Multiplying Complex Numbers (5-6i)(4-4i) In Standard Form

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In the realm of mathematics, complex numbers play a pivotal role, extending the familiar number system to include the imaginary unit, denoted by 'i', where i² = -1. These numbers, expressed in the standard form of a + bi, where 'a' and 'b' are real numbers, find applications in various fields, including electrical engineering, quantum mechanics, and signal processing. Mastering operations with complex numbers, such as multiplication, is crucial for anyone delving into these areas. This comprehensive guide will walk you through the process of multiplying two complex numbers, (5 - 6i) and (4 - 4i), and expressing the result in standard form. Let's embark on this mathematical journey and unravel the intricacies of complex number multiplication.

Understanding Complex Numbers and Their Standard Form

Before we delve into the multiplication process, let's first solidify our understanding of complex numbers and their representation. A complex number is composed of two parts: a real part and an imaginary part. The standard form of a complex number is expressed as a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The imaginary part is always accompanied by the imaginary unit 'i', which, as mentioned earlier, is defined as the square root of -1 (i² = -1). Understanding this standard form is paramount for performing operations with complex numbers, as it provides a clear structure for manipulating the real and imaginary components. For instance, in the complex number 5 - 6i, the real part is 5 and the imaginary part is -6. Similarly, in the complex number 4 - 4i, the real part is 4 and the imaginary part is -4. These distinct components are crucial when performing operations like multiplication, as we need to treat the real and imaginary terms separately and then combine them appropriately to arrive at the final result in standard form. This foundation ensures that we can accurately perform complex number multiplication and other operations, paving the way for more advanced mathematical concepts and applications. Recognizing and correctly identifying the real and imaginary parts allows us to apply the distributive property effectively and simplify the expressions involving complex numbers, ultimately leading to the expression of the result in the desired standard form.

The Distributive Property: The Key to Multiplying Complex Numbers

At the heart of multiplying complex numbers lies the distributive property, a fundamental principle in algebra that allows us to expand expressions involving multiplication over addition or subtraction. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In the context of complex numbers, we apply this property to multiply each term in the first complex number by each term in the second complex number. This systematic approach ensures that we account for all possible combinations of terms, both real and imaginary. For the given problem, (5 - 6i)(4 - 4i), we will distribute each term of the first complex number (5 and -6i) over the terms of the second complex number (4 and -4i). This means we will perform the following multiplications: 5 * 4, 5 * (-4i), -6i * 4, and -6i * (-4i). Each of these multiplications results in a new term, which we then combine to form a preliminary expression. Understanding and applying the distributive property correctly is crucial for accurately multiplying complex numbers. It breaks down a seemingly complex operation into a series of simpler multiplications, which can then be easily managed and combined. The distributive property ensures that every term in the first complex number interacts with every term in the second complex number, thereby accounting for all possible products and leading to the correct expansion of the expression. This step-by-step process is essential for maintaining accuracy and clarity throughout the multiplication process.

Step-by-Step Multiplication of (5 - 6i)(4 - 4i)

Now, let's apply the distributive property to multiply the complex numbers (5 - 6i) and (4 - 4i) step by step. This meticulous process will ensure we accurately account for each term and properly combine them. First, we multiply 5 by both terms in the second complex number: 5 * 4 = 20 and 5 * (-4i) = -20i. Next, we multiply -6i by both terms in the second complex number: -6i * 4 = -24i and -6i * (-4i) = 24i². So far, we have expanded the expression to: 20 - 20i - 24i + 24i². The next critical step is to remember the fundamental property of the imaginary unit: i² = -1. We substitute -1 for i² in our expression: 20 - 20i - 24i + 24(-1). This simplifies to 20 - 20i - 24i - 24. Now, we combine the real terms (20 and -24) and the imaginary terms (-20i and -24i) separately. Combining the real terms, we get 20 - 24 = -4. Combining the imaginary terms, we get -20i - 24i = -44i. Finally, we combine the simplified real and imaginary parts to express the result in standard form: -4 - 44i. This step-by-step breakdown illustrates the importance of careful application of the distributive property and the substitution of i² with -1. By meticulously following these steps, we can confidently multiply complex numbers and express the result in the desired standard form.

Simplifying and Combining Terms: Expressing the Result in Standard Form

After applying the distributive property and substituting i² with -1, the next crucial step is to simplify and combine like terms. This process involves grouping together the real terms and the imaginary terms separately, which allows us to express the final answer in the standard form of a complex number, a + bi. In our example, after expanding (5 - 6i)(4 - 4i), we arrived at the expression 20 - 20i - 24i + 24i². Substituting i² with -1, we got 20 - 20i - 24i - 24. Now, we identify the real terms, which are 20 and -24, and combine them: 20 - 24 = -4. Next, we identify the imaginary terms, which are -20i and -24i, and combine them: -20i - 24i = -44i. Once we have simplified both the real and imaginary parts, we combine them to form the complex number in standard form. In this case, we have -4 (the real part) and -44i (the imaginary part). Therefore, the final result in standard form is -4 - 44i. This step of simplifying and combining terms is vital for presenting the answer in a clear and concise manner. It ensures that the complex number is expressed in its most fundamental form, making it easier to interpret and use in further calculations or applications. By carefully grouping and combining like terms, we maintain accuracy and ensure that the final result accurately represents the product of the original complex numbers. This meticulous approach is essential for mastering operations with complex numbers and building a solid foundation in mathematical concepts.

The Final Result: -4 - 44i

After meticulously applying the distributive property, substituting i² with -1, and simplifying and combining like terms, we arrive at the final result of the multiplication: -4 - 44i. This complex number is expressed in standard form, a + bi, where the real part is -4 and the imaginary part is -44. The journey of multiplying complex numbers (5 - 6i) and (4 - 4i) has led us through several key steps, each essential for achieving the correct answer. Starting with understanding the standard form of a complex number, we then applied the distributive property to expand the product. We carefully multiplied each term in the first complex number by each term in the second, ensuring no term was missed. Next, we remembered the crucial identity i² = -1, which allowed us to convert the imaginary term i² into a real number, thereby simplifying the expression. Following this, we combined the real terms and the imaginary terms separately, grouping like terms together to make the expression more manageable. Finally, we expressed the simplified result in standard form, -4 - 44i, which clearly displays the real and imaginary components of the product. This final result is not just a numerical answer; it represents the culmination of our understanding of complex number operations and our ability to apply mathematical principles systematically. The process we've followed is a testament to the structured approach required for accurate mathematical calculations, especially when dealing with concepts like complex numbers that extend beyond the realm of real numbers. The result -4 - 44i serves as a concrete example of how complex numbers interact under multiplication and provides a solid foundation for further exploration of complex number arithmetic and its applications.

Common Mistakes to Avoid When Multiplying Complex Numbers

When multiplying complex numbers, several common mistakes can lead to incorrect results. Being aware of these pitfalls and taking steps to avoid them is crucial for ensuring accuracy. One of the most frequent errors is forgetting to apply the distributive property fully. As we discussed earlier, each term in the first complex number must be multiplied by each term in the second. Missing even one multiplication can throw off the entire calculation. For instance, in the example (5 - 6i)(4 - 4i), failing to multiply 5 by -4i or -6i by 4 would result in an incomplete expansion and an incorrect final answer. Another common mistake is neglecting to substitute i² with -1. This substitution is a fundamental step in simplifying complex number expressions, and overlooking it will leave the result in an unsimplified and incorrect form. Remember that i² = -1 is a defining property of imaginary numbers, and it plays a critical role in reducing the expression to standard form. A third error arises from incorrectly combining like terms. Real terms should be combined only with real terms, and imaginary terms should be combined only with imaginary terms. Mixing these up can lead to a jumbled and meaningless result. For example, adding a real number directly to an imaginary term without proper consideration of the imaginary unit 'i' is a common mistake. Furthermore, errors can occur due to simple arithmetic mistakes, such as incorrect multiplication or addition of real numbers. These errors, though basic, can significantly impact the final answer. To avoid these mistakes, it's essential to follow a systematic approach: carefully apply the distributive property, remember to substitute i² with -1, accurately combine like terms, and double-check all arithmetic calculations. By being mindful of these common pitfalls and adopting a methodical approach, you can confidently multiply complex numbers and arrive at the correct answer.

Applications of Complex Number Multiplication

The multiplication of complex numbers is not merely a theoretical exercise; it has profound applications in various fields of science and engineering. Understanding how to multiply complex numbers opens doors to solving real-world problems in areas such as electrical engineering, physics, and computer science. In electrical engineering, complex numbers are used extensively to represent alternating current (AC) circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity comprising resistance and reactance. Multiplying complex numbers allows engineers to calculate the total impedance of a circuit, analyze circuit behavior, and design efficient electrical systems. The voltage and current in AC circuits are also often represented as complex numbers, and their multiplication is crucial for determining power and energy consumption. In physics, complex numbers play a vital role in quantum mechanics, where they are used to describe wave functions, which represent the probability amplitude of a quantum particle. The multiplication of complex numbers is essential for calculating probabilities and understanding the behavior of quantum systems. Furthermore, complex numbers are used in signal processing to analyze and manipulate signals, such as audio and video signals. The Fourier transform, a fundamental tool in signal processing, uses complex numbers to decompose a signal into its constituent frequencies. Multiplication of complex numbers is used in filtering, modulation, and demodulation of signals. Additionally, complex number multiplication is used in computer graphics for rotations and scaling of images. By representing points in a 2D or 3D space as complex numbers, transformations like rotations can be efficiently performed using complex number multiplication. The wide range of applications underscores the importance of mastering complex number multiplication. It provides a powerful tool for solving problems in diverse fields and highlights the interconnectedness of mathematical concepts with real-world phenomena. The ability to confidently multiply complex numbers is not just an academic skill; it is a valuable asset for anyone pursuing a career in science, technology, engineering, or mathematics.

Conclusion

In conclusion, multiplying complex numbers is a fundamental operation with far-reaching implications. By mastering the steps involved, from applying the distributive property to simplifying and combining terms, you gain a valuable skill applicable in numerous fields. The process of multiplying (5 - 6i)(4 - 4i) and arriving at the result -4 - 44i in standard form exemplifies the systematic approach required for accurate complex number arithmetic. The key takeaways from this guide include the importance of understanding the standard form of a complex number, the careful application of the distributive property, the crucial substitution of i² with -1, and the accurate combination of like terms. Avoiding common mistakes, such as incomplete distribution or incorrect simplification, is essential for achieving the correct result. Beyond the mechanics of the operation, we have explored the diverse applications of complex number multiplication in areas like electrical engineering, physics, and computer science. This highlights the practical relevance of this mathematical concept and its role in solving real-world problems. The ability to confidently multiply complex numbers empowers you to delve deeper into these fields and tackle more complex challenges. Whether you are a student exploring mathematical concepts or a professional applying these concepts in your work, a solid understanding of complex number multiplication is a valuable asset. The journey through this guide has equipped you with the knowledge and skills to confidently perform complex number multiplication and appreciate its significance in the broader context of mathematics and its applications.