Solving Continued Fractions Find The Value Of X
In mathematics, continued fractions offer a fascinating way to represent numbers. They appear in various fields, from number theory to approximation theory. This article delves into solving an equation involving a continued fraction, providing a step-by-step approach to find the value of the unknown variable. Our goal is to provide a detailed and clear explanation, making it accessible to anyone interested in mathematical problem-solving.
Understanding Continued Fractions
Before diving into the problem, let's briefly explain what continued fractions are. A continued fraction is an expression of the form:
a + 1/(b + 1/(c + 1/d + ...))
where a, b, c, d, ... are integers. These fractions can be finite or infinite, and they provide a unique way to represent both rational and irrational numbers. In our specific problem, we encounter a finite continued fraction, which simplifies the process of finding a solution.
Problem Statement
We are given the equation:
x + 1/(2 + 1/(2 + 1/1)) = 2
Our task is to find the value of x that satisfies this equation. This problem combines basic algebraic principles with the concept of continued fractions. To solve it, we will first simplify the continued fraction and then isolate x.
Simplifying the Continued Fraction
The key to solving this equation lies in simplifying the continued fraction part. We will work from the innermost fraction outwards.
Step 1: Simplify the Innermost Fraction
The innermost fraction is 1/1, which is simply equal to 1. So, we can rewrite the equation as:
x + 1/(2 + 1/(2 + 1)) = 2
Step 2: Simplify the Next Layer
Now, we have 2 + 1 in the denominator, which equals 3. The equation becomes:
x + 1/(2 + 1/3) = 2
Step 3: Continue Simplifying
Next, we simplify the fraction 1/3 within the larger fraction. The expression 2 + 1/3 can be written as a single fraction. To do this, we find a common denominator, which is 3. So, 2 is equivalent to 6/3. Adding 1/3 to 6/3 gives us 7/3. The equation now looks like this:
x + 1/(7/3) = 2
Step 4: Simplify the Main Fraction
We now have a fraction in the denominator. To simplify 1/(7/3), we take the reciprocal of 7/3, which is 3/7. The equation simplifies to:
x + 3/7 = 2
Isolating x
Now that we have simplified the continued fraction, we can easily solve for x. To isolate x, we need to subtract 3/7 from both sides of the equation.
Step 5: Subtract 3/7 from Both Sides
Subtracting 3/7 from both sides gives us:
x = 2 - 3/7
Step 6: Find a Common Denominator
To subtract the fractions, we need a common denominator. We can rewrite 2 as a fraction with a denominator of 7. Since 2 is equivalent to 14/7, the equation becomes:
x = 14/7 - 3/7
Step 7: Perform the Subtraction
Now, we subtract the numerators:
x = (14 - 3)/7
This simplifies to:
x = 11/7
Final Answer
Therefore, the value of x that satisfies the given equation is 11/7. This result demonstrates how continued fractions can be simplified step-by-step to solve algebraic equations. The key is to work from the innermost fraction outwards, simplifying each layer until you arrive at a simple equation that can be solved using basic algebraic techniques.
In this article, we have explored how to solve an equation involving a continued fraction. By systematically simplifying the fraction and using basic algebraic principles, we found the value of x to be 11/7. This problem highlights the importance of understanding continued fractions and their properties. Continued fractions are not only a mathematical curiosity but also a powerful tool in various areas of mathematics and computer science. This step-by-step approach provides a clear methodology for solving similar problems, reinforcing the understanding of both continued fractions and algebraic manipulation.
Let's delve into solving the equation x + 1/(2 + 1/(2 + 1/1)) = 2 to find the value of x. This problem involves a continued fraction, which might seem daunting at first, but can be simplified step-by-step. Understanding continued fractions is crucial here, as they are a unique way to represent numbers and often appear in various mathematical contexts. This detailed explanation aims to provide a clear and understandable solution, making it accessible to anyone interested in mathematics, regardless of their background. Our focus will be on breaking down the problem into manageable steps, simplifying the continued fraction, and then isolating x to find its value.
Breaking Down the Continued Fraction
The given equation is x + 1/(2 + 1/(2 + 1/1)) = 2. The key to solving this is to simplify the continued fraction part first. We will start from the innermost fraction and work our way outwards. This approach ensures that we maintain accuracy and simplify the equation in a logical manner. Each step will be explained in detail, allowing you to follow along and understand the reasoning behind each simplification. The goal is to transform the complex fraction into a simpler form that allows us to easily solve for x.
Step 1: Simplifying the Innermost Fraction
The innermost fraction is 1/1, which is simply equal to 1. This is the first and simplest step in our simplification process. Replacing 1/1 with 1, the equation becomes:
x + 1/(2 + 1/(2 + 1)) = 2
This small change makes the equation slightly more manageable, and we are now ready to tackle the next layer of the continued fraction. Remember, the key to solving these types of problems is to take it one step at a time.
Step 2: Simplifying the Next Layer
Now we need to simplify the expression within the parentheses in the denominator: 2 + 1. This is a straightforward addition, and 2 + 1 = 3. Substituting this back into the equation, we get:
x + 1/(2 + 1/3) = 2
We are making progress! The continued fraction is becoming less complex with each step. Next, we will address the fraction within the larger fraction.
Step 3: Continuing the Simplification
Our next task is to simplify 2 + 1/3. To add these together, we need a common denominator. We can express 2 as a fraction with a denominator of 3, which is 6/3. So, 2 + 1/3 becomes 6/3 + 1/3. Adding these fractions gives us 7/3. The equation now looks like this:
x + 1/(7/3) = 2
The continued fraction is gradually transforming into a more manageable form. We are one step closer to isolating x and finding its value.
Step 4: Simplifying the Main Fraction
We now have a fraction in the denominator: 1/(7/3). To simplify this, we take the reciprocal of 7/3, which is 3/7. The equation now simplifies to:
x + 3/7 = 2
At this stage, the continued fraction is completely simplified, and we are left with a simple algebraic equation. We are now ready to isolate x and solve for its value.
Isolating and Solving for x
Now that we have simplified the continued fraction, we can easily solve for x. The equation is x + 3/7 = 2. To isolate x, we need to subtract 3/7 from both sides of the equation. This will leave x on one side and the numerical value on the other.
Step 5: Subtracting 3/7 from Both Sides
Subtracting 3/7 from both sides of the equation gives us:
x = 2 - 3/7
This is a simple subtraction problem involving fractions. To perform the subtraction, we need a common denominator.
Step 6: Finding a Common Denominator
To subtract the fractions, we need a common denominator. We can rewrite 2 as a fraction with a denominator of 7. Since 2 is equivalent to 14/7, the equation becomes:
x = 14/7 - 3/7
Now that we have a common denominator, we can proceed with the subtraction.
Step 7: Performing the Subtraction
Now, we subtract the numerators:
x = (14 - 3)/7
This simplifies to:
x = 11/7
The Final Solution
Therefore, the value of x that satisfies the given equation is 11/7. This demonstrates how continued fractions can be simplified step-by-step to solve algebraic equations. The key is to work from the innermost fraction outwards, simplifying each layer until you arrive at a simple equation that can be solved using basic algebraic techniques. This comprehensive approach makes the solution clear and easy to follow.
In this article, we have successfully solved for x in an equation involving a continued fraction. By systematically simplifying the fraction and using basic algebraic principles, we found the value of x to be 11/7. This problem highlights the importance of understanding continued fractions and their properties. Continued fractions are not only a fascinating mathematical concept but also a valuable tool in various areas of mathematics and computer science. This step-by-step approach provides a clear methodology for solving similar problems, reinforcing the understanding of both continued fractions and algebraic manipulation. The ability to simplify complex expressions and solve for unknowns is a fundamental skill in mathematics, and this article provides a clear example of how to apply these skills effectively.
In mathematics, solving equations involving continued fractions can often appear intricate, but breaking them down into manageable steps makes the process straightforward. This article provides a comprehensive guide on how to solve for x in such equations, focusing on the given problem: If x + 1/(2 + 1/(2 + 1/1)) = 2, then find the value of x. We will explore the step-by-step simplification of the continued fraction, the algebraic manipulation required to isolate x, and the final solution. Understanding continued fractions is crucial in various fields, including number theory and approximation theory, making this a valuable skill for anyone interested in mathematics. This article aims to provide a clear and accessible explanation, ensuring that readers can confidently approach similar problems in the future. The key to success lies in systematically simplifying the expression and applying basic algebraic principles.
Understanding Continued Fractions and Their Simplification
Before diving into the solution, it's essential to grasp the concept of continued fractions. A continued fraction is an expression of the form a + 1/(b + 1/(c + 1/d + ...)), where a, b, c, d, etc., are integers. These fractions can be finite or infinite and provide a unique way to represent both rational and irrational numbers. In our problem, we have a finite continued fraction, which simplifies the solving process. The simplification strategy involves working from the innermost fraction outwards, progressively reducing the complexity of the expression. This methodical approach ensures accuracy and makes the problem more manageable. Understanding how to simplify continued fractions is fundamental to solving equations that contain them.
Breaking Down the Problem Step-by-Step
The given equation is x + 1/(2 + 1/(2 + 1/1)) = 2. To solve for x, we first need to simplify the continued fraction on the left-hand side of the equation. We begin by focusing on the innermost part of the fraction and working our way outwards. This systematic approach ensures that we address each component in the correct order, leading to a simplified expression. The step-by-step breakdown will illustrate how each simplification contributes to the overall solution, making the process clear and easy to follow.
Step 1: Simplifying the Innermost Fraction (1/1)
The innermost fraction in our expression is 1/1, which simplifies directly to 1. This is the first and simplest step in our simplification journey. Replacing 1/1 with 1, the equation now looks like this:
x + 1/(2 + 1/(2 + 1)) = 2
This seemingly small step is crucial as it sets the stage for further simplification. By addressing the innermost fraction first, we begin to unravel the complexity of the continued fraction.
Step 2: Simplifying the Next Layer (2 + 1)
Moving outwards, we next encounter the expression 2 + 1, which is a straightforward addition. Adding these numbers gives us 3. Substituting this result back into the equation, we get:
x + 1/(2 + 1/3) = 2
With each step, the equation becomes less convoluted. We are progressively simplifying the continued fraction, bringing us closer to isolating x.
Step 3: Simplifying (2 + 1/3)
Now, we need to simplify the expression 2 + 1/3. To add these, we need a common denominator. We can rewrite 2 as a fraction with a denominator of 3, which is 6/3. Thus, 2 + 1/3 becomes 6/3 + 1/3. Adding these fractions results in 7/3. Substituting this back into the equation, we have:
x + 1/(7/3) = 2
This step further simplifies the continued fraction, making it easier to manage and solve. We are consistently reducing the complexity of the expression.
Step 4: Simplifying 1/(7/3)
Our next step is to simplify the fraction 1/(7/3). To do this, we take the reciprocal of 7/3, which is 3/7. The equation now simplifies to:
x + 3/7 = 2
At this point, the continued fraction is completely simplified, and we are left with a basic algebraic equation. We have successfully transformed a complex expression into a manageable form.
Isolating and Solving for x: A Detailed Approach
Having simplified the continued fraction, we now focus on isolating x to find its value. The equation is x + 3/7 = 2. To isolate x, we need to subtract 3/7 from both sides of the equation. This process will leave x on one side and the numerical value on the other, allowing us to determine its value.
Step 5: Subtracting 3/7 from Both Sides of the Equation
Subtracting 3/7 from both sides of the equation x + 3/7 = 2 gives us:
x = 2 - 3/7
This is a standard algebraic manipulation that helps us isolate the variable we are trying to solve for.
Step 6: Finding a Common Denominator for Subtraction
To perform the subtraction, we need a common denominator. We can rewrite 2 as a fraction with a denominator of 7. Since 2 is equivalent to 14/7, the equation becomes:
x = 14/7 - 3/7
This step is crucial for accurately subtracting the fractions. Ensuring a common denominator allows us to combine the fractions correctly.
Step 7: Performing the Subtraction
Now that we have a common denominator, we can subtract the numerators. Subtracting 3 from 14 gives us 11. Therefore, the equation simplifies to:
x = 11/7
The Final Solution for x
Thus, the value of x that satisfies the given equation x + 1/(2 + 1/(2 + 1/1)) = 2 is 11/7. This comprehensive solution demonstrates how continued fractions can be simplified step-by-step using basic algebraic principles. By working from the innermost fraction outwards and applying simple arithmetic operations, we successfully isolated x and determined its value.
In conclusion, this article has provided a detailed explanation of how to solve for x in an equation involving a continued fraction. By systematically simplifying the fraction and applying basic algebraic principles, we found that x = 11/7. This problem illustrates the importance of understanding continued fractions and their properties. Continued fractions are not only a fascinating mathematical concept but also a valuable tool in various areas of mathematics and computer science. The step-by-step approach outlined in this article offers a clear methodology for tackling similar problems, reinforcing the understanding of both continued fractions and algebraic manipulation. The ability to simplify complex expressions and solve for unknowns is a fundamental skill in mathematics, and this article provides a clear example of how to effectively apply these skills. Mastering these techniques empowers individuals to confidently approach and solve a wide range of mathematical problems.
