Subtracting Complex Numbers A Step-by-Step Guide

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Complex numbers, an extension of the real number system, play a crucial role in various fields like mathematics, physics, and engineering. They consist of two parts: a real part and an imaginary part, denoted by the imaginary unit 'i,' where i² = -1. Operations with complex numbers, such as addition, subtraction, multiplication, and division, follow specific rules. In this comprehensive guide, we will delve into the process of subtracting complex numbers, providing a step-by-step explanation and addressing common questions.

Complex Number Basics: A Quick Recap

Before diving into subtraction, let's refresh our understanding of complex numbers. A complex number is generally expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The imaginary part is always multiplied by the imaginary unit 'i'. For instance, in the complex number 3 + 7i, 3 is the real part, and 7i is the imaginary part.

Subtraction of Complex Numbers: The Process

Subtracting complex numbers involves a straightforward process of combining like terms. Like terms are those that share the same variable or, in the case of complex numbers, either both are real or both are imaginary. Here's a detailed breakdown of the steps:

1. Distribute the Negative Sign:

When subtracting one complex number from another, the first step is to distribute the negative sign to each term within the second complex number. This is similar to how you would handle subtraction in algebraic expressions. For example, if we have (3 + 7i) - (-4 - 2i), distributing the negative sign gives us 3 + 7i + 4 + 2i.

2. Combine the Real Parts:

Next, identify the real parts of the complex numbers and combine them. In our example, the real parts are 3 and 4. Adding them together, we get 3 + 4 = 7. This result will be the real part of the final answer.

3. Combine the Imaginary Parts:

Similarly, identify the imaginary parts of the complex numbers and combine them. Remember that the imaginary parts are the terms multiplied by 'i'. In our example, the imaginary parts are 7i and 2i. Adding them together, we get 7i + 2i = 9i. This result will be the imaginary part of the final answer.

4. Write the Result in Standard Form:

Finally, combine the results from steps 2 and 3 to write the final answer in the standard form of a complex number, which is a + bi. In our example, the real part is 7, and the imaginary part is 9i, so the final answer is 7 + 9i.

Solving the Problem: (3 + 7i) - (-4 - 2i)

Now, let's apply these steps to solve the given problem: (3 + 7i) - (-4 - 2i).

Step 1: Distribute the Negative Sign

Distributing the negative sign to the second complex number, we get:

3 + 7i + 4 + 2i

Step 2: Combine the Real Parts

The real parts are 3 and 4. Combining them:

3 + 4 = 7

Step 3: Combine the Imaginary Parts

The imaginary parts are 7i and 2i. Combining them:

7i + 2i = 9i

Step 4: Write the Result in Standard Form

Combining the real and imaginary parts, we get:

7 + 9i

Therefore, the difference between the complex numbers (3 + 7i) and (-4 - 2i) is 7 + 9i.

The Correct Answer: A. 7 + 9i

Based on our step-by-step solution, the correct answer is A. 7 + 9i. This corresponds to the result we obtained by meticulously following the rules of complex number subtraction.

Visualizing Complex Number Subtraction

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. Subtraction of complex numbers can be interpreted geometrically as vector subtraction. If we represent the complex numbers as vectors, subtracting one complex number from another is equivalent to finding the vector that connects the endpoint of the second vector to the endpoint of the first vector.

For instance, in our example, (3 + 7i) can be represented as a vector from the origin to the point (3, 7) on the complex plane, and (-4 - 2i) can be represented as a vector from the origin to the point (-4, -2). Subtracting (-4 - 2i) from (3 + 7i) is equivalent to finding the vector that connects the point (-4, -2) to the point (3, 7). This vector corresponds to the complex number 7 + 9i, which is the result we obtained algebraically.

Common Mistakes to Avoid

When subtracting complex numbers, it's essential to be mindful of potential errors. Here are some common mistakes to avoid:

  • Forgetting to Distribute the Negative Sign: A common mistake is to forget to distribute the negative sign to all terms within the second complex number. This can lead to incorrect results.
  • Incorrectly Combining Like Terms: Ensure that you combine only the real parts with real parts and imaginary parts with imaginary parts. Mixing them up will lead to errors.
  • Sign Errors: Pay close attention to the signs of the terms when combining them. A simple sign error can change the entire result.

Real-World Applications of Complex Number Subtraction

Complex number subtraction, while seemingly abstract, has practical applications in various real-world scenarios. Here are a few examples:

  • Electrical Engineering: Complex numbers are used to represent alternating current (AC) circuits, where voltage and current are sinusoidal functions. Subtracting complex numbers can help analyze the impedance and phase differences in these circuits.
  • Quantum Mechanics: In quantum mechanics, complex numbers are used to describe wave functions, which represent the state of a quantum system. Subtracting complex wave functions can help determine the interference patterns and probabilities of quantum events.
  • Signal Processing: Complex numbers are used to represent signals in signal processing applications. Subtracting complex signals can help filter out noise or extract specific components from a signal.

Practice Problems

To solidify your understanding of complex number subtraction, let's try a few practice problems:

  1. (5 - 2i) - (1 + 3i)
  2. (-2 + 4i) - (3 - i)
  3. (7 + i) - (-2 - 5i)

Work through these problems step-by-step, following the process we outlined earlier. You can check your answers by comparing them with the solutions provided below.

Solutions to Practice Problems

  1. (5 - 2i) - (1 + 3i) = 5 - 2i - 1 - 3i = (5 - 1) + (-2i - 3i) = 4 - 5i
  2. (-2 + 4i) - (3 - i) = -2 + 4i - 3 + i = (-2 - 3) + (4i + i) = -5 + 5i
  3. (7 + i) - (-2 - 5i) = 7 + i + 2 + 5i = (7 + 2) + (i + 5i) = 9 + 6i

Conclusion: Mastering Complex Number Subtraction

Subtracting complex numbers is a fundamental operation in complex number arithmetic. By understanding the process of distributing the negative sign and combining like terms, you can confidently subtract complex numbers and solve related problems. Remember to visualize complex numbers on the complex plane and be mindful of common mistakes to ensure accurate results. With practice, you'll master this essential skill and be well-prepared to tackle more advanced concepts in complex number theory.

By following the step-by-step guide, visualizing the process, and avoiding common mistakes, you can confidently tackle complex number subtraction problems in various contexts. This foundational skill will serve you well in diverse fields that utilize complex numbers, from electrical engineering to quantum mechanics. Continue practicing and exploring the fascinating world of complex numbers!