Solving (13 + 4i) + N = 0: A Step-by-Step Guide To Finding N

This article delves into the realm of complex numbers and provides a comprehensive guide on how to solve a specific equation. We will focus on the equation (13 + 4i) + n = 0, where our goal is to determine the value of n. This seemingly simple equation introduces us to the fundamental concepts of complex number arithmetic and the process of isolating variables within such expressions. Whether you're a student tackling complex numbers for the first time or simply seeking a refresher, this guide will break down the steps involved, ensuring a clear understanding of the solution. By the end of this article, you will confidently solve similar problems and grasp the underlying principles of complex number manipulation.

H2: Introduction to Complex Numbers

Before diving into the solution, let's establish a foundational understanding of complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i = √-1). The real part of the complex number is a, and the imaginary part is b. Complex numbers extend the real number system by incorporating this imaginary unit, allowing us to represent solutions to equations that have no real roots. For instance, the equation x² + 1 = 0 has no real solutions, but it does have complex solutions, namely i and -i. Understanding this basic structure is crucial for manipulating and solving equations involving complex numbers, including the one we are addressing in this article. The ability to identify the real and imaginary components will be key to isolating the variable n and finding its correct value.

H2: The Problem: (13 + 4i) + n = 0

Now, let's revisit the specific problem we aim to solve: (13 + 4i) + n = 0. This equation presents a complex number, 13 + 4i, added to an unknown variable, n, with the result equaling zero. Our task is to isolate n and determine its value. The core principle we'll employ is the concept of additive inverses. In simpler terms, we need to find a complex number that, when added to 13 + 4i, results in zero. This involves understanding how addition and subtraction operate within the complex number system. We'll see that finding n is a matter of identifying the additive inverse of the given complex number. This equation serves as a practical example of how complex numbers are used in mathematical expressions and highlights the importance of understanding their properties for solving problems in algebra and beyond. The systematic approach we'll use here can be applied to a wide range of equations involving complex numbers.

H2: Solving for n: Step-by-Step

To solve for n in the equation (13 + 4i) + n = 0, we need to isolate n on one side of the equation. This can be achieved by using the concept of additive inverses, as mentioned earlier.

H3: Step 1: Isolate n

To isolate n, we subtract the complex number (13 + 4i) from both sides of the equation. This is a fundamental algebraic principle: performing the same operation on both sides maintains the equality. Therefore, we have:

(13 + 4i) + n - (13 + 4i) = 0 - (13 + 4i)

This step is crucial as it sets the stage for directly determining the value of n. By subtracting the complex number from both sides, we effectively move it to the other side, leaving n alone on the left side. This manipulation is based on the additive inverse property, which states that every complex number has an additive inverse that, when added to the original number, results in zero. This principle is the cornerstone of solving equations involving complex numbers, and it's essential to grasp this concept for further work in complex analysis and related fields.

H3: Step 2: Simplify the Equation

Now we simplify the equation. On the left side, (13 + 4i) - (13 + 4i) cancels out, leaving us with just n. On the right side, we have 0 - (13 + 4i), which simplifies to -(13 + 4i). Therefore, the equation becomes:

n = -(13 + 4i)

This simplification step is vital because it directly reveals the value of n. By performing the subtraction, we've effectively found the additive inverse of 13 + 4i. The negative sign in front of the parentheses indicates that we need to distribute the negative sign to both the real and imaginary parts of the complex number. This step showcases the direct application of the additive inverse property, where the sum of a number and its additive inverse equals zero. This principle is not only applicable to complex numbers but also to real numbers and other mathematical entities. The ability to simplify expressions like this is a fundamental skill in algebra and is essential for solving more complex equations and mathematical problems.

H3: Step 3: Distribute the Negative Sign

To further simplify the expression, we distribute the negative sign to both the real and imaginary parts within the parentheses:

n = -13 - 4i

Distributing the negative sign is a key step in fully determining the value of n. It ensures that both the real and imaginary components of the complex number are correctly negated. This process involves applying the distributive property of multiplication over addition (or subtraction), which is a fundamental concept in algebra. The negative sign can be thought of as multiplying the complex number inside the parentheses by -1. This means that each term inside the parentheses, including both the real part (13) and the imaginary part (4i), gets multiplied by -1. This results in the real part becoming -13 and the imaginary part becoming -4i. This step highlights the importance of attention to detail in mathematical operations, as an incorrect sign can lead to a wrong answer. Understanding how to distribute negative signs correctly is crucial for simplifying expressions and solving equations accurately.

H2: The Solution

Therefore, the value of n that satisfies the equation (13 + 4i) + n = 0 is:

n = -13 - 4i

This is the final answer to our problem. We have successfully isolated n and determined its value by applying the principles of complex number arithmetic. The solution n = -13 - 4i is the additive inverse of the complex number 13 + 4i. When these two complex numbers are added together, the real and imaginary parts cancel each other out, resulting in zero. This demonstrates the fundamental property of additive inverses, which is a cornerstone of algebraic manipulation. The process we've followed – isolating the variable, simplifying the equation, and distributing the negative sign – can be applied to solve a variety of equations involving complex numbers. This solution not only answers the specific question but also reinforces the core concepts of complex number arithmetic and provides a framework for tackling similar problems in the future.

H2: Choosing the Correct Option

Now, let's relate our solution to the given options. The options were:

A. -13 - 4i B. 1 C. -13 + 4i D. 0

Comparing our solution, n = -13 - 4i, with the options, we can clearly see that option A is the correct answer. The other options do not represent the additive inverse of 13 + 4i and would not satisfy the original equation. This step is crucial for ensuring that the mathematical solution is correctly translated into the context of the given choices. It reinforces the importance of double-checking the answer against the available options to avoid careless errors. Selecting the correct option also demonstrates a comprehensive understanding of the problem and the solution process, as it requires the ability to recognize the correct form of the answer among distractors. This final step solidifies the learning process and promotes confidence in the problem-solving ability.

H2: Conclusion

In conclusion, we have successfully solved the equation (13 + 4i) + n = 0 and found the value of n to be -13 - 4i. This process involved understanding the basics of complex numbers, applying the concept of additive inverses, and performing algebraic manipulations to isolate the variable. The step-by-step solution provided a clear and concise method for tackling similar problems involving complex numbers. By understanding these fundamental principles, you can confidently approach a wide range of mathematical problems involving complex numbers and algebraic equations. This exploration not only provides the solution to this specific problem but also equips you with the tools and knowledge necessary for further exploration in the realm of complex numbers and their applications in mathematics, physics, and engineering. The ability to manipulate complex numbers is a valuable asset in various fields, and this article serves as a solid foundation for building that skill set.