Mastering Complex Number Multiplication What Is Equivalent To (4 + 7i)(3 + 4i)

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Complex numbers are an essential part of mathematics, extending the real number system with the inclusion of the imaginary unit, denoted as i, where i 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. Operations with complex numbers follow specific rules, particularly in multiplication, which involves the distributive property and the simplification of i² to -1. This article delves into the multiplication of two complex numbers, focusing on the expression (4 + 7i)(3 + 4i), providing a step-by-step explanation to arrive at the correct equivalent expression. Mastering complex number multiplication is crucial for various fields, including electrical engineering, quantum mechanics, and applied mathematics. The ability to accurately perform these calculations ensures a solid foundation for more advanced mathematical concepts and practical applications. This comprehensive guide aims to clarify the process and offer insights into the underlying principles.

The journey through complex number arithmetic starts with a clear understanding of its components and operational rules. When multiplying complex numbers, one must meticulously apply the distributive property, ensuring each term in the first complex number is multiplied by each term in the second complex number. This process generates multiple terms that often include both real and imaginary components. The subsequent step involves simplifying these terms, particularly when encountering i². Since i is the square root of -1, i² equals -1, a critical substitution that transforms imaginary terms into real numbers, thereby consolidating the expression into its simplest form. This simplification is the linchpin in navigating complex number multiplication, enabling the combination of like terms and the derivation of the final complex number in the standard a + bi format. The careful execution of these steps is not just a mathematical exercise but a gateway to understanding the broader applications of complex numbers in diverse scientific and engineering domains.

The multiplication of the complex numbers (4 + 7i) and (3 + 4i) requires careful application of the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first complex number is multiplied by each term in the second complex number. The steps are as follows:

  1. First: Multiply the first terms in each complex number: 4 * 3 = 12.
  2. Outer: Multiply the outer terms: 4 * 4i = 16i.
  3. Inner: Multiply the inner terms: 7i * 3 = 21i.
  4. Last: Multiply the last terms: 7i * 4i = 28i².

Combining these products, we get:

12 + 16i + 21i + 28i²

Now, we simplify the expression by combining like terms and addressing the imaginary unit i². Recall that i² = -1. Substitute -1 for i² in the expression:

12 + 16i + 21i + 28(-1)

This simplifies to:

12 + 16i + 21i - 28

Next, combine the real parts (12 and -28) and the imaginary parts (16i and 21i):

(12 - 28) + (16i + 21i)

This results in:

-16 + 37i

Therefore, the product of (4 + 7i)(3 + 4i) is -16 + 37i. This methodical approach ensures accuracy and clarity in complex number multiplication, highlighting the importance of each step in arriving at the correct solution. Understanding the distributive property and the nature of the imaginary unit are fundamental to mastering this process.

When faced with multiple-choice questions involving complex number multiplication, it's essential to not only perform the calculation accurately but also to carefully analyze the provided options. This ensures that the correct answer is selected and reinforces understanding of the process. In the case of the expression (4 + 7i)(3 + 4i), we calculated the result to be -16 + 37i. Now, let's examine the given options:

  • A) –16 + 37i: This option matches our calculated result exactly. The real part is -16, and the imaginary part is 37i. Therefore, this is the correct answer.
  • B) 12 – 28i: This option is incorrect. While it includes the product of the first terms (4 * 3 = 12), it does not correctly account for the imaginary terms and the simplification of i².
  • C) 16 – 37i: This option has the correct magnitudes for the real and imaginary parts but with incorrect signs. The real part should be negative (-16), and the imaginary part should be positive (37i).
  • D) 37 + 16i: This option incorrectly swaps the real and imaginary parts and also has the wrong sign for the real part. The correct real part is -16, and the imaginary part is 37i.

By comparing our calculated result (-16 + 37i) with each option, it becomes clear that option A is the only one that matches. This analytical approach not only confirms the correctness of our calculation but also helps to identify common errors that can occur during complex number multiplication. Understanding why the other options are incorrect reinforces the importance of each step in the multiplication and simplification process. This thorough analysis is a valuable skill for tackling complex number problems and enhancing mathematical proficiency.

When multiplying complex numbers, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help ensure accuracy and efficiency in solving problems. Here are some of the most frequent errors to watch out for:

  1. Incorrect Application of the Distributive Property: The distributive property (FOIL method) is crucial for multiplying complex numbers. A common mistake is to miss one of the multiplications, such as forgetting to multiply the outer or inner terms. For example, in (4 + 7i)(3 + 4i), failing to multiply 4 by 4i or 7i by 3 will lead to an incomplete expression.
  2. Misunderstanding the Imaginary Unit: The imaginary unit i is defined as the square root of -1, and i² is equal to -1. A frequent error is to forget to substitute -1 for i² when simplifying the expression. This substitution is critical because it transforms an imaginary term into a real number, allowing for the combination of like terms.
  3. Sign Errors: Complex number arithmetic involves both real and imaginary parts, and sign errors can easily occur, especially when dealing with negative numbers. For instance, when substituting i² with -1, the sign of the term changes, and overlooking this can lead to an incorrect result. Careful attention to detail is essential in managing signs throughout the calculation.
  4. Combining Unlike Terms: Another common mistake is attempting to combine real and imaginary terms as if they were like terms. Real and imaginary parts must be kept separate and combined only with their respective counterparts. For example, 12 + 16i cannot be simplified to 28 or 28i; they remain separate terms.
  5. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect multiplication or addition, can derail the entire calculation. Double-checking each step, especially during the initial multiplication and subsequent simplification, can help catch these errors early on.

By being mindful of these common mistakes and practicing meticulous calculation, one can significantly improve their accuracy and confidence in multiplying complex numbers. Understanding the underlying principles and taking a step-by-step approach are key to mastering this skill.

Complex number multiplication isn't just an abstract mathematical concept; it has numerous practical applications in various fields of science and engineering. Understanding these real-world uses can provide a deeper appreciation for the importance of mastering this skill. Here are some notable applications:

  1. Electrical Engineering: In electrical engineering, complex numbers are used extensively to analyze alternating current (AC) circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is represented as a complex number. Multiplying complex impedances is crucial for calculating the total impedance in series circuits and for analyzing circuit behavior. Furthermore, complex numbers simplify the analysis of AC signals, including voltage and current, making circuit design and troubleshooting more efficient.
  2. Quantum Mechanics: Quantum mechanics, the foundation of modern physics, relies heavily on complex numbers. The wave function, which describes the quantum state of a particle, is a complex-valued function. Operations involving wave functions, such as calculating probabilities and expectation values, often require complex number multiplication. The Schrödinger equation, a central equation in quantum mechanics, also involves complex numbers, highlighting their fundamental role in the theory.
  3. Signal Processing: Signal processing, which deals with the analysis and manipulation of signals such as audio and images, utilizes complex numbers extensively. The Fourier transform, a key tool in signal processing, converts signals from the time domain to the frequency domain using complex exponentials. Multiplying complex numbers is essential in performing operations like filtering and modulation, which are fundamental to many communication and media technologies.
  4. Fluid Dynamics: In fluid dynamics, complex potentials are used to describe two-dimensional fluid flow. The complex potential is a complex-valued function whose real and imaginary parts represent the velocity potential and the stream function, respectively. Multiplying complex potentials allows engineers to analyze and design systems involving fluid flow, such as airfoils and hydraulic systems.
  5. Control Systems: Control systems, used in automation and robotics, often employ complex numbers to analyze system stability and performance. Transfer functions, which describe the input-output relationship of a system, are often expressed as complex functions. Multiplying complex transfer functions is necessary for designing controllers and predicting system behavior.

These examples illustrate that complex number multiplication is not just a theoretical exercise but a practical tool with significant implications in various technical fields. Mastering this skill provides a solid foundation for tackling real-world problems and advancing in these domains.

In summary, the multiplication of complex numbers, exemplified by the expression (4 + 7i)(3 + 4i), is a fundamental operation with far-reaching implications. The correct application of the distributive property, combined with the crucial simplification of i² to -1, leads to the accurate result of -16 + 37i. This process not only reinforces basic algebraic principles but also underscores the unique properties of complex numbers. Through careful analysis of each step, including the initial multiplication, term combination, and final simplification, one can avoid common errors and achieve a confident grasp of the subject matter. The step-by-step approach detailed in this guide serves as a robust method for tackling similar problems, ensuring both precision and clarity in calculations.

Furthermore, understanding the real-world applications of complex number multiplication provides a broader context for its importance. From electrical engineering and quantum mechanics to signal processing and fluid dynamics, complex numbers are indispensable tools for solving intricate problems and advancing technological innovation. This article has highlighted not only the mechanics of complex number multiplication but also its relevance in various scientific and engineering disciplines, fostering a deeper appreciation for its significance. The ability to manipulate complex numbers effectively is a valuable asset, empowering individuals to engage with and contribute to a wide range of fields. By mastering this skill, students and professionals alike can unlock new possibilities and tackle complex challenges with confidence and competence.