Simplify -1 - √(-12) A Step-by-Step Guide

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This article provides a detailed, step-by-step explanation of how to simplify the complex number expression -1 - √(-12). We will explore the underlying concepts of imaginary numbers, radicals, and complex number arithmetic to make the process clear and easy to understand. Whether you are a student learning about complex numbers for the first time or someone looking to refresh your knowledge, this guide will provide you with the tools and understanding you need to tackle similar problems.

Understanding Imaginary Numbers

To simplify the expression, we first need to delve into the world of imaginary numbers. These numbers arise from taking the square root of negative numbers, which are not defined within the realm of real numbers. The imaginary unit, denoted by i, is defined as the square root of -1: i = √(-1). This seemingly simple definition opens up a whole new dimension in mathematics, allowing us to work with the square roots of negative numbers.

Imaginary numbers are a fundamental concept in mathematics, extending the number system beyond the familiar real numbers. Real numbers include all rational and irrational numbers, such as integers, fractions, and decimals. However, when we encounter the square root of a negative number, we step into the realm of imaginary numbers. The cornerstone of imaginary numbers is the imaginary unit, denoted as i, which is defined as the square root of -1 (i = √(-1)). This definition allows us to express the square root of any negative number as a multiple of i. For instance, the square root of -4 can be written as √(-4) = √(4 * -1) = √(4) * √(-1) = 2i. Understanding imaginary numbers is crucial for working with complex numbers, which are numbers that have both a real and an imaginary component. Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The ability to manipulate and simplify expressions involving imaginary numbers is essential in various fields, including electrical engineering, quantum mechanics, and signal processing. This foundational understanding will enable us to simplify the given expression, -1 - √(-12), by breaking down the square root of the negative number and expressing it in terms of i. This initial step is crucial for transforming the expression into a standard complex number form, making further calculations and interpretations easier. The concept of imaginary numbers not only enriches our understanding of mathematics but also provides practical tools for solving real-world problems in diverse scientific and engineering domains.

Simplifying the Radical √(-12)

Now, let's focus on simplifying the radical term √(-12). To do this, we can break down -12 into its factors, specifically looking for perfect squares. We can rewrite -12 as -1 * 4 * 3. Taking the square root, we get √(-12) = √(-1 * 4 * 3) = √(-1) * √(4) * √(3). We know that √(-1) = i and √(4) = 2. Therefore, √(-12) simplifies to 2i√(3).

The process of simplifying radicals, especially those involving negative numbers, is a key skill in complex number arithmetic. When faced with a radical like √(-12), the initial step is to recognize the presence of the negative sign, which indicates an imaginary component. We can rewrite √(-12) as √(-1 * 12), which then allows us to separate the square root of -1, denoted as i. Next, we focus on simplifying the remaining radical, √(12). To simplify √(12), we look for perfect square factors within 12. The perfect square factor here is 4, as 12 can be expressed as 4 * 3. Therefore, √(12) can be rewritten as √(4 * 3). Using the property of radicals that states √(a * b) = √(a) * √(b), we can further simplify this to √(4) * √(3). Since √(4) equals 2, we now have 2√(3). Combining the imaginary component, i, with the simplified radical, we get √(-12) = √(-1) * √(4) * √(3) = i * 2 * √(3) = 2i√(3). This simplified form is much easier to work with in further calculations. Understanding how to break down radicals and identify perfect square factors is essential for simplifying expressions involving square roots of negative numbers. This skill not only helps in complex number manipulations but also strengthens the foundation for more advanced mathematical concepts. The ability to simplify radicals efficiently allows for more accurate and streamlined problem-solving in various mathematical and scientific contexts. This step-by-step approach ensures that we handle the imaginary component and the real radical part separately, leading to a clear and concise simplification.

Substituting Back into the Original Expression

With √(-12) simplified to 2i√(3), we can substitute this back into the original expression: -1 - √(-12) becomes -1 - 2i√(3). This is now in the standard form of a complex number, which is a + bi, where a = -1 and b = -2√(3).

The substitution process is a fundamental technique in mathematics that allows us to replace a complex or simplified expression back into its original context. In this case, we simplified √(-12) to 2i√(3), and now we substitute this simplified form back into the original expression, -1 - √(-12). This substitution transforms the expression into -1 - 2i√(3). This step is crucial because it brings us closer to expressing the answer in the standard form of a complex number, which is a + bi. The standard form is essential for further operations and comparisons with other complex numbers. Here, a represents the real part of the complex number, and b represents the imaginary part. By substituting the simplified radical back into the original expression, we can clearly identify the real and imaginary components. In our case, the real part (a) is -1, and the imaginary part (b) is -2√(3). The imaginary part is the coefficient of the imaginary unit i. The ability to identify these components is vital for performing operations such as addition, subtraction, multiplication, and division with complex numbers. Furthermore, expressing complex numbers in the standard form allows for graphical representation on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. This substitution step not only simplifies the expression but also lays the groundwork for further analysis and manipulation of the complex number. The clarity and structure provided by this process are invaluable for both understanding and communicating mathematical solutions effectively.

The Final Simplified Form

Therefore, the simplified form of -1 - √(-12) is -1 - 2i√(3). This expression represents a complex number with a real part of -1 and an imaginary part of -2√(3).

Reaching the final simplified form is the culmination of the step-by-step process of simplifying a mathematical expression. In our case, we started with the expression -1 - √(-12) and, through a series of simplifications, arrived at the complex number -1 - 2i√(3). This final form is crucial because it presents the answer in the most concise and easily understandable manner. The simplified form highlights the two key components of a complex number: the real part and the imaginary part. In our result, the real part is -1, and the imaginary part is -2√(3). These parts are clearly separated, making it easy to visualize and work with the number. Understanding the structure of complex numbers is essential in various fields, including mathematics, physics, and engineering. The ability to identify the real and imaginary parts allows for further operations, such as addition, subtraction, multiplication, and division of complex numbers. Furthermore, this form is ideal for representing complex numbers graphically on the complex plane, where the real part corresponds to the x-axis and the imaginary part corresponds to the y-axis. This graphical representation provides a visual interpretation of complex numbers, aiding in problem-solving and conceptual understanding. The journey to the final simplified form demonstrates the importance of breaking down complex problems into manageable steps. Each step, from understanding imaginary numbers to simplifying radicals and substituting back into the original expression, contributes to the final result. This systematic approach is not only effective for simplifying complex numbers but also for tackling a wide range of mathematical challenges. The clear, concise form of the final answer underscores the elegance and power of mathematical simplification.

Conclusion

In this guide, we have shown how to simplify the complex number expression -1 - √(-12). By understanding imaginary numbers, simplifying radicals, and applying basic arithmetic, we arrived at the simplified form -1 - 2i√(3). This process demonstrates the fundamental principles of complex number manipulation and provides a solid foundation for further exploration in mathematics.