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Chapter 2: Problem 100

The expression \(\left|x_{T}-x\right|\) is defined to be the absolute error in \(x\) where \(x_{T}\) is the true value of a quantity and \(x\) is the measured value or value as stored in a computer. If the true value of a quantity is 0.2 and the approximate value stored in a computer is \(\frac{51}{256},\) find the absolute error.

Short Answer

Expert verified
The absolute error is \(\frac{1}{1280}\).

Step by step solution

01

Understand the Formula for Absolute Error

The absolute error is defined by the formula: \( |x_T - x| \), where \(x_T\) is the true value and \(x\) is the measured or stored value. We need to find the magnitude of the difference between these two values.
02

Identify the Given Values

From the exercise, we know that the true value \(x_T\) is 0.2 and the approximate value stored in a computer \(x\) is \(\frac{51}{256}\).
03

Convert the True Value to a Fraction

To facilitate the subtraction, we need to convert the true value 0.2 to a fraction. Since 0.2 is equivalent to \(\frac{1}{5}\), we use this fraction for further calculations.
04

Perform Subtraction in Fraction Form

To perform the subtraction \(\left| \frac{1}{5} - \frac{51}{256} \right|\), we must first find a common denominator. The least common denominator of 5 and 256 is 1280.
05

Convert Fractions to Common Denominator and Subtract

Convert each fraction: \(\frac{1}{5} = \frac{256}{1280}\) and \(\frac{51}{256} = \frac{255}{1280}\). We can now calculate the difference: \(\left| \frac{256}{1280} - \frac{255}{1280} \right| = \left| \frac{1}{1280} \right|\).
06

Simplify the Result

The absolute value of \(\frac{1}{1280}\) is already in its simplest form, as both the numerator and the denominator are prime numbers with respect to each other.

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Key Concepts

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

Fraction Subtraction
Subtracting fractions involves a few steps to ensure accuracy and simplicity. The key is to have both fractions expressed in terms of a common denominator. This makes the arithmetic straightforward. When you have fractions like \( \frac{1}{5} \) and \( \frac{51}{256} \), their denominators are different, meaning they can't be directly subtracted.To subtract fractions, follow these steps:
  • Identify the denominators of both fractions.
  • Determine the least common denominator (LCD) to create equivalent fractions.
  • Convert each fraction to have the LCD as their new denominator.
  • Subtract the numerators of these equivalent fractions.
  • The result's denominator will be the same LCD you determined earlier.
This step ensures that both pieces are comparable, allowing for straightforward subtraction.
Converting Decimals to Fractions
Converting decimals to fractions is an essential process in mathematics that enables more straightforward calculations. To convert a decimal like 0.2 into a fraction, consider the place value of the decimal point.For example, 0.2 has a '2' in the tenths place. To convert it:
  • Count the decimal places; here, it is one.
  • Rewrite as a fraction with 1 followed by an equal number of zeros as the number of decimal places (i.e., \( \frac{2}{10} \)).
  • Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (here, that would be 2), resulting in \( \frac{1}{5} \).
This process makes it easier to perform operations such as subtraction by using fractions instead of decimal numbers.
Least Common Denominator
Finding the least common denominator (LCD) is a critical step when working with fractions that need to be compared or combined. The LCD is essentially the smallest number that both denominators divide into without leaving a remainder.To calculate the LCD for fractions like \( \frac{1}{5} \) and \( \frac{51}{256} \):
  • List the multiples of each denominator鈥攐r simply factorize them into their prime factors.
  • Combine these prime factors to get the smallest product that can be divided by both original denominators.
  • For 5 and 256, the LCD is their product due to their nature as co-prime numbers, known to be 1280 in this case.
Using the LCD allows you to rewrite the fractions with a common denominator, facilitating operations like addition or subtraction.
Error Analysis
Error analysis, particularly in contexts involving computation and measurement, helps assess the precision and accuracy of results. The absolute error is a common form of error analysis that quantifies the difference between the true value \(x_T\) and an approximation \(x\).For absolute error, the formula \(|x_T - x|\) suggests the magnitude of the error without regard to direction. Steps for error analysis include:
  • Compare the true value with the recorded or estimated value, aware that absolute error can be positive or negative depending on which is larger.
  • Understand that the absolute value in the formula ensures a non-negative result, offering a straightforward decision on precision.
  • Recognize that smaller absolute errors indicate more accurate approximations, crucial when analyzing accuracy in calculations.
This analysis assists in identifying potential discrepancies and understanding their implications on overall computations.

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