Simplifying Complex Numbers Divide 4+i By 2-5i

Complex numbers, a fascinating extension of the real number system, play a crucial role in various fields, including mathematics, physics, and engineering. Operations with complex numbers, such as division, often require careful manipulation to express the result in the standard form of a + bi, where a and b are real numbers and i is the imaginary unit (√-1). In this comprehensive guide, we will delve into the process of simplifying the complex number expression (4 + i) / (2 - 5i). We will explore the fundamental concepts, step-by-step calculations, and the underlying rationale behind each step, providing a clear and concise understanding of the process.

Understanding Complex Numbers and Division

Before we dive into the simplification process, let's briefly revisit the concept of complex numbers and their division. A complex number is generally represented as a + bi, where a is the real part and b is the imaginary part. The imaginary unit, i, is defined as the square root of -1, i.e., i² = -1. This seemingly simple definition opens up a whole new dimension in mathematics, allowing us to solve equations and explore concepts that are impossible within the realm of real numbers alone.

Dividing complex numbers requires a slightly different approach compared to dividing real numbers. The key is to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, effectively removing the imaginary component from the denominator.

The Significance of Complex Conjugates

The concept of complex conjugates is central to the division of complex numbers. A complex conjugate is formed by simply changing the sign of the imaginary part of a complex number. For instance, the complex conjugate of 2 - 5i is 2 + 5i. The magic happens when we multiply a complex number by its conjugate. Let's consider a general complex number a + bi. Its conjugate is a - bi. When we multiply these two, we get:

(a + bi)(a - bi) = a² - abi + abi - b²i²

Since i² = -1, the expression simplifies to:

a² + b²

Notice that the result is a real number, devoid of any imaginary components. This is the core principle we utilize to eliminate the imaginary part from the denominator when dividing complex numbers. The product of a complex number and its conjugate always results in a real number, which is the sum of the squares of the real and imaginary parts.

This property is not just a mathematical curiosity; it has profound implications in various fields. In electrical engineering, complex numbers are used to represent alternating currents and voltages. The impedance of a circuit, which is the opposition to the flow of current, is also a complex quantity. Using complex conjugates simplifies calculations involving impedance and power in AC circuits. In quantum mechanics, complex numbers are fundamental to the description of wave functions, which represent the probability amplitude of a particle's state. The use of complex conjugates ensures that probabilities are real numbers, a necessary condition for physical interpretation.

Step-by-Step Simplification of (4 + i) / (2 - 5i)

Now that we have a firm grasp of the underlying principles, let's proceed with the step-by-step simplification of the complex number expression (4 + i) / (2 - 5i).

Step 1: Identify the Conjugate of the Denominator

The denominator of our expression is 2 - 5i. To find its conjugate, we simply change the sign of the imaginary part. Therefore, the conjugate of 2 - 5i is 2 + 5i. This is a crucial step, as multiplying the denominator by its conjugate will eliminate the imaginary term and allow us to express the result in the standard a + bi form.

Step 2: Multiply Numerator and Denominator by the Conjugate

To eliminate the imaginary part from the denominator, we multiply both the numerator and the denominator of the expression by the conjugate of the denominator, which is 2 + 5i. This is mathematically valid because multiplying both the numerator and denominator by the same value is equivalent to multiplying the entire expression by 1, which does not change its value. This step is the cornerstone of complex number division, allowing us to transform the expression into a form where the denominator is a real number.

(4 + i) / (2 - 5i) * (2 + 5i) / (2 + 5i)

Step 3: Expand the Numerator and Denominator

Next, we expand both the numerator and the denominator by applying the distributive property (also known as the FOIL method). This involves multiplying each term in the first complex number by each term in the second complex number. Careful attention to detail is essential in this step to ensure accurate calculations.

Numerator: (4 + i)(2 + 5i) = 4 * 2 + 4 * 5i + i * 2 + i * 5i = 8 + 20i + 2i + 5i²

Denominator: (2 - 5i)(2 + 5i) = 2 * 2 + 2 * 5i - 5i * 2 - 5i * 5i = 4 + 10i - 10i - 25i²

Step 4: Simplify using i² = -1

Now, we simplify the expressions by substituting i² with -1. This is the key step where the imaginary unit's fundamental property is utilized. Remember that i² = -1 is the cornerstone of complex number arithmetic and allows us to transform imaginary terms into real numbers.

Numerator: 8 + 20i + 2i + 5i² = 8 + 20i + 2i + 5(-1) = 8 + 20i + 2i - 5

Denominator: 4 + 10i - 10i - 25i² = 4 + 10i - 10i - 25(-1) = 4 + 10i - 10i + 25

Step 5: Combine Real and Imaginary Terms

Combine the real and imaginary terms separately in both the numerator and the denominator. This step involves grouping like terms together, making the expression more organized and easier to simplify further.

Numerator: 8 + 20i + 2i - 5 = (8 - 5) + (20i + 2i) = 3 + 22i

Denominator: 4 + 10i - 10i + 25 = (4 + 25) + (10i - 10i) = 29 + 0i = 29

Step 6: Express the Result in Standard Form (a + bi)

Finally, we express the simplified complex number in the standard form a + bi by dividing both the real and imaginary parts of the numerator by the real denominator. This gives us the final answer in the desired format, making it easy to identify the real and imaginary components of the complex number.

(3 + 22i) / 29 = 3/29 + (22/29)i

Therefore, the simplified form of the complex number expression (4 + i) / (2 - 5i) is 3/29 + (22/29)i. This is the standard form representation of the complex number, where 3/29 is the real part and 22/29 is the imaginary part.

Conclusion

Simplifying complex numbers, especially division, might seem daunting at first, but by understanding the underlying principles and following a systematic approach, it becomes a manageable task. The key takeaway is the use of the complex conjugate to eliminate the imaginary part from the denominator. This technique allows us to express the result in the standard form a + bi, which is essential for further calculations and applications. The step-by-step approach outlined in this guide provides a clear and concise method for simplifying complex number expressions, empowering you to confidently tackle similar problems.

Complex numbers are not just abstract mathematical entities; they are powerful tools with applications in various fields. From electrical engineering to quantum mechanics, complex numbers provide a framework for modeling and solving problems that are beyond the scope of real numbers alone. Mastering the manipulation of complex numbers, including division, is therefore a valuable skill for anyone pursuing studies or careers in these fields. By understanding the concepts and practicing the techniques, you can unlock the full potential of complex numbers and their applications in the world around us.