Simplifying Expressions With Complex Numbers A Step By Step Guide

This article provides a detailed walkthrough on how to simplify the expression $(\sqrt{2}-\sqrt{-2})(\sqrt{18}+\sqrt{-2})$. We'll explore the concepts of imaginary numbers, complex numbers, and the distributive property, ensuring a clear and comprehensive understanding of each step. This topic falls under the domain of complex number arithmetic, a crucial area in mathematics with applications in various fields such as physics and engineering. Mastering complex number manipulation not only enhances problem-solving skills but also builds a strong foundation for advanced mathematical concepts.

Understanding the Basics of Complex Numbers

Before diving into the simplification, let's solidify our understanding of complex numbers. A complex number is generally expressed in the form a + bi, where a is the real part and b is the imaginary part. The imaginary unit, denoted by i, is defined as the square root of -1 (i = \sqrt{-1}). This definition is crucial because it allows us to handle square roots of negative numbers, which are not defined within the realm of real numbers. Understanding the imaginary unit is the cornerstone of working with complex numbers, allowing us to extend algebraic operations to a broader set of numbers. The interplay between real and imaginary parts gives complex numbers their richness and versatility, making them indispensable in higher-level mathematics. Grasping the fundamentals of complex numbers is essential for tackling expressions like the one we aim to simplify, as it involves manipulating both real and imaginary components. The ability to seamlessly transition between real and imaginary numbers is a key skill that unlocks a deeper understanding of mathematical structures and their applications.

The Imaginary Unit: i

The imaginary unit i is the foundation of complex numbers, defined as i = \sqrt{-1}. This seemingly simple definition opens up a whole new dimension in mathematics, allowing us to work with the square roots of negative numbers. To further clarify, i² = (\sqrt{-1})² = -1. This property is essential for simplifying expressions involving complex numbers. For example, \sqrt{-4} can be written as \sqrt{4 \cdot -1} = \sqrt{4} \cdot \sqrt{-1} = 2i. When simplifying expressions, we aim to reduce imaginary parts to their simplest form, often using the property of i². This process is crucial in ensuring that complex numbers are expressed in their standard form, making them easier to compare and manipulate. The imaginary unit not only fills a gap in the number system but also provides a powerful tool for solving equations and modeling phenomena that cannot be described using real numbers alone. Understanding i and its properties is the first step in navigating the complex number landscape, setting the stage for more advanced operations and applications.

Complex Numbers in the Form a + bi

A complex number is expressed in the standard form a + bi, where a represents the real part and b represents the imaginary part. This representation allows us to visualize complex numbers as points on a complex plane, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part. This geometric interpretation provides a powerful tool for understanding complex number operations. For instance, the complex number 3 + 2i has a real part of 3 and an imaginary part of 2. Complex numbers encompass both real numbers (when b = 0) and purely imaginary numbers (when a = 0). This inclusivity makes complex numbers a comprehensive number system. Operations such as addition, subtraction, multiplication, and division can be performed on complex numbers by treating i as a variable and then using the property i² = -1 to simplify. Mastery of the a + bi form is crucial for performing complex number arithmetic, ensuring that expressions are simplified and presented in a clear, standard manner. The ability to recognize and manipulate complex numbers in this form is essential for further exploration of complex analysis and its applications in various scientific and engineering fields.

Step-by-Step Simplification of $(\sqrt{2}-\sqrt{-2})(\sqrt{18}+\sqrt{-2})$

Now, let's tackle the given expression step by step: $(\sqrt{2}-\sqrt{-2})(\sqrt{18}+\sqrt{-2})$. We'll break it down into manageable steps, making sure each transformation is clear and justified. This process not only simplifies the expression but also reinforces the principles of complex number arithmetic. The goal is to transform the initial expression into its simplest form, adhering to the rules of algebraic manipulation and complex number properties. By meticulously working through each step, we minimize the chances of error and gain a deeper understanding of the underlying concepts. This detailed approach is invaluable for tackling more complex problems in the future, as it provides a solid framework for problem-solving in complex number algebra. The step-by-step simplification ensures that the final result is both accurate and easily interpretable, demonstrating the power and elegance of mathematical manipulation.

Step 1: Rewrite the Square Roots of Negative Numbers

The first step in simplifying the expression is to rewrite the square roots of negative numbers using the imaginary unit i. Recall that *\sqrt-1} = i*. Therefore, we can rewrite \sqrt{-2} as \sqrt{2 \cdot -1} = \sqrt{2} \cdot \sqrt{-1} = \sqrt{2}i. Now, substitute this back into the original expression $(\sqrt{2 - \sqrt{2}i)(\sqrt{18} + \sqrt{2}i)$. This transformation is crucial because it allows us to work with the expression using standard algebraic rules. By separating the real and imaginary components, we set the stage for applying the distributive property. This step highlights the importance of understanding the definition of i and its role in complex number manipulations. The ability to rewrite square roots of negative numbers in terms of i is a fundamental skill in complex number arithmetic, paving the way for further simplification and calculations. This initial transformation simplifies the problem significantly, making it easier to handle and solve.

Step 2: Simplify $\sqrt{18}$

Next, let's simplify *\sqrt18}*. We can rewrite 18 as 9 * 2, so \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}. Substituting this back into our expression gives us $(\sqrt{2 - \sqrt{2}i)(3\sqrt{2} + \sqrt{2}i)$. Simplifying radicals is a key technique in algebraic manipulation, allowing us to express numbers in their simplest forms. This step reduces the complexity of the expression, making it easier to work with in subsequent steps. By simplifying \sqrt{18}, we avoid dealing with larger numbers, which can lead to errors. This meticulous approach ensures that each component of the expression is in its simplest form before proceeding. The simplification of radicals is a fundamental skill that enhances our ability to manipulate algebraic expressions effectively. This step not only cleans up the expression but also prepares it for the application of the distributive property, the next crucial step in our simplification process.

Step 3: Apply the Distributive Property (FOIL)

Now, we apply the distributive property (often remembered by the acronym FOIL, which stands for First, Outer, Inner, Last) to expand the expression. We multiply each term in the first parenthesis by each term in the second parenthesis:

• First: \sqrt{2} \cdot 3\sqrt{2} = 3 \cdot 2 = 6
• Outer: \sqrt{2} \cdot \sqrt{2}i = 2i
• Inner: -\sqrt{2}i \cdot 3\sqrt{2} = -3 \cdot 2i = -6i
• Last: -\sqrt{2}i \cdot \sqrt{2}i = -2i²

So, the expanded expression is: 6 + 2i - 6i - 2i². The distributive property is a fundamental algebraic technique used to multiply expressions, and its application here is crucial for expanding the product of the complex numbers. By meticulously applying FOIL, we ensure that every term is accounted for and multiplied correctly. This step transforms the product into a sum of terms, making it easier to simplify further. The expansion process is a key step in unraveling the expression, setting the stage for combining like terms and simplifying imaginary units. Mastering the distributive property is essential for algebraic manipulation, allowing us to handle complex expressions with confidence and accuracy. This step significantly progresses our simplification journey, bringing us closer to the final, simplified form of the expression.

Step 4: Simplify and Combine Like Terms

Now, let's simplify the expression 6 + 2i - 6i - 2i². Recall that i² = -1, so we can substitute this into the expression: 6 + 2i - 6i - 2(-1) = 6 + 2i - 6i + 2. Next, combine the real parts (6 and 2) and the imaginary parts (2i and -6i): (6 + 2) + (2i - 6i) = 8 - 4i. This step involves both substitution and combination of like terms, two essential algebraic techniques. Substituting i² = -1 eliminates the imaginary unit squared, simplifying the expression. Combining like terms consolidates the expression, reducing the number of terms and making it easier to interpret. This simplification process is a cornerstone of algebraic manipulation, allowing us to express mathematical expressions in their most concise and understandable forms. By carefully combining real and imaginary parts, we arrive at the standard form of a complex number, making it easy to compare with other complex numbers. This step brings us to the final stage of simplification, where the expression is in its simplest and most elegant form.

Final Answer

Therefore, the simplified form of the expression $(\sqrt{2}-\sqrt{-2})(\sqrt{18}+\sqrt{-2})$ is 8 - 4i. This final answer encapsulates all the steps we've taken, demonstrating the power of complex number arithmetic and algebraic manipulation. The journey from the initial expression to this simplified form highlights the importance of understanding complex number properties and applying algebraic techniques systematically. The result, 8 - 4i, is a complex number in its standard form, with a real part of 8 and an imaginary part of -4. This simplified form is not only aesthetically pleasing but also facilitates further calculations and comparisons with other complex numbers. Mastering the simplification of such expressions builds confidence in handling more complex mathematical problems, solidifying the foundation for advanced studies in mathematics, physics, and engineering. The final answer serves as a testament to the elegance and efficiency of mathematical methods, showcasing how seemingly complex expressions can be reduced to simple, understandable forms.