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Chapter 7: Problem 7

Compute the absolute and relative errors in using c to approximate \(x\). \(x=\pi ; c=3.14\)

Short Answer

Expert verified
In the step-by-step solution above, the true value (pi) and the approximate value (3.14) are used to calculate the absolute and relative errors. The absolute error comes out to be 0.00159265359 and the relative error is approximately 0.0507%.

Step by step solution

01

Identify true and approximate values

In order to compute the absolute and relative errors, we first need to identify the true value, x, and the approximate value, c. In this case, the true value, x, is pi, and the approximate value, c, is 3.14.
02

Calculate the absolute error

The absolute error is the difference between the true value, x, and the approximate value, c. We can calculate it using the formula: Absolute error = \(|x - c|\) In our case: Absolute error = \(|\pi - 3.14|\) At this point, we can calculate the absolute error: Absolute error = \(|3.14159265359 - 3.14|\) Absolute error = \(0.00159265359\)
03

Calculate the relative error

The relative error is the ratio of the absolute error to the true value, x. We can calculate the relative error using the formula: Relative error = \(\frac{|x - c|}{|x|}\) In our case: Relative error = \(\frac{|\pi - 3.14|}{|\pi|}\) Now we substitute the previously calculated absolute error and true value into the equation: Relative error = \(\frac{0.00159265359}{3.14159265359}\) Relative error 鈮� 0.000507292
04

Express the relative error as a percentage

To express the relative error as a percentage, multiply it by 100: Relative error (in percentage) = 0.000507292 脳 100 Relative error 鈮� 0.0507% The absolute and relative errors in using c (3.14) to approximate x (pi) are 0.00159265359 and 0.0507% respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Error
Absolute error is a crucial concept when dealing with approximations. It gives us a direct and straightforward measure of how much an approximation deviates from the true value. In essence, the absolute error tells us the magnitude of the difference between our calculated or measured value and the actual or true value.

To calculate the absolute error, use the formula:
  • Absolute error = \(|x - c| \)
Here, \(x\) represents the true value, and \(c\) stands for the approximate value.

For example, when approximating \(pi\) with 3.14, we find the absolute error by calculating \(|3.14159265359 - 3.14|\), which results in 0.00159265359. This tells us that there is a slight difference between our approximation and the true value.

Absolute error is essential for understanding the precision of your calculations or measurements, especially when small differences can significantly impact the outcome.
Relative Error
Relative error offers a different perspective by considering the size of the error relative to the true value. This is particularly useful for comparing errors in different contexts, as it scales the absolute error by the magnitude of the value being measured.

The formula to calculate the relative error is as follows:
  • Relative error = \(\frac{|x - c|}{|x|} \)
In this equation, \(|x - c|\) represents the absolute error, while \(|x|\) is the true value.

Using our example of approximating \(\pi\) with 3.14, we divide the absolute error 0.00159265359 by the true value of \(\pi\), which is approximately 3.14159265359. The resulting relative error is around 0.000507292.

To further enhance understanding, it is common to express the relative error as a percentage. This can be achieved by multiplying the relative error by 100, giving us 0.0507%. A smaller relative error percentage indicates a more accurate approximation in context.
Approximation
Approximation is the process of finding values that are close enough to the correct value for a specific purpose. It is widely used in mathematics, engineering, and science where exact values are either impossible or unnecessary to determine.

Approximations are often necessary because:
  • It might be computationally expensive to find the exact value.
  • In real-world situations, measurements and true values can be inexact due to limitations of instruments or methods.
In the context of the given exercise, using 3.14 to approximate \(\pi\) is a simple form of approximation. Although 3.14 is not equal to \(\pi\), it is used frequently when a simple and adequate representation of \(\pi\) is needed for calculations.

Approximations should balance between accuracy and simplicity. It is important to understand that every approximation carries an inherent error, which needs to be considered, especially in precise fields. This is where understanding and calculating absolute and relative errors becomes invaluable.

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