Rebecca Zeier and Audrey Grammas
Math 388 – History of Math
History of Pi
Buffon’s Needle Experiment
Discovery of Pi
Objectives:
Students will have a cursory knowledge of the history of pi .
Students will obtain firsthand experience with Buffon’s Needle Experiment.
Students will make a connection with the calculation of pi .
Students will recognize the applications of p in everyday life.
Students will have access to technological resources demonstrating pi .
NCTM Curriculum Standards for grades 6-9
Standard #1: Problem solving
Standard #3: Reasoning
Standard #13: Measurement
Illinois Standards: (Goal 6)
Investigate, represent and solve problems using number facts, operations
and their properties, algorithms and relationships. Compute and estimate using
mental mathematics, paper-and-pencil methods, calculators and computers.
Rational: Students will have a deeper understanding of the mathematical function of p , it’s history and some of the great mathematicians working on p . Additionally, they will have an understanding of practical applications and the importance of its use in calculating circumference and diameter as used here and in astronomical configurations.
Materials Needed:
Computer for technology demonstration and projection capability, list of chronology of Pi, lengths of ribbon or string for circumference lab, round or cylindrical objects for measurement, measuring devise (ruler, tape measure etc.), chalk, blackboard (for recording findings), Homework for entire class, various applicable overheads.
Content/Procedure:
Introduction to Pi: (Rebecca … 5 minutes)
A little known verse of the bible reads 'and he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. The same verse can be found in II Chronicles 4,2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and it's interest here is that it gives pi = 3. With that, we will be discussing pi today.
Pi is an infinite decimal. Unlike numbers such as 3, 9.876, and 4.5, which have finitely many nonzero numbers to the right of the decimal place, pi has infinitely many numbers to the right of the decimal point. Although many mathematicians have tried to find it, no repeating pattern for pi has been discovered.
Brief History of Pi: (Audrey … 15 minutes)
Pi is a very old number. The earliest values of pi including the 'biblical' value of 3, were most certainly found by measurement. We know that the Egyptians and the Babylonians knew about the existence of the constant ratio pi, although they didn't know its value nearly as well as we do today. They had figured out that it was a little bigger than 3. The Egyptians calculated it to be approximately (4/3)^4 with equals 3.1604 and the Babylonians had an approximation of 3 1/8 which is about 3.125.
The earliest known reference to pi occurs in a Middle Kingdom papyrus scroll, written around 1650 BC by a scribe named Ahmes. He began the scroll with the words "The entrance into the knowledge of all existing things" and remarked in passing that he composed the scroll "in likeness to writings made of old." Toward the end of the scroll, which is composed of various mathematical problems and their solutions, the area of a circle is found using a rough sort of pi.
Around 200 BC, Archimedes of Syracuse found that pi is somewhere about 3.14. He wrote a book called "The Measurement of a Circle." In the book he states that Pi is a number between 3 10/71 and 3 1/7. He found this out by taking a polygon with 96 sides and inscribing a circle inside the polygon. That was Archimedes' concept of pi.
In the 1800's people sat down for years on end to find the values of pi to about 100 places. Imagine doing this by hand with no calculators. This has become a thing of the past, since the tedium that used to be done by hand is now done by computer.
This is of course just a brief history of how people studied pi … the study of pi has gone on for centuries upon centuries and if we covered it all, we'd probably bore you to death … so we'd like to give you this more detailed chronology of the history of pi for you to study at your own leisure.
Buffon's Needle Experiment: (Rebecca … 10 minutes)
One of the famous mathematicians we would like to point out to you on your chronology list is Compte De Buffon. Compete de Buffon was a French aristocrat with a courageous way of looking at the world. During the 18th Century most biological questions were answered by looking at biblical doctrine. Buffon rejected these ideas and dogmas and even went so far as to publish them in a book entitled "Historie Naturelle", which superceded the writings of Darwin by 100 years. He had some definite ideas about environment and organisms, which were in direct opposition to those ideas enforced by the church. In addition to his interest in nature, medicine and law, Buffon was a revolutionary thinker and because of this he was able to come up with an experiment known as the Buffon’s Needle Experiment which curiously proves the accuracy of Pi.
Buffon's needle experiment is one of the oldest problems in the field of geometrical probability. It was first stated in 1777. The idea is simple. Suppose you have a tabletop with a number of parallel lines drawn on it, which are equally spaced. Suppose you also have a pin or needle. If you drop the needle on the table, you will find that one of two things happen: 1) The needle crosses or touches one of the lines, or 2) the needle crosses no lines. The idea now is to keep dropping this needle over and over on the table, and to record the statistics. Namely, we want to keep track of both the total number of times that the needle is randomly dropped on the table, call that N, and the number of times that it crosses a line, call it C. The remarkable result is that if you keep dropping the needle, eventually you will find that the number 2N/C approaches the value of pi. Another way to look at it is that the probability is directly related to the value of pi.
We would like to show you three simulations to this experiment on the computer screen. The first simulation displays the idea of the number 2N/C approaching pi. This means that the more and more times we perform the experiment, the closer and closer the value gets to the value of pi. This is known as the limit. The second simulation shows the probability idea of the needle dropping. The third ….
Now isn't that interesting?
Discovering Pi: (Audrey and Rebecca … 20 minutes)
While many applications can be associated with pi, who knows the most common use of pi today? The most common use of pi is to calculate the circumference and area of a circle. In fact, by definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it. Believe us? Well let's test that out….
Learners will get into groups. In these groups, the learners will complete the Discovering Pi guide sheet (except for the Concluding Questions section). When each group has completed its worksheet, a member of each group will go to the board and add their answers to a chart on the board. After the data is on the board, the whole class will answer the Concluding Questions section together. See guide sheet for more information.
*** Another idea, which we included in our lesson, was to read the story of Sir Cumference and the Knight of Pi. We then followed up with practical applications and conclusions ***
Practical Applications of Pi: (Audrey)
We have demonstrated several different ways that we can prove Pi. We have even showed how we can use Pi to calculate circumference, diameter and radius of circles. We have some really important calculations that could not be done without the use of Pi.
A Greek philosopher named Eratosthenes used simple geometric reasoning to calculate the size of our planet in about 200 BC using a formulation including Pi. Using the formulation Erastosthenes was able to estimate the circumference of the Earth to within one percent accuracy using only simple geometry. The formula coupled with the angle at which the Sun passed overhead and the distance between Alexandria a city to the South of where he lived and his location allowed Erastothenes to do his detailed calculations.
Additionally Greek Astronomers Ptolemy, Aristotle a Greek Philosopher, were able to hypothesize the Geocentric model of our solar system and Isaac Newton, was able to create his laws of planetary motion reinforced later by Johannes Kepler in his laws of planetary motion and elliptic orbit in addition to escape speed.
Conclusion: (Rebecca)
Four thousand years ago, people discovered that the ratio of the circumference of a circle to its diameter was about 3. In nature people saw circles, great and small, and they realized that this ratio was an important tool.
This tool was used by the Babylonians and the Egyptians. Reference is made to the concept of pi in the bible. The Chinese found a value of pi that stood for one thousand years. One man felt the accomplishment of taking pi to 35 places was the most important achievement of his life, so much so, that he had it inscribed on his epitaph. With the help of computers, pi has been taken to over 6 billion places. People have been fascinated by pi, and irrational number, throughout history.
Worksheet:
Names ____________________________
Discovering Pi
Please carefully follow the directions!
1. Get a circular cover and piece of ribbon.
2. Use the ribbon to measure the circumference of the circular object. With a pencil, make a mark on the ribbon to record the measurement of the circumference of the circular object. Using the ruler, measure the marked off length of the ribbon to the nearest centimeter. This is the circumference of your circular object. Record that measurement here:
Circumference of Object (C) = ______________
Diameter of Object (d) = ________________
C + d = _________________
C – d = _________________
C * d = _________________
C / d = _________________
Concluding Questions
We will answer these questions as a class, but record your answers since this sheet will be collected for a grade.
C + d C – d C * d C / d