Numbers that are sometimes used as approximations of πare 3.14 and 3 (1)/(7) . To four decimal p

step 1 355 divided by 11313 is 3 .14159. So this actually gives the same decimal places up to four, bec

Numbers that are sometimes used as approximations of πare...

Numbers that are sometimes used as approximations of $\pi$ are 3.14 and $3 \frac{1}{7} .$ To four decimal places, $\pi$ is 3.1416
An especially good approximation of $\pi$ is the fraction $\frac{355}{113}$. To how many decimal places does it give the correct value?

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Key Concepts

Rational Approximations

This concept involves representing an irrational number, which has an infinite non?repeating decimal expansion, using a fraction (a ratio of two integers) to simplify calculations and estimations. Approximations like 355/113 are used because they yield values close to the true value of ? while being easier to work with in computations.

Decimal Place Accuracy

Decimal place accuracy refers to the number of digits in the approximated number that exactly match the corresponding digits of the true value when written in decimal form. This concept is used to assess and express the precision of an approximation by counting how many leading decimal digits are correct before a discrepancy occurs.

Error Analysis in Approximations

This concept involves comparing an approximated value with the true value to determine the magnitude of error, typically by looking at their decimal representations. In the context of ?, error analysis helps quantify how close a rational approximation, like 355/113, is to the actual value by identifying the number of matching digits.

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