Simplifying Complex Numbers The Expression $(7+\sqrt{-4})-(-9+\sqrt{-36})$

Understanding Complex Number Arithmetic

In the realm of mathematics, complex numbers extend the familiar real number system by incorporating the imaginary unit, denoted as i, where i is defined as the square root of -1. This inclusion allows us to address equations that have no solutions within the real number domain, such as the square root of negative numbers. Complex numbers are expressed in the form a + bi, where a represents the real part and bi represents the imaginary part. Operations involving complex numbers follow specific rules that blend real number arithmetic with the unique properties of i. For instance, when adding or subtracting complex numbers, we combine the real parts separately and the imaginary parts separately. Multiplication and division involve applying the distributive property and employing the fact that i² = -1 to simplify expressions. The complex conjugate, formed by changing the sign of the imaginary part, plays a crucial role in dividing complex numbers, as it helps eliminate the imaginary component from the denominator. Mastering these operations is fundamental for solving complex equations, exploring advanced mathematical concepts, and applying complex numbers in various scientific and engineering fields.

Breaking Down the Expression

When dealing with mathematical expressions involving complex numbers, a systematic approach is crucial for accuracy and clarity. The expression $(7+\sqrt{-4})-(-9+\sqrt{-36})$ presents a combination of real and imaginary components that requires careful manipulation. Our initial step involves simplifying the square roots of negative numbers. Recognizing that $\sqrt{-4}$ can be rewritten as $\sqrt{4 \cdot -1}$, we can separate this into $\sqrt{4} \cdot \sqrt{-1}$, which equals 2i. Similarly, $\sqrt{-36}$ becomes $\sqrt{36 \cdot -1}$, simplifying to $\sqrt{36} \cdot \sqrt{-1}$, or 6i. Substituting these simplified imaginary terms back into the original expression, we now have (7 + 2i) - (-9 + 6i). This transformation is pivotal as it allows us to work with the expression using standard complex number arithmetic rules. Next, we address the subtraction operation between the two complex number terms. To do this, we distribute the negative sign across the second term, effectively changing the subtraction into an addition of the negative. This step is crucial for correctly combining like terms and arriving at the final simplified form. By meticulously breaking down each component and applying the fundamental principles of complex number manipulation, we pave the way for a seamless simplification process and ensure the accuracy of our result.

Step-by-Step Solution

To effectively solve the complex number expression $(7+\sqrt{-4})-(-9+\sqrt{-36})$, a methodical, step-by-step approach is essential. We begin by simplifying the square roots of the negative numbers, as these introduce the imaginary unit i. Recall that $\sqrt{-1}$ is defined as i. Therefore, $\sqrt{-4}$ can be expressed as $\sqrt{4} \cdot \sqrt{-1}$, which simplifies to 2i. Similarly, $\sqrt{-36}$ is equivalent to $\sqrt{36} \cdot \sqrt{-1}$, simplifying to 6i. Now, substitute these simplified forms back into the original expression, resulting in $(7 + 2i) - (-9 + 6i)$. The next critical step involves distributing the negative sign across the terms inside the second parentheses. This transforms the expression into 7 + 2i + 9 - 6i. By correctly distributing the negative sign, we ensure that the subsequent combination of like terms is accurate. Now, we group the real and imaginary parts together: (7 + 9) + (2i - 6i). This grouping allows for straightforward addition and subtraction of the real and imaginary components. Adding the real parts, 7 + 9, gives us 16. For the imaginary parts, 2i - 6i results in -4i. Combining these, we arrive at the simplified complex number 16 - 4i. This step-by-step breakdown ensures clarity and reduces the chance of errors, providing a clear pathway to the final solution. By meticulously applying the rules of complex number arithmetic, we can confidently simplify and solve such expressions.

Final Result and Interpretation

After meticulously following the step-by-step solution, we arrive at the simplified form of the expression $(7+\sqrt{-4})-(-9+\sqrt{-36})$. As demonstrated earlier, the expression simplifies to 16 - 4i. This final result is a complex number, consisting of a real part and an imaginary part. The real part, 16, represents the component along the real number line, while the imaginary part, -4i, represents the component along the imaginary axis. In the complex plane, this number can be visualized as a point with coordinates (16, -4), where the x-coordinate corresponds to the real part and the y-coordinate corresponds to the imaginary part. Understanding the structure of this result is crucial for various applications in mathematics, physics, and engineering. In electrical engineering, for instance, complex numbers are used to represent alternating current (AC) circuits, where the real part might represent resistance and the imaginary part represents reactance. Similarly, in quantum mechanics, complex numbers are fundamental to describing wave functions and quantum states. The ability to simplify complex expressions and interpret the resulting complex numbers is therefore not just an academic exercise but a practical skill with wide-ranging implications. Furthermore, the result 16 - 4i highlights the importance of adhering to the rules of complex number arithmetic. Each step, from simplifying square roots of negative numbers to distributing signs and combining like terms, contributes to the accuracy of the final answer. Any deviation from these rules can lead to an incorrect result, underscoring the need for careful and precise calculations when working with complex numbers.

Conclusion

In summary, the simplification of the expression $(7+\sqrt{-4})-(-9+\sqrt{-36})$ demonstrates the fundamental principles of complex number arithmetic. By meticulously breaking down the problem into manageable steps, we transformed the original expression into its simplest form: 16 - 4i. This process involved recognizing and simplifying the square roots of negative numbers, understanding the role of the imaginary unit i, distributing negative signs, and combining like terms. The resulting complex number, 16 - 4i, consists of a real part (16) and an imaginary part (-4i), which can be visually represented as a point in the complex plane. This exercise underscores the importance of adhering to the rules of complex number arithmetic to ensure accurate calculations and interpretations. Complex numbers are not merely abstract mathematical concepts; they have significant applications in various fields, including electrical engineering and quantum mechanics. Their ability to represent phenomena that cannot be described by real numbers alone makes them indispensable tools in these disciplines. Therefore, mastering the manipulation and simplification of complex expressions is crucial for anyone pursuing studies or careers in these areas. The step-by-step approach used in solving this expression serves as a model for tackling more complex problems involving complex numbers. By consistently applying these principles, one can confidently navigate the world of complex numbers and their applications.