Subtracting Complex Numbers (6-5i) - (7+6i) A Step-by-Step Guide

Jul 6, 2025 by Admin 65 views

Unraveling Complex Number Subtraction (6-5i) - (7+6i)

In the realm of mathematics, complex numbers extend the familiar number line by incorporating an imaginary component. These numbers, expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1), play a crucial role in various fields, including electrical engineering, quantum mechanics, and signal processing. One of the fundamental operations involving complex numbers is subtraction, which involves finding the difference between two complex numbers. In this comprehensive exploration, we will delve into the intricate details of subtracting the complex number (7 + 6i) from the complex number (6 - 5i), providing a step-by-step guide to ensure a thorough understanding of the process.

Understanding Complex Number Subtraction

Before we embark on the actual subtraction, it is imperative to grasp the underlying principles of complex number subtraction. Complex numbers, as the name suggests, consist of two distinct parts: the real part and the imaginary part. When subtracting complex numbers, we essentially subtract the real parts separately and the imaginary parts separately. This can be represented mathematically as follows:

(a + bi) - (c + di) = (a - c) + (b - d)i

In this equation, a and c represent the real parts of the two complex numbers, while b and d represent the imaginary parts. The result of the subtraction is a new complex number whose real part is the difference between the original real parts (a - c) and whose imaginary part is the difference between the original imaginary parts (b - d) multiplied by the imaginary unit i.

Step-by-Step Subtraction of (6 - 5i) - (7 + 6i)

Now that we have a firm grasp of the principles of complex number subtraction, let's apply this knowledge to the specific problem at hand: subtracting (7 + 6i) from (6 - 5i). We will meticulously follow a step-by-step approach to ensure clarity and accuracy.

Step 1: Identify the Real and Imaginary Parts

The first step in subtracting complex numbers is to identify the real and imaginary parts of each number. In the complex number (6 - 5i), the real part is 6 and the imaginary part is -5. Similarly, in the complex number (7 + 6i), the real part is 7 and the imaginary part is 6.

Step 2: Subtract the Real Parts

Next, we subtract the real parts of the two complex numbers. This involves subtracting the real part of the second complex number (7) from the real part of the first complex number (6):

6 - 7 = -1

The result of this subtraction is -1, which will be the real part of the resulting complex number.

Step 3: Subtract the Imaginary Parts

Following the subtraction of the real parts, we subtract the imaginary parts. This involves subtracting the imaginary part of the second complex number (6) from the imaginary part of the first complex number (-5):

-5 - 6 = -11

The result of this subtraction is -11, which will be the coefficient of the imaginary unit i in the resulting complex number.

Step 4: Combine the Results

Finally, we combine the results from the previous steps to form the resulting complex number. The real part is -1, and the imaginary part is -11, so the resulting complex number is:

-1 - 11i

Therefore, the result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i. This complex number represents the difference between the two original complex numbers in the complex plane.

Visualizing Complex Number Subtraction

Complex numbers can be visualized as points in a two-dimensional plane called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Subtracting complex numbers can be interpreted geometrically as finding the vector that connects the two complex numbers in the complex plane.

In our example, the complex number (6 - 5i) can be represented as the point (6, -5) in the complex plane, and the complex number (7 + 6i) can be represented as the point (7, 6). Subtracting (7 + 6i) from (6 - 5i) is equivalent to finding the vector that starts at the point (7, 6) and ends at the point (6, -5). This vector corresponds to the complex number -1 - 11i, which is the result we obtained through algebraic subtraction.

Practical Applications of Complex Number Subtraction

Complex number subtraction, although seemingly abstract, has numerous practical applications in various fields. One notable application is in electrical engineering, where complex numbers are used to represent alternating current (AC) circuits. Subtracting complex impedances, which represent the opposition to current flow in AC circuits, allows engineers to analyze circuit behavior and design efficient electrical systems. For instance, calculating the difference between two impedances in a series circuit helps determine the overall impedance, which is crucial for understanding voltage drops and current flow.

In quantum mechanics, complex numbers are fundamental to describing the behavior of quantum systems. Wave functions, which describe the probability of finding a particle in a particular state, are often complex-valued. Subtracting wave functions allows physicists to analyze interference phenomena and understand the superposition of quantum states. For example, when two wave functions describing different possible states of a particle are subtracted, the resulting wave function can reveal regions where the probability of finding the particle is significantly reduced due to destructive interference.

Signal processing, a field that deals with the manipulation and analysis of signals, also relies heavily on complex numbers. Signals, such as audio or video signals, can be represented as complex-valued functions. Subtracting signals can be used for noise cancellation, where unwanted noise is subtracted from the original signal to improve clarity. In audio processing, subtracting a noise signal from a recorded sound can significantly enhance the listening experience by reducing background disturbances.

Common Pitfalls and How to Avoid Them

While the process of complex number subtraction is relatively straightforward, there are a few common pitfalls that students and practitioners should be aware of. One common mistake is forgetting to distribute the negative sign when subtracting a complex number. For example, when subtracting (7 + 6i) from (6 - 5i), it is crucial to remember that the negative sign applies to both the real and imaginary parts of (7 + 6i). Failure to distribute the negative sign can lead to incorrect results.

Another common pitfall is incorrectly combining real and imaginary parts. When subtracting complex numbers, it is essential to remember that real parts should be subtracted from real parts, and imaginary parts should be subtracted from imaginary parts. Mixing these parts can lead to erroneous answers. For instance, mistakenly subtracting the real part of one complex number from the imaginary part of another will result in an incorrect complex number.

To avoid these pitfalls, it is advisable to carefully follow the step-by-step approach outlined earlier. Double-checking each step and paying close attention to the signs can help minimize errors. Additionally, practicing complex number subtraction with various examples can solidify understanding and reduce the likelihood of making mistakes.

Conclusion: Mastering Complex Number Subtraction

In conclusion, subtracting complex numbers involves subtracting the real parts separately and the imaginary parts separately. This process, while fundamental, has far-reaching implications in various fields, including electrical engineering, quantum mechanics, and signal processing. By understanding the underlying principles, following a step-by-step approach, and visualizing the operation in the complex plane, one can master complex number subtraction and apply it effectively in diverse contexts. The result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i, a complex number that encapsulates the difference between the two original complex numbers. This operation not only forms a cornerstone of complex number arithmetic but also serves as a gateway to understanding more advanced mathematical and scientific concepts.

Delving into the intricacies of complex number arithmetic, we encounter the operation of subtraction, a fundamental process with far-reaching implications in various scientific and engineering disciplines. This article serves as a comprehensive guide to understanding and executing complex number subtraction, focusing on the specific example of subtracting (7 + 6i) from (6 - 5i). We will break down the process into manageable steps, elucidate the underlying principles, and explore the practical significance of this operation.

The Essence of Complex Numbers

Before we embark on the subtraction process, it is essential to establish a firm understanding of complex numbers themselves. A complex number is a mathematical entity that extends the realm of real numbers by incorporating an imaginary component. It is conventionally expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, defined as the square root of -1. The real part of the complex number is denoted by a, while the imaginary part is denoted by b.

Complex numbers find applications in a plethora of fields, including electrical engineering, quantum mechanics, fluid dynamics, and signal processing. Their ability to represent quantities with both magnitude and phase makes them indispensable tools for analyzing and modeling complex phenomena. For example, in electrical engineering, complex numbers are used to represent alternating current (AC) circuits, where voltage and current fluctuate sinusoidally over time. The magnitude of a complex number can represent the amplitude of the AC signal, while the phase represents the timing relationship between voltage and current.

The Mechanics of Complex Number Subtraction

The subtraction of complex numbers follows a straightforward procedure rooted in the fundamental principles of arithmetic. When subtracting one complex number from another, we independently subtract the real parts and the imaginary parts. Mathematically, this can be expressed as follows:

(a + bi) - (c + di) = (a - c) + (b - d)i

Here, (a + bi) and (c + di) represent two complex numbers, where a and c are the real parts, and b and d are the imaginary parts. The result of the subtraction is a new complex number whose real part is the difference between the original real parts (a - c) and whose imaginary part is the difference between the original imaginary parts (b - d) multiplied by the imaginary unit i.

A Step-by-Step Subtraction of (6 - 5i) - (7 + 6i)

Having laid the groundwork for complex number subtraction, let us now apply this knowledge to the specific problem at hand: subtracting (7 + 6i) from (6 - 5i). We will meticulously follow a step-by-step approach to ensure clarity and accuracy.

Step 1: Identify the Components

The first step in subtracting complex numbers is to identify the real and imaginary parts of each number. In the complex number (6 - 5i), the real part is 6, and the imaginary part is -5. Similarly, in the complex number (7 + 6i), the real part is 7, and the imaginary part is 6. Accurate identification of these components is crucial for the subsequent steps.

Step 2: Subtract the Real Parts

Next, we subtract the real part of the second complex number (7) from the real part of the first complex number (6):

6 - 7 = -1

The result of this subtraction is -1, which will form the real part of the resultant complex number.

Step 3: Subtract the Imaginary Parts

Following the subtraction of the real parts, we subtract the imaginary part of the second complex number (6) from the imaginary part of the first complex number (-5):

-5 - 6 = -11

The result of this subtraction is -11, which will be the coefficient of the imaginary unit i in the resultant complex number.

Step 4: Combine the Results

Finally, we combine the results from the previous steps to construct the resulting complex number. The real part is -1, and the imaginary part is -11, yielding the following complex number:

-1 - 11i

Therefore, the result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i. This complex number represents the difference between the two original complex numbers in the complex plane.

Visualizing Complex Number Subtraction in the Complex Plane

Complex numbers can be visually represented as points in a two-dimensional plane known as the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Subtracting complex numbers can be geometrically interpreted as finding the vector that connects the two complex numbers in the complex plane.

In our illustrative example, the complex number (6 - 5i) can be represented as the point (6, -5) in the complex plane, while the complex number (7 + 6i) can be represented as the point (7, 6). Subtracting (7 + 6i) from (6 - 5i) is equivalent to finding the vector that originates at the point (7, 6) and terminates at the point (6, -5). This vector corresponds to the complex number -1 - 11i, which aligns with the result we obtained through algebraic subtraction. This visual representation provides an intuitive understanding of the subtraction process and its geometric interpretation.

The Significance of Complex Number Subtraction

Complex number subtraction, while seemingly abstract, plays a crucial role in various scientific and engineering domains. In electrical engineering, complex numbers are used to represent impedances in AC circuits. Subtracting complex impedances allows engineers to analyze circuit behavior, calculate voltage drops, and design efficient electrical systems. For instance, determining the difference between two impedances in a series circuit aids in calculating the overall impedance, which is critical for understanding voltage distribution and current flow.

In quantum mechanics, complex numbers are indispensable for describing the behavior of quantum systems. Wave functions, which represent the probability amplitude of finding a particle in a particular state, are often complex-valued. Subtracting wave functions allows physicists to analyze interference phenomena and explore the superposition of quantum states. When two wave functions representing different possible states of a particle are subtracted, the resulting wave function can reveal regions where the probability of finding the particle is significantly reduced due to destructive interference.

Signal processing, a field dedicated to the manipulation and analysis of signals, also leverages complex numbers extensively. Signals, such as audio or video signals, can be represented as complex-valued functions. Subtracting signals finds applications in noise cancellation, where unwanted noise is subtracted from the original signal to enhance clarity. In audio processing, subtracting a noise signal from a recorded sound can significantly improve the listening experience by mitigating background disturbances.

Common Errors and Mitigation Strategies

While the procedure for complex number subtraction is relatively straightforward, certain common errors can arise if care is not exercised. One frequent mistake is neglecting to distribute the negative sign correctly when subtracting a complex number. For instance, when subtracting (7 + 6i) from (6 - 5i), it is crucial to remember that the negative sign applies to both the real and imaginary parts of (7 + 6i). Failure to distribute the negative sign can lead to incorrect results.

Another prevalent pitfall is incorrectly combining real and imaginary parts. When subtracting complex numbers, it is imperative to remember that real parts should be subtracted from real parts, and imaginary parts should be subtracted from imaginary parts. Mixing these parts can lead to erroneous outcomes. For example, mistakenly subtracting the real part of one complex number from the imaginary part of another will yield an incorrect complex number.

To minimize these errors, it is advisable to adhere meticulously to the step-by-step approach outlined earlier. Double-checking each step and paying close attention to signs can help prevent mistakes. Furthermore, engaging in practice with various examples of complex number subtraction can reinforce understanding and reduce the likelihood of errors.

Concluding Remarks: Mastering Complex Number Subtraction

In summary, subtracting complex numbers involves subtracting the real parts independently and the imaginary parts independently. This operation, while seemingly simple, has profound implications across diverse scientific and engineering disciplines. By comprehending the underlying principles, adhering to a step-by-step approach, and visualizing the operation in the complex plane, one can master complex number subtraction and apply it effectively in a multitude of contexts. The result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i, a complex number that encapsulates the difference between the two original complex numbers. This operation not only forms a fundamental building block in complex number arithmetic but also serves as a gateway to understanding more advanced mathematical and scientific concepts.

In the fascinating world of complex numbers, subtraction stands as a fundamental operation, paving the way for more intricate calculations and applications across various scientific and engineering domains. This comprehensive guide delves into the process of subtracting complex numbers, specifically focusing on the example of (6 - 5i) - (7 + 6i). We will dissect the steps involved, clarify the underlying principles, and highlight the real-world significance of this operation.

Grasping the Essence of Complex Numbers

Before we embark on the subtraction process, it is crucial to establish a solid understanding of what complex numbers are. A complex number is a mathematical entity that extends the realm of real numbers by incorporating an imaginary component. It is conventionally expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, defined as the square root of -1. The real part of the complex number is denoted by a, while the imaginary part is denoted by b. For example, in the complex number 3 + 4i, 3 is the real part, and 4 is the imaginary part.

Complex numbers are not just abstract mathematical concepts; they have practical applications in numerous fields, including electrical engineering, quantum mechanics, fluid dynamics, and signal processing. Their ability to represent quantities with both magnitude and phase makes them invaluable tools for analyzing and modeling complex phenomena. In electrical engineering, for instance, complex numbers are used to represent alternating current (AC) circuits, where voltage and current fluctuate sinusoidally over time. The magnitude of a complex number can represent the amplitude of the AC signal, while the phase represents the timing relationship between voltage and current. Similarly, in quantum mechanics, complex numbers are used to describe wave functions, which represent the probability amplitude of finding a particle in a particular state.

Unveiling the Mechanics of Complex Number Subtraction

The subtraction of complex numbers follows a straightforward procedure rooted in the fundamental principles of arithmetic. When subtracting one complex number from another, we independently subtract the real parts and the imaginary parts. Mathematically, this can be expressed as follows:

(a + bi) - (c + di) = (a - c) + (b - d)i

In this equation, (a + bi) and (c + di) represent two complex numbers, where a and c are the real parts, and b and d are the imaginary parts. The result of the subtraction is a new complex number whose real part is the difference between the original real parts (a - c) and whose imaginary part is the difference between the original imaginary parts (b - d) multiplied by the imaginary unit i. This process ensures that the real and imaginary components are treated separately, maintaining the integrity of the complex number structure.

A Step-by-Step Guide to Subtracting (6 - 5i) - (7 + 6i)

Now that we have laid the theoretical foundation, let us apply our knowledge to the specific problem at hand: subtracting (7 + 6i) from (6 - 5i). We will meticulously follow a step-by-step approach to ensure clarity and accuracy.

Step 1: Identifying the Components

The initial step in subtracting complex numbers is to identify the real and imaginary parts of each number. In the complex number (6 - 5i), the real part is 6, and the imaginary part is -5. Likewise, in the complex number (7 + 6i), the real part is 7, and the imaginary part is 6. Accurate identification of these components is paramount for the subsequent steps, as it ensures that the correct values are used in the subtraction process.

Step 2: Subtracting the Real Parts

Following the identification of components, we proceed to subtract the real part of the second complex number (7) from the real part of the first complex number (6):

6 - 7 = -1

The result of this subtraction is -1, which will form the real part of the resultant complex number. This step is a straightforward application of basic arithmetic principles.

Step 3: Subtracting the Imaginary Parts

Having subtracted the real parts, we now turn our attention to the imaginary parts. We subtract the imaginary part of the second complex number (6) from the imaginary part of the first complex number (-5):

-5 - 6 = -11

The result of this subtraction is -11, which will be the coefficient of the imaginary unit i in the resultant complex number. This step mirrors the subtraction of the real parts, ensuring that the imaginary components are treated appropriately.

Step 4: Combining the Results

With the real and imaginary parts subtracted separately, we combine the results to construct the resulting complex number. The real part is -1, and the imaginary part is -11, yielding the following complex number:

-1 - 11i

Therefore, the result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i. This complex number represents the difference between the two original complex numbers in the complex plane.

Visualizing Complex Number Subtraction in the Complex Plane

To gain a more intuitive understanding of complex number subtraction, it is helpful to visualize the process in the complex plane. The complex plane is a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Each complex number can be represented as a point in this plane, with the real part corresponding to the x-coordinate and the imaginary part corresponding to the y-coordinate.

Subtracting complex numbers can be geometrically interpreted as finding the vector that connects the two complex numbers in the complex plane. In our example, the complex number (6 - 5i) can be represented as the point (6, -5) in the complex plane, while the complex number (7 + 6i) can be represented as the point (7, 6). Subtracting (7 + 6i) from (6 - 5i) is equivalent to finding the vector that originates at the point (7, 6) and terminates at the point (6, -5). This vector corresponds to the complex number -1 - 11i, which aligns with the result we obtained through algebraic subtraction. This visual representation provides a powerful tool for understanding the geometric implications of complex number subtraction.

The Wide-Ranging Significance of Complex Number Subtraction

Complex number subtraction, while seemingly an abstract mathematical operation, has a profound impact on various scientific and engineering disciplines. In electrical engineering, complex numbers are used to represent impedances in AC circuits. Subtracting complex impedances allows engineers to analyze circuit behavior, calculate voltage drops, and design efficient electrical systems. For example, determining the difference between two impedances in a series circuit aids in calculating the overall impedance, which is critical for understanding voltage distribution and current flow.

Common Pitfalls and Strategies for Mitigation

While the procedure for complex number subtraction is relatively straightforward, certain common errors can arise if care is not exercised. One frequent mistake is neglecting to distribute the negative sign correctly when subtracting a complex number. For instance, when subtracting (7 + 6i) from (6 - 5i), it is crucial to remember that the negative sign applies to both the real and imaginary parts of (7 + 6i). Failure to distribute the negative sign can lead to incorrect results. To avoid this pitfall, it is essential to pay close attention to the signs and ensure that the negative sign is correctly applied to both the real and imaginary components of the complex number being subtracted.

Another prevalent pitfall is incorrectly combining real and imaginary parts. When subtracting complex numbers, it is imperative to remember that real parts should be subtracted from real parts, and imaginary parts should be subtracted from imaginary parts. Mixing these parts can lead to erroneous outcomes. For example, mistakenly subtracting the real part of one complex number from the imaginary part of another will yield an incorrect complex number. To mitigate this error, it is crucial to maintain the separation between real and imaginary components throughout the subtraction process.

To minimize these errors, it is advisable to adhere meticulously to the step-by-step approach outlined earlier. Double-checking each step and paying close attention to signs can help prevent mistakes. Furthermore, engaging in practice with various examples of complex number subtraction can reinforce understanding and reduce the likelihood of errors. Practice is key to mastering this operation and avoiding common pitfalls.

Concluding Thoughts: Mastering Complex Number Subtraction

In conclusion, subtracting complex numbers involves subtracting the real parts independently and the imaginary parts independently. This operation, while seemingly simple, has profound implications across diverse scientific and engineering disciplines. By comprehending the underlying principles, adhering to a step-by-step approach, and visualizing the operation in the complex plane, one can master complex number subtraction and apply it effectively in a multitude of contexts. The result of subtracting (7 + 6i) from (6 - 5i) is -1 - 11i, a complex number that encapsulates the difference between the two original complex numbers. This operation not only forms a fundamental building block in complex number arithmetic but also serves as a gateway to understanding more advanced mathematical and scientific concepts. Mastering complex number subtraction is a valuable skill that opens doors to a deeper understanding of the mathematical world.