Subtract Complex Numbers (3+9i) From (-2+17i) - Step-by-Step Guide

Understanding Complex Number Subtraction

In the realm of mathematics, particularly within complex number theory, performing arithmetic operations such as subtraction is a fundamental skill. Complex numbers, which extend the real number system by including an imaginary unit denoted as i (where i² = -1), are expressed in the form a + bi, where a and b are real numbers. The a component is referred to as the real part, while b is the imaginary part. Subtracting complex numbers involves treating the real and imaginary parts separately, much like combining like terms in algebraic expressions. To effectively subtract complex numbers, one must first grasp the concept of the imaginary unit and its properties. The imaginary unit i is defined as the square root of -1, and it plays a crucial role in extending the number system to include solutions for equations that have no real solutions. Operations with complex numbers often involve manipulating expressions containing i, keeping in mind that i² is always -1. This property is essential when simplifying expressions or performing multiplications and divisions of complex numbers. For instance, when subtracting one complex number from another, we subtract the real parts from each other and the imaginary parts from each other separately. This process ensures that the result is also a complex number in the standard form a + bi. Complex numbers find applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. Their ability to represent quantities with both magnitude and direction makes them indispensable in analyzing alternating currents, wave phenomena, and other complex systems. Understanding how to perform basic operations such as subtraction is therefore crucial for anyone working in these areas. In the context of subtraction, complex numbers behave in a predictable and consistent manner, allowing for straightforward calculations once the underlying principles are understood. The process of complex number subtraction is analogous to vector subtraction, where each component (real and imaginary) is treated independently. This analogy provides a visual and intuitive way to grasp the concept. In essence, subtracting complex numbers is an elegant extension of real number subtraction, incorporating the imaginary unit to expand the mathematical toolkit. The ability to manipulate complex numbers effectively opens doors to solving a wider range of problems and understanding more intricate mathematical models.

Step-by-Step Subtraction Process

To subtract the complex number (3 + 9i) from (-2 + 17i), we follow a straightforward process that involves subtracting the real parts and the imaginary parts separately. This method ensures that we maintain the complex number's structure, which consists of a real component and an imaginary component. The real parts are subtracted from each other, and the imaginary parts are subtracted from each other. This approach aligns with the fundamental principles of complex number arithmetic, which treats real and imaginary components as distinct entities. The first step in this process is to identify the real and imaginary parts of each complex number. In the complex number (3 + 9i), the real part is 3, and the imaginary part is 9. Similarly, in the complex number (-2 + 17i), the real part is -2, and the imaginary part is 17. Once these parts are identified, we can proceed with the subtraction. Subtracting the real parts involves taking the real part of the second complex number (-2) and subtracting from it the real part of the first complex number (3). This operation is represented as -2 - 3, which equals -5. This result will be the real part of our answer. Next, we subtract the imaginary parts. This involves taking the imaginary part of the second complex number (17) and subtracting from it the imaginary part of the first complex number (9). This operation is represented as 17 - 9, which equals 8. Since we are dealing with imaginary parts, this result is 8i. By performing these two subtractions separately, we ensure that we are only combining like terms, which is a crucial aspect of complex number arithmetic. The result of subtracting the real parts is the real part of the final answer, and the result of subtracting the imaginary parts is the imaginary part of the final answer. After performing the subtractions, we combine the results to form the final complex number. In this case, we have -5 as the real part and 8i as the imaginary part. Therefore, the result of subtracting (3 + 9i) from (-2 + 17i) is -5 + 8i. This method of subtraction is consistent with the way we perform other arithmetic operations on complex numbers, such as addition, where we also combine real parts with real parts and imaginary parts with imaginary parts. The step-by-step approach not only simplifies the calculation but also reinforces the understanding of how complex numbers behave under arithmetic operations. This understanding is crucial for more advanced topics in complex analysis and related fields.

Detailed Calculation

To further illustrate the subtraction process, let's break down the calculation of subtracting complex numbers (3 + 9i) from (-2 + 17i) into a detailed, step-by-step explanation. This will provide a clear understanding of each operation performed and how the final result is obtained. The problem at hand is to find the result of (-2 + 17i) - (3 + 9i). The first step in solving this problem is to distribute the negative sign to both terms within the second complex number. This means we are essentially changing the sign of each term inside the parentheses. Distributing the negative sign, we get: -2 + 17i - 3 - 9i. Now that we have removed the parentheses, we can proceed to combine the real and imaginary parts separately. This is analogous to combining like terms in algebraic expressions, where we group similar terms together. We group the real parts together: -2 and -3. Adding these together, we get: -2 - 3 = -5. This result is the real part of the final complex number. Next, we group the imaginary parts together: 17i and -9i. Combining these, we get: 17i - 9i. To subtract these terms, we subtract the coefficients of i: 17 - 9 = 8. Therefore, the imaginary part of the final complex number is 8i. Now that we have both the real and imaginary parts, we can combine them to form the final complex number. The real part is -5, and the imaginary part is 8i. Combining these, we get: -5 + 8i. This is the result of subtracting (3 + 9i) from (-2 + 17i). The process of separating real and imaginary parts and then combining them is a fundamental technique in complex number arithmetic. It allows us to perform operations such as subtraction (and addition) in a systematic and straightforward manner. By breaking down the calculation into these detailed steps, it becomes easier to understand the logic and reasoning behind each operation. This approach is particularly useful for students learning complex numbers for the first time, as it provides a clear and structured method for solving problems. Furthermore, this detailed calculation reinforces the understanding that complex number operations are based on the same principles as algebraic operations, with the added consideration of the imaginary unit i.

Final Result and Interpretation

After performing the detailed calculation, the final result of subtracting (3 + 9i) from (-2 + 17i) is -5 + 8i. This complex number represents a point in the complex plane, where the real part (-5) corresponds to the horizontal axis, and the imaginary part (8) corresponds to the vertical axis. The interpretation of this result involves understanding its components and their significance in the context of complex numbers. The complex number -5 + 8i has a real part of -5 and an imaginary part of 8. The real part, -5, indicates the position of the complex number along the real axis, which is the horizontal axis in the complex plane. A negative real part means that the number is located to the left of the origin. The imaginary part, 8, indicates the position of the complex number along the imaginary axis, which is the vertical axis in the complex plane. A positive imaginary part means that the number is located above the origin. Therefore, the complex number -5 + 8i is located in the second quadrant of the complex plane, since it has a negative real part and a positive imaginary part. The complex plane is a graphical representation of complex numbers, where each complex number a + bi is represented as a point (a, b). This visual representation provides an intuitive way to understand the magnitude and direction of complex numbers, similar to how vectors are represented in a two-dimensional plane. The magnitude (or modulus) of the complex number -5 + 8i can be calculated using the Pythagorean theorem, which is the square root of the sum of the squares of the real and imaginary parts. In this case, the magnitude is √((-5)² + 8²) = √(25 + 64) = √89. The magnitude represents the distance of the complex number from the origin in the complex plane. The argument (or phase) of the complex number -5 + 8i is the angle that the line connecting the complex number to the origin makes with the positive real axis. This angle can be calculated using trigonometric functions, specifically the arctangent function. In this case, the argument is arctan(8/-5), which gives an angle in the second quadrant. The argument represents the direction of the complex number in the complex plane. In summary, the final result -5 + 8i is a complex number that can be visualized as a point in the second quadrant of the complex plane. It has a magnitude of √89 and an argument that can be calculated using the arctangent function. Understanding these interpretations is crucial for applying complex numbers in various fields, such as electrical engineering, physics, and computer science. The complex number interpretation helps in visualizing mathematical concepts and their practical applications.

Practical Applications of Complex Number Subtraction

Complex number subtraction is not merely an abstract mathematical exercise; it has a wide range of practical applications in various fields of science and engineering. The ability to subtract complex numbers allows for the manipulation and analysis of systems and phenomena that can be represented using these numbers. Understanding these applications provides a deeper appreciation for the importance of complex number arithmetic. One of the most significant applications of complex numbers is in electrical engineering, particularly in the analysis of alternating current (AC) circuits. In AC circuits, voltage and current vary sinusoidally with time, and complex numbers provide a convenient way to represent these quantities. Impedance, which is the opposition to the flow of current in an AC circuit, is also a complex quantity, with a real part representing resistance and an imaginary part representing reactance (due to capacitors and inductors). When analyzing AC circuits, it is often necessary to subtract complex impedances to determine the equivalent impedance of a circuit or to calculate voltage drops across components. For example, if two impedances, Z1 and Z2, are connected in series, the total impedance is the sum of the two impedances: Z_total = Z1 + Z2. However, if we want to find the difference between two impedances, we use complex number subtraction: Z_difference = Z1 - Z2. This operation can be crucial in designing filters or matching networks in electrical circuits. Another important application of complex number subtraction is in signal processing. Signals, such as audio or radio waves, can be represented as complex functions of time. Subtracting two complex signals can reveal important information about the differences between them, such as phase shifts or frequency components. This technique is used in various signal processing applications, including noise cancellation, signal filtering, and modulation/demodulation. For instance, in noise cancellation systems, the noise signal is subtracted from the desired signal to improve the quality of the received signal. This subtraction is often performed in the complex domain to account for phase and amplitude differences. In quantum mechanics, complex numbers are fundamental to the mathematical formulation of the theory. Wave functions, which describe the state of a quantum system, are complex-valued functions. Operations on wave functions, including subtraction, are used to calculate probabilities and other physical quantities. The superposition principle in quantum mechanics states that the state of a system can be a linear combination of multiple states. Subtracting wave functions allows physicists to analyze the interference and interactions between different quantum states. Fluid dynamics is another field where complex numbers and their subtraction play a role. Complex potential functions are used to describe two-dimensional fluid flows. Subtracting complex potentials can help analyze the flow around obstacles or the interaction between different flow patterns. This technique is particularly useful in aerodynamics and hydrodynamics for designing airfoils and analyzing wave propagation. In control systems engineering, complex numbers are used to analyze the stability and performance of feedback control systems. The transfer function of a control system, which relates the output to the input, is often a complex function. Subtracting transfer functions can help in designing controllers that achieve desired system behavior. These are just a few examples of the practical applications of complex number subtraction. The ability to manipulate complex numbers through arithmetic operations like subtraction enables the analysis and design of complex systems in various fields of science and engineering. The versatility of complex numbers in representing physical phenomena makes them an indispensable tool for researchers and engineers.

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