Simplifying Complex Numbers Performing Operations On (-12 + √-45) / 24

Understanding Complex Numbers and Standard Form

When faced with mathematical expressions involving square roots of negative numbers, we delve into the realm of complex numbers. These numbers extend the real number system by incorporating the imaginary unit, denoted as i, where i is defined as the square root of -1 (i = √-1). This allows us to express solutions to equations that were previously deemed unsolvable within the real number system.

Complex numbers are typically written in the standard form of a + bi, where a represents the real part and b represents the imaginary part. This standard form provides a clear and concise way to represent and manipulate complex numbers, making it easier to perform arithmetic operations and analyze their properties. The real part, a, is a real number, and the imaginary part, b, is also a real number that is multiplied by the imaginary unit i. Understanding this standard form is crucial for expressing the result of the given operation in a manner that is both mathematically accurate and easily interpretable.

Before we tackle the specific problem, let’s reiterate the importance of simplifying expressions involving square roots of negative numbers. The key lies in recognizing that the square root of a negative number can be rewritten using the imaginary unit. For instance, √-4 can be expressed as √(4 * -1) = √4 * √-1 = 2i. This transformation is fundamental to working with complex numbers and is the first step in converting expressions into standard form. When simplifying, always look for perfect square factors within the radicand (the number under the square root symbol) to further reduce the expression. This process ensures that the complex number is represented in its most simplified and standard form, a + bi.

This conversion to standard form is not merely an aesthetic preference; it serves a crucial purpose in facilitating further mathematical operations. Once a complex number is in standard form, arithmetic operations such as addition, subtraction, multiplication, and division can be performed using well-defined rules. For example, adding two complex numbers involves adding their real parts and their imaginary parts separately. Similarly, multiplication involves using the distributive property and remembering that i² = -1. Thus, standard form provides a consistent and structured framework for manipulating complex numbers, enabling us to solve complex equations and explore advanced mathematical concepts.

Step-by-Step Solution for $\frac{-12+\sqrt{-45}}{24}$

Now, let's apply this understanding to the problem at hand: $\frac{-12+\sqrt{-45}}{24}$. This expression involves a complex number in the numerator, and our goal is to simplify it and write the result in the standard form a + bi. We'll proceed step-by-step to ensure clarity and accuracy.

Step 1: Simplify the Square Root of a Negative Number The first step is to address the square root of the negative number, √-45. As discussed earlier, we can rewrite this using the imaginary unit i. We have √-45 = √(45 * -1) = √45 * √-1. Now, we need to simplify √45. We look for perfect square factors of 45. Since 45 = 9 * 5, we can write √45 = √(9 * 5) = √9 * √5 = 3√5. Therefore, √-45 = 3√5 * i = 3i√5. This step is crucial because it transforms the expression from involving the square root of a negative number to a form where we can clearly see the imaginary component.

Step 2: Substitute the Simplified Square Root Back into the Expression Substitute the simplified square root back into the original expression: $\frac{-12 + 3i\sqrt{5}}{24}$. This substitution replaces √-45 with its simplified equivalent, 3i√5, making the expression easier to manage and putting us closer to the standard form.

Step 3: Separate the Real and Imaginary Parts To write the complex number in standard form (a + bi), we need to separate the real and imaginary parts. We can do this by dividing both terms in the numerator by the denominator: $\frac{-12 + 3i\sqrt{5}}{24}$ = $\frac{-12}{24}$ + $\frac{3i\sqrt{5}}{24}$. This separation is a key step in achieving the a + bi format, as it isolates the real component and the imaginary component.

Step 4: Simplify Each Fraction Now, simplify each fraction individually. The real part is $\frac{-12}{24}$, which simplifies to -\frac{1}{2}. The imaginary part is $\frac{3i\sqrt{5}}{24}$, which simplifies to $\frac{i\sqrt{5}}{8}$. This simplification involves reducing each fraction to its lowest terms, making the expression more concise and easier to interpret. The imaginary part is simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3 in this case.

Step 5: Write the Result in Standard Form Finally, write the simplified expression in the standard form a + bi. Combining the simplified real and imaginary parts, we get -\frac{1}{2} + \frac{\sqrt{5}}{8}i. This is the final answer, expressed in the standard form, where a = -\frac{1}{2} and b = \frac{\sqrt{5}}{8}. This step completes the process, presenting the solution in a clear and easily recognizable format that adheres to the conventions of complex number representation.

Common Mistakes and How to Avoid Them

When working with complex numbers, several common mistakes can occur. Being aware of these pitfalls and understanding how to avoid them is crucial for accurate problem-solving. Let's discuss some frequent errors and the strategies to prevent them.

Mistake 1: Incorrectly Simplifying Square Roots of Negative Numbers A common error is mishandling the square root of a negative number. For instance, incorrectly stating that √-4 = -2 instead of 2i. The key is to remember that √-1 = i, and this must be correctly applied when simplifying. Always factor out -1 from the radicand and replace √-1 with i. For example, √-16 = √(16 * -1) = √16 * √-1 = 4i. This careful application of the definition of i will prevent errors in simplification.

Mistake 2: Forgetting to Simplify Radicals Another frequent mistake is failing to simplify the radical completely. For example, if you have √45, you should simplify it to 3√5. Leaving the radical in a non-simplified form can lead to incorrect final answers and a loss of precision. Always look for perfect square factors within the radicand and simplify accordingly. This ensures that the complex number is in its most reduced and accurate form.

Mistake 3: Errors in Arithmetic Operations Arithmetic errors, such as mistakes in adding, subtracting, multiplying, or dividing, can easily occur, especially when dealing with fractions and radicals. Double-checking each step of the calculation is essential. When adding or subtracting complex numbers, ensure that you combine the real parts and the imaginary parts separately. In multiplication, remember the distributive property and the fact that i² = -1. Careful attention to these details will minimize the chances of arithmetic errors.

Mistake 4: Not Writing the Final Answer in Standard Form The final common mistake is not expressing the answer in the standard form a + bi. While you may have correctly performed all the operations, failing to present the result in standard form can be considered incomplete. Always separate the real and imaginary parts and write the answer in the a + bi format. This ensures clarity and adherence to mathematical conventions.

How to Avoid These Mistakes To avoid these mistakes, follow these best practices:

• Practice regularly: Consistent practice is the best way to solidify your understanding and improve accuracy.
• Show your work: Writing down each step helps you track your progress and identify errors more easily.
• Double-check each step: Take the time to review your calculations and simplifications to catch any mistakes.
• Understand the definitions and rules: Make sure you have a solid understanding of complex number definitions and arithmetic rules.
• Use online calculators or tools: Tools like complex number calculators can help you verify your answers and identify errors.
• Seek help when needed: Don't hesitate to ask your teacher, classmates, or online forums for assistance if you're struggling with a concept or problem.

By being aware of these common mistakes and adopting effective problem-solving strategies, you can improve your accuracy and confidence in working with complex numbers.

Conclusion

In conclusion, simplifying expressions involving complex numbers requires a systematic approach and a clear understanding of the properties of i. By following the steps outlined above—simplifying radicals, separating real and imaginary parts, and writing the result in standard form—you can confidently solve these types of problems. Remember to be mindful of common mistakes, and always double-check your work to ensure accuracy. With practice, working with complex numbers will become second nature, opening the door to more advanced mathematical concepts and applications.