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How many times have students approached you to ask, What is pi ? Similar questions usually arise during class discussions about the areas and circumferences of circles. When given the definition of ir as the ratio of the circumference of the circle to the diameter of the circle, many students perceive this definition as "circular" in nature. They often ask, After all, do we not use ir to calculate the circumference? Whether it is of historical significance, mathematical importance, or personal interest, (pi) has universal appeal, in and out of the classroom. The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that its discovery is difficult to trace.
Since pi is an irrational number, it cannot be expressed as a ratio of whole numbers. Hence, better decimal approximations of (pi) have been explored from biblical times to the present. Evidence suggests that around 2000 B.C., the Babylonians and the Egyptians had considered approximations for 7. Beckmann (1971) suggests that the Babylonians had approximated pi with 3 1/8 and that Egyptians used 4(8/9)2. One of the first theoretical calculations was carried out by Archimedes of Syracuse (287-212 B.C.), when he obtained the approximation 223/71 < pi < 22/7 by using regular polygons. Other mathematicians from antiquity to more recent times who are given credit for results related to the concept of pi include Ptolemy (ca. A.D. 150), Tsu Chung Chi (A.D. 430-501), al-Khwarizmi (ca. A.D. 800), Al-Kashi (ca. A.D. 1430), Viete (A.D. 1540-1603), Rooman (A.D. 1561-1615), and van Ceulen (ca. A.D. 1600).
This article describes a classroom inquiry into expressing pi as the limit of a sequence of different ratios...