Dividing Complex Numbers A Step-by-Step Guide To (1+6i) / (3+5i)
Complex numbers, an extension of the real number system, play a pivotal role in various fields, including mathematics, physics, and engineering. Understanding complex number operations, such as division, is essential for anyone delving into these disciplines. In this comprehensive guide, we will explore the process of dividing complex numbers, focusing on the specific example of $\frac{1+6i}{3+5i}\$ while providing a deeper understanding of the underlying principles and practical applications.
Understanding Complex Numbers
Before diving into the division process, it is crucial to grasp the fundamental concepts of complex numbers. A complex number is expressed in the form a + bi, where a represents the real part, b represents the imaginary part, and i is the imaginary unit, defined as the square root of -1 (i.e., $i = \sqrt{-1}\$). Complex numbers can be visualized on a complex plane, with the real part plotted on the horizontal axis and the imaginary part on the vertical axis. This geometric representation allows for a more intuitive understanding of complex number operations.
Complex numbers, while seemingly abstract, have profound real-world applications. They are indispensable in electrical engineering for analyzing alternating current circuits, where voltage and current can be represented as complex quantities. In quantum mechanics, complex numbers are the foundation for describing wave functions, which govern the behavior of subatomic particles. Signal processing relies heavily on complex numbers for analyzing and manipulating signals in various domains. Even in computer graphics, complex numbers are employed to perform transformations such as rotations and scaling. This ubiquitous presence underscores the importance of mastering complex number arithmetic.
Consider the complex numbers 3 + 4i and -2 + i. To add these numbers, we simply add the real parts and the imaginary parts separately: (3 + 4i) + (-2 + i) = (3 - 2) + (4 + 1)i = 1 + 5i. Subtraction follows the same principle: (3 + 4i) - (-2 + i) = (3 - (-2)) + (4 - 1)i = 5 + 3i. Multiplication involves the distributive property and the fact that $i^2 = -1$: (3 + 4i)(-2 + i) = 3(-2) + 3(i) + 4i(-2) + 4i(i) = -6 + 3i - 8i + 4i^2 = -6 - 5i - 4 = -10 - 5i. These basic operations lay the groundwork for understanding the more complex operation of division.
The Division Challenge
Dividing complex numbers is not as straightforward as dividing real numbers. The challenge arises from the presence of the imaginary unit i in the denominator. To perform division, we need to eliminate the imaginary part from the denominator, a process known as rationalizing the denominator. This is achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi, which essentially means changing the sign of the imaginary part. The crucial property of complex conjugates is that their product is always a real number, effectively eliminating i from the denominator.
The need to rationalize the denominator stems from the conventional representation of complex numbers. Just as we prefer to express fractions with rational denominators, complex numbers are typically written with real denominators. This simplifies further calculations and makes it easier to compare and interpret complex numbers. The process of multiplying by the conjugate might seem like an algebraic trick, but it's a fundamental technique for maintaining this standard form and ensuring clarity in complex number manipulations.
For instance, consider the simple division $\frac1}{i}$. Directly dividing by i doesn't lead to a simplified form. However, if we multiply both the numerator and denominator by the conjugate of i, which is -i, we get{i} \times \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{-(-1)} = -i$. This demonstrates how multiplying by the conjugate transforms the expression into a standard complex number form. This same principle applies to more complex divisions, ensuring that we always end up with a complex number expressed as a + bi.
Step-by-Step Solution for $ rac{1+6i}{3+5i}$
Now, let's apply this principle to the specific problem: dividing (1 + 6i) by (3 + 5i). We will follow a step-by-step approach to ensure clarity and understanding.
Step 1: Identify the complex conjugate of the denominator. The denominator is 3 + 5i, so its complex conjugate is 3 - 5i. Remember, we simply change the sign of the imaginary part.
Step 2: Multiply both the numerator and the denominator by the complex conjugate. This is the crucial step in rationalizing the denominator. We multiply both the top and bottom of the fraction by (3 - 5i):
This step is algebraically sound because multiplying a fraction by a form of 1 (in this case, $\frac{3-5i}{3-5i}\$ ) does not change its value. We are essentially rewriting the expression in an equivalent form that allows us to perform the division.
Step 3: Expand the numerator and the denominator. We use the distributive property (also known as FOIL - First, Outer, Inner, Last) to multiply the complex numbers in both the numerator and the denominator.
- Numerator: (1 + 6i )(3 - 5i) = 1(3) + 1(-5i) + 6i(3) + 6i(-5i) = 3 - 5i + 18i - 30i^2
- Denominator: (3 + 5i)(3 - 5i) = 3(3) + 3(-5i) + 5i(3) + 5i(-5i) = 9 - 15i + 15i - 25i^2
The expansion process highlights the importance of careful algebraic manipulation. Each term must be multiplied correctly, and the signs must be tracked accurately. This meticulous approach ensures that we arrive at the correct simplified expression.
Step 4: Simplify using i^2 = -1. Recall that i is the imaginary unit, and by definition, i^2 = -1. Substitute -1 for i^2 in both the numerator and the denominator.
- Numerator: 3 - 5i + 18i - 30(-1) = 3 - 5i + 18i + 30
- Denominator: 9 - 15i + 15i - 25(-1) = 9 - 15i + 15i + 25
This substitution is the key to eliminating the imaginary unit from the denominator. By replacing i^2 with -1, we transform the expression into one involving only real numbers and the imaginary unit i.
Step 5: Combine like terms. Combine the real terms and the imaginary terms separately in both the numerator and the denominator.
- Numerator: (3 + 30) + (-5i + 18i) = 33 + 13i
- Denominator: (9 + 25) + (-15i + 15i) = 34 + 0i = 34
Notice that the imaginary terms in the denominator cancel out, leaving us with a real number, which is exactly what we aimed for when rationalizing the denominator.
Step 6: Write the result in the standard form a + bi. Divide both the real and imaginary parts of the numerator by the real denominator.
Therefore, the result of dividing (1 + 6i) by (3 + 5i) is $\frac{33}{34} + \frac{13}{34}i$. This final step expresses the result in the standard complex number form, a + bi, where a = 33/34 and b = 13/34.
Practice Problems
To solidify your understanding, try these practice problems:
Working through these problems will reinforce the steps involved in dividing complex numbers and help you develop confidence in your abilities. Remember to follow the same methodical approach: identify the conjugate, multiply, expand, simplify, and express the result in standard form.
Conclusion
Dividing complex numbers is a fundamental operation in mathematics with wide-ranging applications. By mastering the process of rationalizing the denominator, you can confidently perform complex number division and tackle more advanced problems in various fields. The key takeaway is the importance of multiplying by the conjugate, which elegantly eliminates the imaginary part from the denominator and allows us to express the result in the standard complex number form. Understanding complex number operations is not just an academic exercise; it's a crucial skill for anyone working in science, engineering, or technology. This guide has provided a comprehensive explanation and step-by-step solution, empowering you to confidently divide complex numbers and delve deeper into the fascinating world of complex analysis.
