The value of the irrational number π correct to ten decimal places (without rounding off...

The value of the irrational number π 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 π.

(a) (4/3)⁴: This is the value used for π. in the Rhind papyrus, an ancient Babylonian text written about 1650 B.C.

(b) 22/7: Archimedes (287–212 B.C.) showed that The use of the approximation 22/7 for π 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 π.

(c) 355/113: This approximation of π 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) This approximation was also discovered by Ramanujan. For a fascinating account of the history of π, see the book by Petr Beckmann, A History of π, 16th ed. (New York: Barnes & Noble Books, 1989), and for a more modern look at π, see Richard Preston’s article, “The Mountains of Pi,” in The New Yorker (March 2, 1992, pp. 36–67).