  1. Apply the distributive property:
    • (-1 + 4i)(5 - 3i) = -1(5) - 1(-3i) + 4i(5) + 4i(-3i)
    • = -5 + 3i + 20i - 12i²
  2. Simplify by combining like terms and substituting i² = -1:
    • = -5 + 23i - 12(-1)
    • = -5 + 23i + 12
  3. Combine the real terms:
    • = 7 + 23i

Therefore, the simplified complex number is 7 + 23i.

Problem 3: Multiply the complex numbers (7 + 2i) and (7 - 2i) and express the result as a simplified complex number.

Solution:

  1. Apply the distributive property:
    • (7 + 2i)(7 - 2i) = 7(7) + 7(-2i) + 2i(7) + 2i(-2i)
    • = 49 - 14i + 14i - 4i²
  2. Simplify by combining like terms and substituting i² = -1:
    • = 49 - 4(-1)
    • = 49 + 4
  3. Combine the real terms:
    • = 53

Therefore, the simplified complex number is 53. Note that in this case, the imaginary terms canceled out, resulting in a purely real number. This occurs when multiplying a complex number by its conjugate.

These practice problems illustrate the process of complex number multiplication in action. By working through these examples and attempting similar problems on your own, you can reinforce your understanding and develop the skills necessary to confidently tackle more complex calculations. Remember to pay close attention to the distributive property, the simplification of i², and the combination of like terms. With practice, you'll master the art of complex number multiplication.

In conclusion, mastering the multiplication of complex numbers is a fundamental skill in mathematics, with far-reaching applications in various fields. Throughout this comprehensive guide, we have meticulously dissected the process of multiplying complex numbers, focusing on the specific example of (6 + 3i)(4 - i). We began by laying the groundwork, defining complex numbers and highlighting the importance of understanding their operations. We then embarked on a step-by-step journey through the multiplication process, emphasizing the application of the distributive property, the simplification using i² = -1, and the combination of like terms. This systematic approach ensures accuracy and clarity in our calculations.

We further underscored the significance of expressing the result as a simplified complex number in the standard form of a + bi. This format not only adheres to mathematical conventions but also facilitates clear communication and comparison with other complex numbers. Moreover, this simplified form is essential for performing further operations and visualizing complex numbers on the complex plane. The ability to confidently manipulate complex numbers in this way is a cornerstone of complex number arithmetic and unlocks a wide range of applications.

To further enhance your understanding, we addressed common mistakes that students often encounter during complex number multiplication. These include errors in applying the distributive property, mishandling the imaginary unit 'i', incorrectly combining like terms, and carelessness with signs. By being aware of these potential pitfalls and implementing strategies to avoid them, you can significantly improve your accuracy and confidence in your calculations. Practice and attention to detail are paramount in mastering this skill.

Finally, we provided a set of practice problems, complete with step-by-step solutions, to solidify your understanding and provide opportunities for application. These problems illustrate the concepts we've discussed in action and allow you to identify any areas where further clarification may be needed. By working through these examples and attempting similar problems on your own, you can reinforce your knowledge and develop the skills necessary to confidently tackle more complex calculations.

In essence, the multiplication of complex numbers, while seemingly intricate at first glance, can be mastered with a structured approach, consistent practice, and a solid understanding of the underlying principles. This guide has provided you with the tools and knowledge necessary to confidently navigate the world of complex number multiplication. As you continue your mathematical journey, remember that practice and perseverance are the keys to unlocking the power of complex numbers and their applications.