Simplify Complex Numbers Express In A + Bi Form

Understanding Complex Numbers

To simplify complex numbers, we first need a solid understanding of what they are. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit i is defined as the square root of -1, meaning i^2 = -1. The a part is the real component, and the bi part is the imaginary component of the complex number.

Operations with complex numbers involve treating i as a variable, with the crucial distinction that whenever i^2 appears, it should be replaced with -1. Complex numbers are used extensively in various fields, including mathematics, physics, engineering, and computer science, to represent quantities that involve both magnitude and direction. For example, in electrical engineering, complex numbers are used to describe alternating current circuits, where the real part represents resistance and the imaginary part represents reactance. In quantum mechanics, complex numbers are essential for describing wave functions, which are mathematical representations of the state of a quantum system. The ability to simplify complex numbers is crucial for solving problems in these fields and for developing a deeper understanding of the underlying concepts. When dealing with complex numbers, it's important to remember the rules for performing arithmetic operations such as addition, subtraction, multiplication, and division. Addition and subtraction are straightforward, involving combining the real and imaginary parts separately. Multiplication requires using the distributive property and the fact that i^2 = -1. Division involves multiplying both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. Mastering these operations is essential for working effectively with complex numbers in a variety of contexts. In many real-world applications, complex numbers provide a powerful tool for modeling and analyzing systems that cannot be adequately described using real numbers alone. Their ability to represent both magnitude and phase makes them indispensable in fields where oscillations, waves, and rotations are involved. For students and professionals alike, a strong foundation in complex numbers is essential for success in these areas.

Problem Statement: Subtracting Complex Numbers

The problem we're tackling today involves subtracting one complex number from another. We are given the expression (-20 + 10i) - (-13 - 8i). Our goal is to simplify this expression and write the result in the standard form of a complex number, which is a + bi. This means we need to combine the real parts and the imaginary parts separately to arrive at a single complex number in the desired format. Subtraction of complex numbers follows a similar principle to addition, where we treat the real and imaginary components independently. The key difference, of course, is that we are subtracting the corresponding parts instead of adding them. Understanding how to perform subtraction correctly is fundamental for more advanced operations with complex numbers, such as multiplication, division, and exponentiation. Moreover, it lays the groundwork for solving equations involving complex variables and for understanding concepts like complex functions and transformations. In the context of various applications, such as signal processing and control systems, complex number subtraction is often used to find the difference between two signals or to analyze the response of a system to a particular input. A thorough understanding of this operation is therefore essential for students and professionals working in these fields. As we proceed with the solution, we will carefully apply the rules of arithmetic to ensure that we arrive at the correct result in the form a + bi. This will involve distributing the negative sign, combining like terms, and expressing the final answer in the required format. By breaking down the problem into smaller steps, we can gain a clear understanding of the process and avoid common errors. This approach will not only help us solve this specific problem but also provide a solid foundation for tackling more complex problems involving complex numbers.

Step-by-Step Solution

1. Distribute the Negative Sign: The first step is to distribute the negative sign in front of the second complex number. This means we change the sign of each term inside the parentheses:

(-20 + 10i) - (-13 - 8i) = -20 + 10i + 13 + 8i

Distributing the negative sign correctly is a crucial step in simplifying complex number expressions. It ensures that we account for the subtraction of both the real and imaginary components of the second complex number. By carefully applying the distributive property, we can avoid errors and accurately combine like terms in the subsequent steps. This process is similar to subtracting algebraic expressions, where we distribute the negative sign to each term within the parentheses before combining like terms. In the context of complex numbers, this step is particularly important because it sets the stage for correctly adding the real and imaginary parts separately. A common mistake is to forget to distribute the negative sign to both terms, which can lead to an incorrect final answer. Therefore, it's essential to pay close attention to this step and double-check that the signs of all terms have been correctly adjusted. Mastering the distribution of the negative sign not only simplifies this specific type of problem but also builds a solid foundation for more advanced operations with complex numbers, such as polynomial manipulation and solving complex equations. By consistently applying this rule, we can ensure accuracy and efficiency in working with complex expressions.

2. Combine Real Parts: Next, we combine the real parts of the complex numbers:

-20 + 13 = -7

Combining the real parts is a fundamental step in simplifying complex numbers. It involves adding or subtracting the real components of the complex numbers to obtain a single real number. This process is analogous to combining like terms in algebraic expressions, where we group the constant terms together. In the context of complex numbers, the real part represents the component that does not involve the imaginary unit i. Therefore, we can treat the real parts as regular numbers and perform the arithmetic operations accordingly. This step is crucial because it helps us isolate the real component of the final complex number, which is necessary for expressing the result in the standard a + bi form. Errors in combining the real parts can lead to an incorrect final answer, so it's important to pay close attention to the signs and values involved. In this specific problem, we are subtracting the real part of the second complex number from the real part of the first complex number. This requires careful attention to the signs to ensure that we perform the subtraction correctly. Once we have combined the real parts, we move on to combining the imaginary parts, which will complete the process of simplifying the complex number expression. By following these steps methodically, we can accurately simplify complex numbers and express them in the desired format.

3. Combine Imaginary Parts: Now, we combine the imaginary parts of the complex numbers:

10i + 8i = 18i

Combining the imaginary parts is the counterpart to combining the real parts in simplifying complex numbers. It involves adding or subtracting the imaginary components, which are the terms that include the imaginary unit i. Just as we treated the real parts as regular numbers, we can treat the imaginary parts as like terms and perform the arithmetic operations accordingly. In this step, we are adding the imaginary parts of the two complex numbers, which means we are combining the coefficients of i. This process is similar to combining like terms in algebraic expressions, where we group the terms with the same variable together. The result of combining the imaginary parts will give us the imaginary component of the final complex number, which is necessary for expressing the result in the standard a + bi form. Accuracy in combining the imaginary parts is crucial for obtaining the correct final answer. A common mistake is to forget to include the imaginary unit i in the final term, which would change the nature of the number. Therefore, it's essential to ensure that we correctly combine the coefficients of i and include i in the resulting term. Once we have combined both the real and imaginary parts, we can write the final answer in the a + bi form, which completes the simplification process. By following these steps systematically, we can confidently simplify complex number expressions and solve problems involving complex numbers.

4. Write in a + bi Form: Finally, we write the result in the standard form a + bi:

-7 + 18i

Writing the result in the a + bi form is the final step in simplifying complex numbers. This form clearly separates the real and imaginary parts of the complex number, making it easy to identify and work with. In this form, a represents the real part, and b represents the imaginary part, which is the coefficient of the imaginary unit i. By expressing the complex number in this standard form, we ensure that it is in a format that is universally recognized and can be easily used in further calculations or applications. The a + bi form is particularly useful when performing arithmetic operations with complex numbers, as it allows us to treat the real and imaginary parts separately and combine them according to the rules of complex number arithmetic. It also makes it easier to visualize complex numbers in the complex plane, where the real part is plotted along the horizontal axis and the imaginary part is plotted along the vertical axis. In many fields, such as electrical engineering and quantum mechanics, complex numbers are frequently expressed in the a + bi form to facilitate calculations and analysis. Therefore, mastering the ability to write complex numbers in this form is essential for anyone working with complex numbers. In this specific problem, we have combined the real and imaginary parts to obtain the final result, which we then express in the a + bi form. This completes the simplification process and provides us with the answer in the required format.

Final Answer

Therefore, (-20 + 10i) - (-13 - 8i) = -7 + 18i. This result is in the required a + bi form, where a = -7 and b = 18.

Conclusion

In conclusion, we have successfully simplified the expression (-20 + 10i) - (-13 - 8i) and expressed the result in the standard complex number form a + bi. This process involved distributing the negative sign, combining the real parts, and combining the imaginary parts. Understanding how to simplify complex numbers is essential for various applications in mathematics, physics, and engineering. Complex numbers provide a powerful tool for representing and manipulating quantities that involve both magnitude and direction. Their applications range from electrical circuit analysis to quantum mechanics, where they are used to describe wave functions. The ability to perform arithmetic operations with complex numbers, such as addition, subtraction, multiplication, and division, is crucial for solving problems in these fields. In this article, we focused on the subtraction of complex numbers, which is a fundamental operation that forms the basis for more advanced concepts. By carefully following the steps outlined in the solution, we can accurately simplify complex number expressions and obtain the results in the required format. The a + bi form is particularly important because it allows us to clearly identify the real and imaginary components of a complex number, making it easier to work with in further calculations. Mastering the simplification of complex numbers is not only valuable for academic purposes but also for practical applications in various scientific and engineering disciplines. By developing a strong understanding of complex number arithmetic, we can enhance our ability to solve complex problems and gain insights into the behavior of systems that can be modeled using complex numbers. The concepts and techniques discussed in this article provide a solid foundation for further exploration of complex analysis and its applications.