Drag The Tiles To The Correct Boxes Matching Complex Number Expressions

In the realm of mathematics, complex numbers extend the familiar number line into a two-dimensional plane, opening up a world of fascinating properties and applications. Mastering complex number operations is crucial for success in various fields, including electrical engineering, quantum mechanics, and signal processing. This article delves into the intricacies of complex number arithmetic, providing a step-by-step guide to solving complex number problems. Let's consider the given problem: given that ${ x = 3 + 8i }$ and ${ y = 7 - i }$, we aim to match the equivalent expressions for ${ 2x - 3y }$ and ${ -x \cdot y }$. This exercise provides an excellent opportunity to reinforce fundamental operations such as addition, subtraction, and multiplication within the complex number system. Let's embark on this mathematical journey together.

Understanding Complex Numbers

Before diving into the calculations, it’s essential to grasp the fundamental nature of complex numbers. A complex number is expressed in the form ${ a + bi }$, where ${ a }$ and ${ b }$ are real numbers, and ${ i }$ is the imaginary unit, defined as ${ \sqrt{-1} }$. The real part of the complex number is ${ a }$, while the imaginary part is ${ b }$. Complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. This geometric representation allows for a deeper understanding of complex number operations.

Complex numbers are more than just abstract mathematical concepts; they have significant real-world applications. In electrical engineering, complex numbers are used to analyze alternating current circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity that combines resistance and reactance. By using complex numbers, engineers can simplify circuit analysis and design more efficient systems. In quantum mechanics, complex numbers are fundamental to describing the wave function of particles. The wave function, which is a complex-valued function, provides information about the probability of finding a particle in a particular state. Complex numbers allow physicists to accurately model quantum phenomena and make predictions about the behavior of particles at the atomic and subatomic levels. Signal processing, a field concerned with analyzing and manipulating signals, also relies heavily on complex numbers. Signals can be represented as complex-valued functions, and various signal processing techniques, such as Fourier analysis, utilize complex numbers to decompose signals into their constituent frequencies. This allows engineers to filter noise, compress data, and perform other essential signal processing tasks. The widespread use of complex numbers in these diverse fields underscores their importance in both theoretical and applied mathematics.

Calculating 2x - 3y

The expression ${ 2x - 3y }$ involves scalar multiplication and subtraction of complex numbers. To find ${ 2x - 3y }$, we first multiply the complex numbers ${ x }$ and ${ y }$ by the scalars 2 and 3, respectively. Then, we subtract the resulting complex numbers. Let’s break down the steps:

1. Scalar Multiplication: Multiply ${ x }$ by 2: ${ 2x = 2(3 + 8i) = 6 + 16i }$ Similarly, multiply ${ y }$ by 3: ${ 3y = 3(7 - i) = 21 - 3i }$

2. Subtraction: Subtract ${ 3y }$ from ${ 2x }$: ${ 2x - 3y = (6 + 16i) - (21 - 3i) }$ To subtract complex numbers, we subtract the real parts and the imaginary parts separately: ${ 2x - 3y = (6 - 21) + (16i - (-3i)) }$ ${ 2x - 3y = -15 + 19i }$

Thus, the equivalent expression for ${ 2x - 3y }$ is ${ -15 + 19i }$. The process of scalar multiplication involves multiplying each part of the complex number (both the real and imaginary parts) by the scalar. This is a straightforward application of the distributive property. Subtraction of complex numbers follows a similar principle: we subtract the real parts from each other and the imaginary parts from each other. This ensures that we are combining like terms, which is crucial for maintaining the correct result. The result, ${ -15 + 19i }$, is another complex number, with a real part of -15 and an imaginary part of 19. Understanding these basic operations is foundational for more complex calculations involving complex numbers.

Calculating -x ⋅ y

Now, let's compute the expression ${ -x \cdot y }$, which involves the multiplication of complex numbers. We'll first find ${ -x }$ and then multiply it by ${ y }$. The multiplication of complex numbers involves using the distributive property and the fact that ${ i^2 = -1 }$. Here’s the step-by-step calculation:

1. Find -x: Multiply ${ x }$ by -1: ${ -x = -1(3 + 8i) = -3 - 8i }$

2. Multiplication: Multiply ${ -x }$ by ${ y }$: ${ -x \cdot y = (-3 - 8i)(7 - i) }$ Apply the distributive property (also known as the FOIL method) to multiply the two complex numbers: ${ -x \cdot y = (-3)(7) + (-3)(-i) + (-8i)(7) + (-8i)(-i) }$ ${ -x \cdot y = -21 + 3i - 56i + 8i^2 }$

3. Simplify: Recall that ${ i^2 = -1 }$, so substitute ${ -1 }$ for ${ i^2 }$: ${ -x \cdot y = -21 + 3i - 56i + 8(-1) }$ ${ -x \cdot y = -21 + 3i - 56i - 8 }$

4. Combine Like Terms: Combine the real parts and the imaginary parts: ${ -x \cdot y = (-21 - 8) + (3i - 56i) }$ ${ -x \cdot y = -29 - 53i }$

Therefore, the equivalent expression for ${ -x \cdot y }$ is ${ -29 - 53i }$. The multiplication of complex numbers is a key operation that extends the rules of algebra to the complex plane. The distributive property ensures that each term in the first complex number is multiplied by each term in the second complex number. The crucial step in simplifying the product is recognizing that ${ i^2 = -1 }$. This allows us to convert the imaginary term ${ 8i^2 }$ into a real number, which can then be combined with the other real terms. The final result, ${ -29 - 53i }$, is again a complex number, demonstrating that the product of two complex numbers is also a complex number. This operation is fundamental in many areas of mathematics and physics, particularly in the study of alternating current circuits and quantum mechanics.

Matching the Expressions

Based on our calculations:

• ${ 2x - 3y = -15 + 19i }$
• ${ -x \cdot y = -29 - 53i }$

Therefore, we can match the expressions as follows:

• ${ 2x - 3y }$ corresponds to ${ -15 + 19i }$
• ${ -x \cdot y }$ corresponds to ${ -29 - 53i }$

This exercise highlights the importance of meticulously following the order of operations and paying close attention to the signs. Complex number arithmetic, while governed by the same algebraic rules as real number arithmetic, requires careful handling of the imaginary unit ${ i }$. By breaking down the problem into smaller steps, we can minimize errors and arrive at the correct solutions. The process of matching expressions not only reinforces our understanding of complex number operations but also enhances our problem-solving skills. This type of exercise is particularly valuable in preparing for exams and for applying complex number concepts in real-world applications.

Conclusion

In this article, we explored complex number operations by calculating ${ 2x - 3y }$ and ${ -x \cdot y }$, given ${ x = 3 + 8i }$ and ${ y = 7 - i }$. We found that ${ 2x - 3y = -15 + 19i }$ and ${ -x \cdot y = -29 - 53i }$. These calculations underscore the fundamental principles of complex number arithmetic, including scalar multiplication, subtraction, and multiplication. Mastering these operations is crucial for success in various mathematical and scientific disciplines.

The significance of complex numbers extends far beyond the classroom. They are essential tools in electrical engineering, where they are used to analyze alternating current circuits; in quantum mechanics, where they describe the wave function of particles; and in signal processing, where they facilitate the analysis and manipulation of signals. By understanding and applying complex number operations, students and professionals alike can tackle complex problems and develop innovative solutions in their respective fields. This article serves as a comprehensive guide to understanding and performing basic complex number operations, laying the foundation for more advanced topics in complex analysis and its applications.

As a final thought, remember that practice is key to mastering complex number operations. By working through various problems and exercises, you can build confidence in your abilities and develop a deeper understanding of the fascinating world of complex numbers. Whether you are a student preparing for an exam or a professional working on a real-world application, a solid foundation in complex number arithmetic will undoubtedly serve you well.