Complex Number Solution Finding Values For Real Results

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Introduction: Delving into the Realm of Complex Numbers

At the heart of mathematics lies the fascinating world of complex numbers, numbers that extend the familiar realm of real numbers by incorporating an imaginary unit, denoted by i, which is defined as the square root of -1. Complex numbers, expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, open up a whole new dimension of mathematical possibilities. Understanding how complex numbers interact and the conditions under which they yield real results is crucial in various fields, including physics, engineering, and computer science.

In this article, we embark on a journey to explore the intricacies of complex numbers, focusing on a specific problem that requires us to determine the values of two unknowns, c and d, such that a given expression involving complex numbers results in a real number. This problem serves as an excellent illustration of the properties of complex numbers and the techniques used to manipulate them. Our investigation will involve delving into the arithmetic of complex numbers, understanding the concept of conjugate pairs, and strategically applying these concepts to isolate the real part of the expression.

Let's dive into the heart of the problem and begin our quest to unravel the values of c and d that will unlock the real nature of the expression.

The Problem at Hand: Deciphering the Expression

Our mission is to determine the specific values of c and d that will transform the complex expression i(2 + 3i) (c + di) into a real number. To accomplish this, we must first unravel the expression by performing the necessary operations and then carefully analyze the resulting terms to identify the conditions that eliminate the imaginary component. This requires a solid understanding of how complex numbers interact under multiplication and the role of the imaginary unit i in shaping the final result.

The expression i(2 + 3i) (c + di) is a product of three complex terms: the imaginary unit i, the complex number (2 + 3i), and the complex number (c + di). To simplify this expression, we will perform the multiplications step by step, keeping in mind the fundamental property of the imaginary unit: i² = -1. This property is the cornerstone of complex number arithmetic, and its application is essential in simplifying expressions and isolating real and imaginary components.

As we proceed with the multiplication, we will pay close attention to how the terms involving i interact and how they contribute to the overall imaginary component of the expression. Our ultimate goal is to identify the values of c and d that will nullify this imaginary component, leaving us with a purely real number. This strategic approach will guide our calculations and ensure that we arrive at the correct solution.

The Solution: A Step-by-Step Unraveling

To solve this problem, we will embark on a step-by-step approach, carefully applying the rules of complex number arithmetic. First, we multiply the first two terms: i(2 + 3i). This gives us:

  • i(2 + 3i) = 2i + 3i²

Now, we invoke the fundamental property of the imaginary unit, i² = -1, to simplify further:

  • 2i + 3i² = 2i + 3(-1) = -3 + 2i

Next, we multiply the result (-3 + 2i) by the complex number (c + di):

  • (-3 + 2i) (c + di) = -3c - 3di + 2ci + 2di²

Again, we apply the property i² = -1:

  • -3c - 3di + 2ci + 2di² = -3c - 3di + 2ci + 2d(-1) = -3c - 2d + (-3d + 2c)i

Now, we have the expression in the form a + bi, where a = -3c - 2d and b = -3d + 2c. For this expression to represent a real number, the imaginary part, b, must be equal to zero. Therefore, we set -3d + 2c = 0.

This equation gives us a relationship between c and d: 2c = 3d. We can rewrite this as c = (3/2)d. Now, we examine the given options to see which pair of values satisfies this relationship.

  • A. c = -2, d = 3: Substituting these values, we get -2 = (3/2)(3), which simplifies to -2 = 9/2. This is not true, so option A is incorrect.
  • B. c = 2, d = 3: Substituting these values, we get 2 = (3/2)(3), which simplifies to 2 = 9/2. This is also not true, so option B is incorrect.
  • C. c = -3, d = -2: Substituting these values, we get -3 = (3/2)(-2), which simplifies to -3 = -3. This is true, so option C is a potential solution.
  • D. c = 3, d = -2: Substituting these values, we get 3 = (3/2)(-2), which simplifies to 3 = -3. This is not true, so option D is incorrect.

Therefore, the only option that satisfies the condition for the expression to be a real number is option C, where c = -3 and d = -2.

Verification: Confirming the Solution

To ensure the correctness of our solution, we will substitute the values c = -3 and d = -2 back into the original expression and verify that the result is indeed a real number. This step provides a crucial confirmation of our calculations and reasoning.

Substituting c = -3 and d = -2 into the expression i(2 + 3i) (c + di), we get:

  • i(2 + 3i) (-3 - 2i)

We already know that i(2 + 3i) = -3 + 2i, so we have:

  • (-3 + 2i) (-3 - 2i)

Multiplying these two complex numbers, we get:

  • (-3 + 2i) (-3 - 2i) = (-3)(-3) + (-3)(-2i) + (2i)(-3) + (2i)(-2i)

Simplifying, we have:

  • 9 + 6i - 6i - 4i²

Applying the property i² = -1, we get:

  • 9 + 6i - 6i - 4(-1) = 9 + 4 = 13

The result is 13, which is a real number. This confirms that our solution, c = -3 and d = -2, is correct.

Conclusion: The Answer Revealed

Through a meticulous step-by-step approach, we have successfully determined the values of c and d that transform the complex expression i(2 + 3i) (c + di) into a real number. Our journey involved unraveling the expression using the rules of complex number arithmetic, setting the imaginary part to zero, and solving the resulting equation. We then verified our solution by substituting the values back into the original expression and confirming that the result was indeed a real number.

The values that satisfy the condition are c = -3 and d = -2, corresponding to option C. This problem serves as a valuable exercise in understanding the properties of complex numbers and the techniques used to manipulate them. By mastering these concepts, we can unlock a deeper understanding of mathematics and its applications in various fields.

Understanding the behavior of complex numbers, like in this problem, is not just an academic exercise. Complex numbers are the bedrock of many technologies we use daily, from medical imaging to digital signal processing. By solving this problem, we've not only found a mathematical solution but also glimpsed the profound ways complex numbers shape our world.

In conclusion, the problem provides a comprehensive understanding of the manipulation and simplification of complex numbers, which is a fundamental concept in mathematics. The correct answer is C. c = -3, d = -2. This exercise emphasizes the importance of careful calculation and verification in mathematical problem-solving.