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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
Question: Calculate the absolute and relative errors when using the value 3.14 to approximate 蟺. Solution: Step 1: Compute the Absolute Error The absolute error can be denoted as: $$AE = |x - c|$$ Here, \(x\) is 蟺 and \(c\) is 3.14. Substitute the values and compute the absolute error: $$AE = |蟺 - 3.14| = |0.00159265359|$$ Step 2: Compute the Relative Error The relative error is the absolute error divided by the actual value (\(x\)). $$RE = \frac{AE}{|x|}$$ Substitute the absolute error value found in Step 1 and the actual value of \(x\) (蟺) to compute the relative error: $$RE = \frac{0.00159265359}{|蟺|} = 0.000506957\dots$$ Step 3: Express Results The absolute error (AE) is approximately 0.0016 (rounded to four decimal places), and the relative error (RE) is approximately 0.0507% (rounded to four decimal places or 0.000507 as a decimal).

Step by step solution

01

Compute the Absolute Error

To compute the absolute error, we need to find the difference between the actual value (\(x\)) and the approximation (\(c\)). The absolute error can be denoted as: $$AE = | x - c |$$ Here, \(x\) is 蟺 and \(c\) is 3.14. Substitute the values and compute the absolute error.
02

Compute the Relative Error

The relative error is the absolute error divided by the actual value (\(x\)). It is represented as: $$RE = \frac{AE}{|x|}$$ Substitute the absolute error value found in Step 1 and the actual value of \(x\) (蟺) to compute the relative error.
03

Express Results

Lastly, express the results for absolute error (AE) and relative error (RE) appropriately. The absolute error can be expressed as a number, and the relative error can be expressed as a percentage or decimal.

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Key Concepts

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

Absolute Error
When you work with measurements and calculations, precise values can be challenging to capture. This is where understanding the concept of **Absolute Error** becomes important. Absolute error provides a way to express how far off our approximate value is from the exact value.

Let's break it down further:
  • The actual value is the real number that should be the result of our measurement or calculation. In our example, it is 蟺 (pi).
  • The approximate value is what you have estimated this real number to be, and in our example, this is 3.14.
  • To find the absolute error, you simply subtract the approximate value from the actual value and ignore any negative sign. This is because you're only interested in how big the difference is, not in which direction.
Mathematically, this is expressed as:

\[ AE = | x - c | \]

Here, \( x \) represents the exact value (蟺), and \( c \) is the approximation (3.14). By filling in the numbers, you get the absolute error for your measurement.
Relative Error
The **Relative Error** gives you an idea of the size of the absolute error compared to the true value. It's useful, especially when dealing with larger numerical values, because it provides context about the scale of the error. A small absolute error can be significant or insignificant, depending on the magnitude of the actual value.

Here鈥檚 how to calculate it:
  • First, find the absolute error, as discussed earlier.
  • Then, divide this absolute error by the actual value.
The formula for relative error is:
\[ RE = \frac{AE}{|x|} \]

This gives you a decimal value that you can also convert into a percentage by multiplying by 100, making it easy to understand and compare. So, in our example where \( x \) is 蟺, you substitute to find the relative error with respect to the approximation 3.14. This provides a better understanding of how significant the error is in relative terms.
Approximation
In mathematics and sciences, **Approximation** refers to finding a value that is close to, but not exactly, the true value of a number or measurement. Approximations are essential when dealing with irrational numbers, like 蟺, or when the exact measurement is not feasible.

Here are some points to remember about approximations:
  • Approximations are often necessary in real-world calculations because they simplify the numbers without heavily impacting the results.
  • They enable easier mathematical operations and quicker estimations, which are useful in various applications like engineering, science, and daily life.
  • The goal of approximation is to create a value that is "good enough" for the intended purpose, balancing simplicity and accuracy.
In our example, using 3.14 instead of 蟺 is an approximation that works well for practical applications. However, it's important to recognize the potential errors involved and understand how these errors might affect the results of calculations. This understanding includes recognizing how absolute and relative errors are also involved when dealing with approximations.

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Most popular questions from this chapter

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