Problem 14 - College Physics

Question

Express each of the following approximations of \(\pi\) to six significant figures: (a) \(22 / 7,\) (b) \(355 / 113 .\) (c) Are these approximations accurate to that precision?

Step-by-Step Solution

Answer

(a) 3.14286, (b) 3.14159. Only (b) is accurate to six significant figures.

Step 1: Calculate the value of 22/7

Divide 22 by 7 to find the first approximation of \(\pi\). Use a calculator to find: \(\frac{22}{7} = 3.142857142857...\).

Step 2: Round 22/7 to Six Significant Figures

Round the result from Step 1 to six significant figures. \(3.142857142857...\) rounds to \(3.14286\) for the first approximation.

Step 3: Calculate the value of 355/113

Divide 355 by 113 to find the second approximation of \(\pi\). Use a calculator to find: \(\frac{355}{113} = 3.14159292035398...\).

Step 4: Round 355/113 to Six Significant Figures

Round the result from Step 3 to six significant figures. \(3.14159292035398...\) rounds to \(3.14159\) for the second approximation.

Step 5: Determine Accuracy to Six Significant Figures

Compare each approximation to the actual value of \(\pi\), which is \(3.14159265358979...\). The approximation \(3.14159\) from \(355/113\) matches the first six digits of \(\pi\), making it accurate to six significant figures. The approximation \(3.14286\) from \(22/7\) does not match the digits of \(\pi\), and so it is not accurate to six significant figures.

Key Concepts

Approximations of Pi

Approximations of \(\pi\) have fascinated mathematicians for centuries. \(\pi\) itself, the ratio of a circle's circumference to its diameter, is an irrational number. This means its decimal form never ends and does not repeat. Since \(\pi\) is not a simple fraction, we often use rational approximations to work with it easily. Some of the commonly used approximations of \(\pi\) are \(\frac{22}{7}\) and \(\frac{355}{113}\). These fractions have been chosen because they offer close values to true \(\pi\).
* \(\frac{22}{7}\) approximates \(\pi\) as \(3.142857...\), often used for its simplicity.
* \(\frac{355}{113}\) is a more precise approximation, yielding a value of \(3.14159292035398...\).
Both serve as useful tools in calculations, making seemingly complex math problems simpler to solve.

Accuracy in Mathematics

Accuracy in mathematics is about how close an estimated value is to the true value. When working with numbers, especially approximations, it's crucial to understand how accurate they are. The precision can greatly affect the outcome of calculations. In the context of \(\pi\), an accurate approximation is one that closely matches the first several digits of the true \(\pi\) value \(3.14159265358979...\). Each digit in these numbers represents a level of precision, or accuracy, we aim to achieve.
For instance, we see that \(\frac{355}{113}\) provides a highly accurate result coinciding with the first six digits of \(\pi\). In contrast, \(\frac{22}{7}\) diverges after relatively few digits. It's essential to choose the right approximation depending on the level of precision required for a mathematical problem.

Rational Approximations

Rational approximations are fractions that closely represent irrational numbers like \(\pi\). These approximations simplify complex calculations by providing a number that's close enough to the actual value.A rational number is any number that can be represented as the quotient of two integers. \(\frac{22}{7}\) and \(\frac{355}{113}\) are clear examples where rational numbers are used to approximate irrational numbers.
* \(\frac{22}{7}\) is widely known though not the most accurate, yet it is easy to remember and use.
* \(\frac{355}{113}\) offers a much closer approximation to \(\pi\). It was discovered by the Chinese mathematician Zu Chongzhi and remains popular for precise calculations.
Understanding these approximations allows us to perform mathematical operations with reasonable precision without delving into infinite non-repeating decimals of irrational numbers. It’s like having a practical shorthand for an otherwise complex value.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify numbers while retaining their approximate value. When rounding, you decide how many digits are needed, usually depending on the required precision for a task. Significant figures are used to indicate the precision of a number. When dealing with approximations of \(\pi\), like \(\frac{22}{7}\) or \(\frac{355}{113}\), we round the unending decimals to a manageable number of significant figures. * \(\frac{22}{7}\) is rounded to six significant figures as \(3.14286\).
* \(\frac{355}{113}\) is rounded more precisely to \(3.14159\), aligning closely with \(\pi\).Using significant figures ensures you maintain the essential accuracy without unnecessary complexity. This practice is invaluable in scientific calculations where precision is crucial but ease of use is also necessary. Rounding helps to communicate and work with numbers more effectively in real-world situations.