The Value of p |
| It is surprising, but many college students, when asked
which is greater, 3.14 or 22/7, will say that the two numbers are the same.
That is because they think that both numbers are the same as the number
p. Neither number is actually
p. Since 22/7 starts off 3.1428571..., it is the
greater number. There is some evidence that the ancient Hebrews and Babylonians were even less accurate than today's college students. In the Bible (First Kings 7:23) we learn of the model of a sea made by Hiram of Tyre for King Solomon: "it was round, ten cubits from brim to brim.. . and a line of thirty cubits measured its circumference." The implication is that p is 3, since p is the ratio of the circumference of a circle to its diameter. There is some evidence that Babylonian mathematicians used a better value for p, namely 3.125. It was clear to the ancient Greeks and Chinese that one could get a good approximation of p by comparing a circle to a straight-sided figure that was approximately a circle. It is relatively easy to find the length of the perimeter of such a regular polygon if you know the distance from the polygon's centre to one of its sides or to one of its vertices. Using this method, Archimedes calculated that p is between 3-10/71 and 3-10/70 (22/7), while Chinese scholars around 500 AD showed that p is between 3.1415926 and 3.14152927. In 1596, Ludolph of Cologne used this method to calculate p to 32 places. His result was engraved on his tombstone and to this day Germans call p the Ludolphine number. Although everyone knew that these values for p were not exact (since they were based on perimeters of polygons, not the circumference of a circle), it was not clear whether an exact value could be found. Around the time of Ludolph, Vieta developed the first simple numerical expression for p. It was not expressed as a decimal numeral or as a fraction, however. It was an infinite product. Later mathematicians also found other infinite products and infinite series (sums) for p. Two with especially easy patterns are actually for p/2 and p/4. In the seventeenth century, John Wallis discovered p/2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 )( ..., in which the numerators are the even numbers from 2 given twice, while the denominators are a similar pattern of odd numbers. James Gregory and Wilhelm Gottfried Leibniz discovered an even simpler pattern for an infinite sum: p/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9-1/11 + . . .. This pattern is known as the Leibniz series, although Gregory was the first to find it. Note that these patterns carried to infinity yield exact values for p, but hey still do not tell whether p can be expressed as a finite decimal. Many infinite products and series converge to finite decimals. These infinite products and series, and others like them, however, provided an easier way to compute pproximations to p than using polygons. At the end of the seventeenth century, Abraham Sharp found 71 decimal places. In the nineteenth century, p was gradually extended, reaching 707 places in the calculation of William Shanks in 1853 that took him 15 years to complete. When computers were invented, however,it was found that Shanks had made a mistake in 528th place, causing every place afterward to be wrong. In the meantime, in the eighteenth century, Johann Lambert finally solved one of the problems connected with p. He showed that p is irrational; in other words it cannot be expressed as a finite decimal, nor can it have a simple repeating pattern as a decimal. A related problem was still unsolved. Since the time of Anaxagoras at least, in the fifth century BC, people had been trying to use a straightedge and compass construct a square the same area as a given circle. By 1775 the ranks of people trying to solve this famous problem were so great that the Academy of Paris passed a resolution that it could no longer examine purported successes. This problem was effectively solved in 1882, when the mathematician Ferdinand Lindemann showed p is a member of a large class of numbers of which only a few are commonly known. These numbers called transcendental. There are more of them than any of the more familiar numbers. Their defining characteristic is that they are not the solutions to algebraic equations with integer coefficients. Constructing a line with a straightedge and compass implies that its length is the solution to such an equation. Since p istranscendental, it cannot be that kind of solution. Squaring the circle is impossible. This did not stop people from calculating the value of p to more and more decimal places. When electronic computers became available in the 1940s and 1950s some people used the calculation of p as a kind demonstration of how powerful these computers were. By 1949, in 70 hours of computer time (as opposed toShanks's 15 years of paper-and-pencil time), p was extended to 2037 places. By 1988 Japanese computer scientist Yasumasa Kanada had reached 201 ,326,000 decimal places, and he was planning to go further. The 1988 computation only took six hours of supercomputer time. |
Alexander Hellemans and Bryan Bunch
"Timetables of Science" p360
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