What They Don't Tell You About Pi in High School
In the instruction of mathematics, students usually are taught the generic definition of pi. They are told that pi is the ratio of the circumference of a circle to its diameter. They are told that it is an irrational number approximately equal to or 3.14159. It is very common for high school students to miss out on several conceptual components and other useful information due to lack of time or indifference on the part of the teacher.
Through this project, we are going to explore some concepts of pi beyond the typical geometric applications of finding circumferences and areas of circles. It is our intention that readers of this work will become enlightened to the depth and versatility of this transcendental number. The historical component of this project will inform the reader of the laborious efforts that were required to find increasingly more accurate representations for pi.
Pi serves several functions in mathematics beyond geometry. It appears repeatedly in calculus, especially during discussions concerning series. We have also discovered some interesting uses of pi. We will share these as we move through the project.
II. A Brief History of Pi
At some point in history, man came to recognize that the relation between the size of the circumference and the diameter of all circles was a constant ratio. This knowledge was displayed in the earliest recorded mathematical documents of Egypt and Babylon over 2000 years ago. They did not use the p symbol as representation (that came much later), but they had established that the ratio was equal to , where C is the circumference and D is the diameter of any given circle. Even at this early stage, Babylonian and Egyptian mathematicians had come up with numerical approximations to , the number we now call pi. Their methods are unknown and must have been rather crude, lacking the modern number system or even pencil and paper. Beckman suggests that they may have used sticks and rope to draw circles in the sand, and used the rope to measure how many diameters made up a circumference (13). Early documents use the numbers 3, 3, and 3 as values for pi. There is a Biblical reference to a circular, molten sea ten cubits across and thirty cubits around, suggesting the value pi = 3 (I Kings 7:23).
Once the existence of the ratio pi was established, it could be used to determine the area of a circle. This could be done approximately by a method of rearrangement. If a circle of diameter D and circumference C was divided into equal wedge shaped sections, the sections could be rearranged to form a near parallelogram with length approximately and width approximately. The area of the circle would be approximately the same as the area of the parallelogram. This, of course, would be the product of the length and width, ()() = p ()2. The accuracy can be improved by increasing the number of wedges in the circle. A pictorial representation using eight wedges follows:
Early Greeks originated the notion that an approximation of a circle (and its circumference and area) could be made by constructing regular polygons. The more sides on the polygon, the closer the figure resembled the circle (and its circumference and area). Archimedes used this notion to obtain a geometrically based estimate of pi. He inscribed and circumscribed polygons of 96 sides within and about a circle. Then, calculating the areas of the two polygons and determining the area of the circle to be between these two values, he approximated pi to be between and . In today’s notation, these values are accurate to two decimal places. In about 150 A.D., Alexandrian mathematician Claudius Ptolemy was able to compute the length of a side of a polygon of 360 sides. Using this length, he knew the circumference of the 360-gon and used it to calculate pi to be approximately (or about 3.14167). In 500 A.D. Arya-Bhata of India calculated the perimeter of a polygon of 384 sides to estimate pi as , which also equals 3.1416. Similar methods of calculating pi using polygons of increasing number of sides were employed up until the Middle Ages, when there was very little progress made in the determination of pi, or for that matter, in all of mathematics.
The establishment of the modern decimal notation and the discovery of logarithms greatly facilitated Archimedean methods of calculating pi. In the sixteenth century, Francois Vieta of Paris found pi to nine decimal places using polygons of 393,216 sides. In 1596 Ludolph Van Ceulen of Germany calculated pi to 20 places and in 1610 to 35 places using Archimedean methods. This degree of accuracy was beyond all practical use, but nevertheless, mathematicians continued to calculate pi to more and more decimal places.
Vieta had a much more significant contribution to the determination of pi. In 1593 he became the first person to express pi as an infinite series of algebraic operations. He came up with the expression: pi =
In 1650 John Wallis of England discovered the following infinite series:
Or as the infinite fraction:
In 1668 James Gregory of Scotland introduced the infinite series:
Arctan (x) = x - + - +…
When x = 1, this becomes:
Gottfried W. Leibniz of Germany also discovered this series in 1673. Although this expression of pi was of great mathematical importance, it was not useful in calculating pi to great accuracy. It took over 300 terms in the series to get the two decimal accuracy of . Of greater importance was Isaac Newton's use of calculus to approximate pi. He used his binomial theorem to expand the formula for a circle. He then calculated the area of the circle by using calculus and then by geometrical methods. When he combined the two, he had a method that calculated pi to seven decimal places with only nine terms of his binomial expression. Abraham Sharp of England used x = in Gregory's series to calculate pi to 72 decimal places in 1699. John Machin, also of England, used Gregory's series to calculate pi to 100 decimal places in 1706. He substituted the series for arctan () and arctan () into the expression: = 4arctan() - arctan (). In 1798 Leonard Euler used pi = 20arctan() + 8arctan(). These series converged to pi much faster than Gregory's original series.
In 1760 Count de Buffon demonstrated an interesting method of estimating pi in his famous Needle Problem. His method relied on probability. He set up a series of parallel lines on a plane a distance of "A" units from each other. Then he dropped a needle of length L (L less than A) onto the surface. He proved that the probability of the needle landing so that it crossed one of the lines was equal to . By repeating the experiment a large number of times, he could approximate pi relatively accurately. This later became known as a Monte Carlo method.
A big step in understanding the nature of pi was made in 1767 by Johann Heinrich Lambert. He proved that pi was an irrational number, meaning that it is not expressible in terms of a quotient of integers. This proof was extremely important because it implied that pi did not have a terminating or repeating decimal expansion. Thus it was insured that the exact calculation of pi was a hopeless endeavor.
In 1882 Ferdinand Lindemann proved that pi was not only an irrational number but also a transcendental number. This meant that pi could not be the solution to any polynomial expression of rational numbers. Consequently, he also proved that the circle could not be squared, something mathematicians have tried to do for over 2000 years. Squaring a circle means to construct a square with the same area as a given circle using only a straightedge and a compass. Srinivasa Ramanujan of India introduced many other interesting formulas approximating pi around the beginning of the twentieth century. One of his approximations was pi = , for which he produced a geometrical construction.
Meanwhile, calculations of pi were being made to greater and greater degrees of accuracy. Using Machin's method, William Shanks of England determined pi to 607 places in 1853 and to 707 places in 1873. These figures later proved to be accurate only to 527 places, but were corrected in 1946 by D. F. Ferguson. After 1949, hand calculations of pi became obsolete. It was then that the computer first found the value of pi to 2037 places. The computer has taken this problem beyond imagination. By 1990, pi had been calculated to over half a billion decimal places.
This has been a brief overview of how mathematicians have come to look at pi over the last three or four thousand years. Although the nature of pi is mysterious and difficult to understand, it has helped man to understand many mathematical concepts. Just as there is no limit to the measure of its precise calculation, there is no limit to the measure of importance pi has had in the study of the mathematical world.
A. Ribbon Problem
Suppose you had a ribbon wrapped around a sphere the size of the earth. Now add six feet to the length of the ribbon. The ribbon should be the same distance from the sphere at all points. a) Calculate the distance between the sphere and the ribbon. b) Now suppose you had a ribbon wrapped around a sphere the size of a basketball (radius 5 inches). Add six feet to the length of this ribbon. Calculate the distance between the ball and ribbon. Compare your results.
Let r be the radius of a sphere. Then the length (L) of a ribbon wrapped tight around the sphere is 2p r. Add six feet to the length of the ribbon to get a new length, L’. L’=2p r+6. Now call the new radius r’. L’ = 2p r’= 2p r+6. Divide both sides by 2p to get r’= r +. The change in radius is , regardless of the original size of the sphere! See another example in the attached visual representation.
B. Pie slicing
Imagine a round pizza pie. Try to cut it into four pieces of equal area, using only one cut and without taking the knife around the edge. Equivalent problem: draw a circle on a piece of paper. Now divide it into four parts of equal area using a pencil, without lifting the pencil off the paper and without drawing around the outside of the circle.
1. Explanation: see the attached explanation for one solution.
C. Buffon's Needle
1. What you need:
a. A needle or toothpick (call the length one unit).
b. A piece of paper with several parallel lines one unit apart from each other.
2. What to do:
The probability of a hit is . Therefore = (# of Hits)/(# of Drops). Rearranging, we get:
Pi = 2(# of Drops)/(# of Hits)
For a complete explanation of the problem and how the probability is justified, visit the Buffon’s Needle website in the list of pi links.
D. Construct line segment with length of pi
1. How to construct a straight line segment with a length that differs from pi by less than .0000003 using only a compass and straightedge:
a. Construct a quadrant of unit radius. Draw the line segments as shown so that:
i. bc is of the radius
ii. dg is of the radius
iii. de is parallel to ac
iv. df is parallel to be
IV. Pi Links
1. http://pauillac.inria.fr/algo (Algorithms Project at INRIA)
2. http://pauillac.inria.fr/algo/bsolve/constant/constant.html (favorite math constants)
3. http://pauillac.inria.fr/algo/bsolve/constant/pi/pi.html (Archimedes’ constant pi)
4. http://www.mste.uiuc.edu/reese/buffon/buffon.html#intro (Buffon's needle)
5. http://pauillac.inria.fr/algo/bsolve/constant/pi/srs.html (infinite series involving pi)
6. http://www.yahoo.com.au/Science/Mathematics/Numbers/Specific_Numbers/Pi/ (search page)
1. Beckman, Petr. A History of Pi. Boulder: Golem Press, 1982.
2. Dunham, William. Journey Through Genius. New York: Penguin Books, 1991.
3. Schepler, Herman C. "The Chronology of Pi." Mathematics Magazine 23 (1950): 165-170, 216-228, and 279-283.
4. Gardner, Martin. The Transcendental Number Pi, "Martin Gardner's New Mathematical Diversions from Scientific American." Simon and Schuster, New York, 1966.
D’Andre D. Jenkins and James D. Pett