Problem 16 - College Physics

Question

\(\cdot\) Express each of the following approximations of \(\pi\) to six significant figures: (a) \(22 / 7,(\) b) 35\(/ 113 .\) (c) Are these approximations accurate to that precision?

Step-by-Step Solution

Answer

Both 22/7 and 35/113 are not accurate to six significant figures.

Step 1: Understand Significant Figures

Significant figures reflect the precision of a number. Here, six significant figures mean that you need to round or express the number to contain exactly six digits that contribute to its precision, starting from the first non-zero digit.

Step 2: Calculate 22/7

Divide 22 by 7 to find this approximation of \(\pi\). This yields a repeating decimal \(3.142857\). The significant figures round this to \(3.14286\).

Step 3: Calculate 35/113

Divide 35 by 113 to find this approximation of \(\pi\). The result is approximately \(0.3106\) repeating. Expressed to six significant figures, it is \(0.310618\).

Step 4: Compare with True Value of π

The true value of \(\pi\) is \(3.14159\) approximately (to six significant figures). Compare this with the resulting approximations from steps 2 and 3.

Step 5: Analyze the Precision

\(22/7\)'s approximation \(3.14286\) is not accurate to six significant figures because it's different from \(\pi\) after the fourth digit. \(35/113\) as \(0.310618\) is not accurate either because it does not represent the right precision for \(\pi\).

Key Concepts

Approximations of Pi

Approximating π can be quite fun and enlightening because it involves comparing different fractions and seeing how close they get to \(\pi\). Everyone's favorite infinite number is essential in mathematics for measurements involving circles. Commonly known approximations are \(\frac{22}{7}\) and \(\frac{35}{113}\).

• When we calculate \(\frac{22}{7}\), we get approximately \(3.142857\), which rounds to \(3.14286\) when considering six significant figures.
• Similarly, \(\frac{35}{113}\) yields approximately \(0.3106\) repeating, which rounds to \(0.310618\) when expressed to six significant figures.

It's fascinating to analyze these values because they showcase different levels of closeness to the actual numerical value of \(\pi\), which is approximately \(3.141592\) when expressed to six significant figures. Despite being historical and educational, both of these fractions fail to match the actual precision required for six significant figures of \(\pi\).
Understanding approximations helps solidify the concept that while certain fractions can be close representations, they may not perfectly encapsulate the precision of the irrational number \(\pi\). It's a vivid example of how infinite decimals differ from fixed ratios.

Precision and Accuracy

Precision refers to the level of detail in the representation of a number, while accuracy describes how close this number comes to the true or accepted value. In the context of approximations, these two play a critical role.
When you look at numbers like \(\frac{22}{7}\) and \(\frac{35}{113}\), they have specific levels of precision depending on how they are expressed. Precision is all about the significant figures we select. For example:

• With \(3.14286\), there are six significant figures, but this doesn't necessarily guarantee it's accurate compared to the true value of \(\pi\).
• Similarly, \(0.310618\) maintains six significant figures, but its distance from \(\pi\) tells a different story in terms of accuracy.

Sometimes, numbers can be highly precise but still lack accuracy if they don't align closely with true values. In summary, understanding both precision and accuracy in mathematical operations ensures clarity in how numbers and results are interpreted, aiding in better decision-making and communication.
Moreover, inaccuracies often emerge when approximation methods, like converting infinite decimals, inherently limit the accuracy due to their inability to capture every nuance of irrational or complex numbers.

Mathematical Calculations

Performing mathematical calculations with fractions and significant figures presents an opportunity to refine one's mathematical prowess. The process involves working through steps systematically and involves competing considerations of precision and the absolute representing number's closeness to the actual target.
Here's an engaging refresher on what to keep in mind when handling mathematics with approximations and calculations:

• First, converting fractions into decimal form is the initial step. This helps in visualizing and understanding the number's proximity to constants like π.
• Next, rounding off to a specific number of significant figures embodies the intricacies involved in the computation, providing insight into how nicer or simpler forms can sometimes overlook important details.
• Analyzing and comparing these rounded off values with true mathematical constants stresses fundamental concepts and encourages curiosity.

Learning these mathematical calculation techniques, especially in terms of handling constants like π using fractions, can enrich students' understanding of how theoretical numbers are applied practically. The careful mix of precision in prediction and accuracy check in results encourages patience and attention to detail.