Problem 55 - Precalculus

Question

The value of the irrational number \(\pi\), correct to ten decimal places (without rounding off), is \(3.1415926535 .\) By using a calculator, determine to how many decimal places each of the following quantities agrees with \(\pi\) (a) \((4 / 3)^{4}:\) This is the value used for \(\pi\) in the Rhind papyrus, an ancient Babylonian text written about 1650 B.C. (b) \(22 / 7:\) Archimedes \((287-212 \text { B.C. })\) showed that \(223 / 71<\pi<22 / 7 .\) The use of the approximation \(22 / 7\) for \(\pi\) was introduced to the Western world through the writings of Boethius (ca. \(480-520\) ), a Roman philosopher, mathematician, and statesman. Among all fractions with numerators and denominators less than 100 , the fraction \(22 / 7\) is the best approximation to \(\pi\) (c) \(355 / 113:\) This approximation of \(\pi\) was obtained in fifth-century China by Zu Chong-Zhi (430-501) and his son. According to David Wells in The Penguin Dictionary of Curious and Interesting Numbers (Harmondsworth, Middlesex, England: Viking Penguin, Ltd., 1986 ), "This is the best approximation of any fraction below \(103993 / 33102\) " (d) \(\frac{63}{25}\left(\frac{17+15 \sqrt{5}}{7+15 \sqrt{5}}\right):\) This approximation for \(\pi\) was obtained by the Indian mathematician Scrinivasa Ramanujan (1887-1920).

Step-by-Step Solution

Answer

(a) 1 decimal place; (b) 2 decimal places; (c) 6 decimal places; (d) 10 decimal places.

Step 1: Calculate (4/3)^4

Begin by calculating \((4 / 3)^{4}\). \( (4/3)^4 = \left( \frac{4}{3} \right)^4 = \frac{256}{81} \approx 3.1604938272 \). This value should be compared to \(\pi = 3.1415926535\).

Step 2: Determine Agreement for (4/3)^4

Compare the decimal places of \(3.1604938272\) to \(3.1415926535\). The values agree only up to one decimal place (3.1), as they start to differ from the second decimal place onwards.

Step 2: Calculate 22/7

Next, calculate \( \frac{22}{7} \approx 3.1428571429 \). This should be compared to \(\pi = 3.1415926535\).

Step 4: Determine Agreement for 22/7

Compare the decimal places of \(3.1428571429\) to \(3.1415926535\). The values agree up to two decimal places (3.14), as they differ at the third decimal place.

Step 3: Calculate 355/113

Calculate \( \frac{355}{113} \approx 3.1415929204 \). Compare this with \(\pi = 3.1415926535\).

Step 6: Determine Agreement for 355/113

Compare the decimal places of \(3.1415929204\) to \(3.1415926535\). The values agree up to six decimal places (3.141592), as they start to differ at the seventh decimal place.

Step 4: Calculate Ramanujan's Approximation

Calculate \( \frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right) \). First, find \( \sqrt{5} \approx 2.236067977 \), then evaluate each part inside the fraction. Compute the expression, resulting approximately in \(3.1415926535\).

Step 8: Determine Agreement for Ramanujan's Approximation

Ramanujan's expression evaluates to exactly \(3.1415926535\), matching \(\pi\) in all places to the tenth decimal. Therefore, the two values agree over all ten decimal places.

Key Concepts

Rhind Papyrus Approximation

The Rhind Papyrus, dating back to approximately 1650 B.C., presents one of the earliest known approximations of \(\pi\). It gives us \(\pi\) as \((4/3)^4\). When calculated, \((4/3)^4\) results in a value of approximately 3.1604938272. This means the Egyptians saw \(\pi\) as a close value to 3.16. Why is this interesting? Ancient Babylonians and Egyptians used this approach for practical purposes like architecture and astronomy, without the precision of modern day. Hence, while only accurate to one decimal place, it was quite commendable for its time.

Archimedean Ratio

The well-known Greek mathematician Archimedes proposed a method for calculating \(\pi\) that placed it between two values, \(223/71\) and \(22/7\). The fraction \(22/7\) became a widely used approximation due to its simplicity and fairly good accuracy.Calculating \(22/7\) results in 3.1428571429, which is a decent approximation, accurate to two decimal places for \(\pi\). This value was so influential that it persisted into Western thought through the work of Boethius. Archimedes’ approach laid foundational work in mathematical analysis and approximation techniques.

Zu Chong-Zhi Approximation

In fifth-century China, mathematician Zu Chong-Zhi, alongside his son, found an even more precise fraction to approximate \(\pi\). They realized that \(355/113\) resulted in a number very close to \(\pi\), about 3.1415929204.What is remarkable about \(355/113\) is its precision, matching \(\pi\) up to six decimal places. This was an extraordinary feat for the time, considering it remained the best approximation involving simple fractions until many years later. Zu Chong-Zhi's approximation demonstrated the heights of ancient mathematical understanding.

Ramanujan's Approximation

The genius Indian mathematician Srinivasa Ramanujan developed a range of formulas for \(\pi\) during the early 20th century. One such approximation is \(\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)\), which directly evaluates to \(\pi\) up to ten decimal places: 3.1415926535.Ramanujan's methods were unique. They included rapidly converging series and highly accurate fractional representations. This specific expression showcases his deep insights and profound understanding of mathematical concepts. His approximations continue to be celebrated for their precision and elegance.