Subtracting Complex Numbers (3+2i) From (-9-8i) A Step-by-Step Guide

Understanding Complex Number Subtraction

When dealing with complex numbers, subtraction, like addition, is a straightforward process that involves manipulating the real and imaginary parts separately. Complex number subtraction is a fundamental operation in mathematics, especially within the realm of algebra and number theory. To effectively subtract complex numbers, one must first grasp the basic structure of a complex number. A complex number is expressed in the form $a + bi$, where $a$ represents the real part and $bi$ represents the imaginary part. The symbols $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which is defined as the square root of -1 (i.e., $i = \sqrt{-1}$). This imaginary unit is the cornerstone of complex numbers, allowing us to express numbers that extend beyond the real number line. Subtraction of complex numbers is performed by subtracting the real parts and the imaginary parts separately. If we have two complex numbers, say $z_1 = a + bi$ and $z_2 = c + di$, the subtraction $z_1 - z_2$ is calculated as $(a - c) + (b - d)i$. This means we subtract the real part of the second complex number from the real part of the first, and similarly, we subtract the imaginary part of the second complex number from the imaginary part of the first. The result is a new complex number whose real part is $(a - c)$ and whose imaginary part is $(b - d)i$. This process ensures that we are treating real and imaginary components as distinct entities, adhering to the algebraic rules that govern complex numbers. Understanding this method is crucial for solving problems involving complex numbers, particularly in fields such as electrical engineering, quantum mechanics, and advanced mathematics, where complex numbers are frequently used to model and solve intricate problems. In essence, complex number subtraction is a critical skill for anyone delving into higher-level mathematics and its applications.

Step-by-Step Subtraction Process

To subtract $(3+2i)$ from $(-9-8i)$, we follow the standard procedure for complex number subtraction. The problem requires us to find the difference between two complex numbers: $(-9 - 8i)$ and $(3 + 2i)$. We express this subtraction as $(-9 - 8i) - (3 + 2i)$. The first step in this process is to distribute the negative sign across the second complex number. This means we change the sign of both the real and imaginary parts of $(3 + 2i)$. Distributing the negative sign gives us $-3 - 2i$. Now, the expression becomes $(-9 - 8i) + (-3 - 2i)$. The next step involves combining the real parts and the imaginary parts separately. We group the real parts together: $-9$ and $-3$. Adding these gives us $-9 + (-3) = -12$. Similarly, we group the imaginary parts together: $-8i$ and $-2i$. Adding these gives us $-8i + (-2i) = -10i$. Therefore, by combining the real and imaginary parts, we get the result of the subtraction. The complex number we obtain is $-12 - 10i$. This is the final answer, representing the difference between the two original complex numbers. This step-by-step approach ensures accuracy and clarity in handling complex number subtractions. Understanding how to meticulously combine real and imaginary components is essential for avoiding common mistakes and for achieving proficiency in complex number arithmetic. This method not only applies to simple subtractions but also forms the basis for more complex operations involving complex numbers, such as division, exponentiation, and finding complex roots. Mastering this fundamental skill is invaluable for anyone studying mathematics, engineering, or physics, where complex numbers are widely used.

Detailed Solution: Subtracting the Complex Numbers

Let's delve into the detailed solution for subtracting the complex number $(3 + 2i)$ from $(-9 - 8i)$. This process involves a methodical application of the rules of complex number arithmetic. Our initial expression is $(-9 - 8i) - (3 + 2i)$. As previously mentioned, the first critical step is to distribute the negative sign across the second complex number. This means we multiply both the real and imaginary parts of $(3 + 2i)$ by $-1$. When we distribute the negative sign, we get $-1 * (3) = -3$ for the real part and $-1 * (2i) = -2i$ for the imaginary part. The expression then transforms to $(-9 - 8i) + (-3 - 2i)$. Now that we have eliminated the parentheses and the subtraction sign, we can proceed with combining like terms. We group the real parts together: $-9$ and $-3$. Adding these real parts gives us $-9 + (-3) = -12$. Next, we group the imaginary parts together: $-8i$ and $-2i$. Adding these imaginary parts gives us $-8i + (-2i) = -10i$. By combining the real and imaginary parts, we arrive at our final complex number: $-12 - 10i$. This complex number represents the difference between the original two complex numbers. The solution $-12 - 10i$ indicates that the real part of the difference is $-12$, and the imaginary part is $-10i$. To summarize, the detailed solution involves distributing the negative sign, combining the real parts, and combining the imaginary parts. This methodical approach ensures accuracy and provides a clear understanding of how complex number subtraction works. This process is not only essential for solving basic problems but also for tackling more advanced mathematical challenges involving complex numbers.

Final Result and Interpretation

After performing the subtraction, we have arrived at the final result: $-12 - 10i$. This complex number represents the difference between $(-9 - 8i)$ and $(3 + 2i)$. The final result, $-12 - 10i$, consists of two parts: the real part, which is $-12$, and the imaginary part, which is $-10i$. It's important to understand what each part signifies in the context of complex numbers. The real part, $-12$, is a standard real number and can be visualized on the real number line. It represents the component of the complex number that does not involve the imaginary unit $i$. The imaginary part, $-10i$, is a multiple of the imaginary unit $i$, which is defined as the square root of $-1$. This part represents the component of the complex number that lies along the imaginary axis in the complex plane. Together, the real and imaginary parts define a point in the complex plane. The complex plane is a two-dimensional plane with the horizontal axis representing real numbers and the vertical axis representing imaginary numbers. The complex number $-12 - 10i$ can be plotted as a point in this plane, with coordinates $(-12, -10)$. The x-coordinate is the real part $(-12)$, and the y-coordinate is the imaginary part $(-10)$. This visual representation can be particularly useful in applications such as electrical engineering and physics, where complex numbers are used to represent phenomena like alternating current circuits and quantum mechanical states. In these fields, the real and imaginary parts often have physical interpretations, such as resistance and reactance in electrical circuits. Therefore, understanding the final result, $-12 - 10i$, not only involves knowing the numerical answer but also comprehending its representation and significance in both mathematical and applied contexts.

Practical Applications and Importance

Complex number subtraction, while a fundamental operation, has significant practical applications in various fields, highlighting its importance in mathematics and beyond. Complex numbers, in general, are instrumental in many areas of science and engineering, and the ability to perform basic operations like subtraction is essential for solving complex problems. One of the most prominent applications of complex numbers is in electrical engineering. In the analysis of alternating current (AC) circuits, complex numbers are used to represent impedances, which combine resistance and reactance (the opposition to the flow of current caused by inductors and capacitors). Subtracting complex impedances allows engineers to calculate the net impedance in a circuit, which is crucial for designing and analyzing electrical systems. For instance, when dealing with series circuits, the total impedance is found by adding the individual impedances, and in parallel circuits, more complex calculations involving subtraction and division are necessary. Another significant application is in signal processing. Signals, such as audio or radio waves, can be represented as complex functions. Operations on these signals, like filtering or modulation, often involve complex number arithmetic, including subtraction. Signal processing is vital in telecommunications, audio engineering, and image processing, where complex numbers help in manipulating and analyzing signals effectively. In quantum mechanics, complex numbers are fundamental. The state of a quantum system is described by a complex-valued wave function, and operations on these wave functions often involve complex number arithmetic. Subtraction is used in calculations involving interference and superposition of quantum states, which are key concepts in understanding the behavior of quantum systems. Beyond these specific fields, complex numbers and their operations, including subtraction, are used in fluid dynamics, control theory, and various branches of physics and applied mathematics. The ability to perform complex number subtraction accurately and efficiently is therefore a crucial skill for professionals and researchers in these domains. Its importance lies not only in its mathematical simplicity but also in its broad applicability to real-world problems, making it a cornerstone of quantitative analysis and problem-solving.