Explain where π comes from and its history?
Allison Silvaggio
June 16, 2010
Discovery and Uses of the History and Uses of Mathematics
The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long
that it is quite untraceable.
C=πd
A=πr2
π occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is
that it gives π = 3.
The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a
value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer
approximation.
Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162
In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4
value for π.
(8/9)2 = 3.16 as a
Archimedes of Syracuse (287-212 BC). He obtained the approximation 223/71 < π < 22/7.
Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2,
calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so
on until he had calculated a6 and b6. His conclusion was that
b6 < π < a6 .
It is important to realize that the use of trigonometry here is unhistorical: Archimedes
did not have the advantage of an algebraic and trigonometrical notation and had to
derive (1) and (2) by purely geometrical means.
He stopped with polygons of 96 sides.
Liu Hui was a Chinese mathematician. He calculated π using an inscribed regular polygon with 192 sides.
The European Renaissance brought about in due course a whole new mathematical world. Among the first
effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of
Wallis (1616-1703)
2/π = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...) and one of the best-known is π/4 = 1 - 1/3 + 1/5 - 1/7 + ....
This formula is sometimes attributed to Leibniz (1646-1716)
discovered by James Gregory (1638- 1675).
but is seems to have been first
They show the surprising results that infinite processes can achieve and point the way to the wonderful richness
of modern mathematics.
The difficulty in computing π is the sheer boredom of continuing the calculation. A few people were silly
enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit.
Shanks knew that π was irrational since this had been proved in 1761 by Lambert, that if x is rational, tan(x)
must be irrational. It follows that if tan(x) is rational, x must be irrational. Since tan(pi/4)=1, pi/4 must be
irrational; therefore, pi must be irrational. In 1794, however, the A. M. Legendre found another proof which
backed Lambert up. This proof also went as far as to prove that pi^2 was also irrational.
Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the
solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that
'squaring the circle' is impossible. The transcendentality of π implies that there is no ruler and compass
construction to construct a square equal in area to a given circle.
Ferdinand von Lindemann proved that π is transcendental in 1882, he based his proof on the works of two other
mathematicians: Charles Hermite and Euler. In 1873, Hermite proved that the constant e was transcendental.
Combining this with Euler's famous equation e^(i* π)+1=0, Lindemann proved that since e^x+1=0, x is
required to be transcendental. Since it was accepted that i was algebraic, π had to be transcendental in order to
make i* π transcendental.
1949 a computer was used to calculate π to 2000 places.
Oughtred in 1647 used the symbol
David Gregory (1697) used
for the ratio of the diameter of a circle to its circumference.
for the ratio of the circumference of a circle to its radius.
The first to use π with its present meaning was a Welsh mathematician William Jones in 1706 when
he states "3.14159... = π".
Euler adopted the symbol in 1737 and it quickly became a standard notation.
Webster's Collegiate Dictionary defines π as "1: the 16th letter of the Greek alphabet... 2 a: the symbol pi
denoting the ratio of the circumference of a circle to its diameter b: the ratio itself: a transcendental number
having a value to eight decimal places of 3.14159265"
Today Pi is known to more than 10 billion decimal places.
Connections to Middle School Curriculum:
What is Pi?
This is a cool lesson. Basically, you just need a collection of jar lids, of varying sizes, some string, scissors,
rulers, and calculators. I also use chart paper to post and analyze the data.
I have the students work in pairs. They each gather a few jar lids (you could even limit it to one). The more lids
you have them measure, the larger the sampling size (another cool concept), and the more accurate the
results.
Then, I ask them to respond in their logs to the follow questions: What is pi? Tell me everything you know.
Then we share their thoughts at this point. Then, I give them these directions for each jar lid. I caution them to
be as accurate as possible.
1. cut a piece of string equal to the circumference of your lid
2. cut a piece of string equal to the diameter of your string
3. Measure each.
4. Figure out how many diameter strings it would take to equal your
circumference string in length. (C divided by d, but I don't tell them
that). GIve your answer in decimal form
5. Post your answer on the chart when you are done.
Those who finish early can always measure another lid to add to the data base.
Then, when everyone is done, ask them what they notice about the data.
Analysis
They'll mention that most are in the ballpark. You can talk about why some aren't; what might have happened.
Then, you can allow students to check their data and re--post.
Then, average the numbers. You should come up with a figure very close to pi, 3.14.
Students may decide during analysis to throw out the highest and lowest numbers. That's one way to deal with
the data. I prefer to have them check their measurements and change them if THEY feel it’s appropriate.
When we are done, we talk again about why everyone didn't come up with exactly the same comparison, since
pi IS 3.14. (Accuracy of cutting the string, stretching the string, reading the measurement, etc.)
Finally, I ask them to write what they know about pi now and how it might be useful. From now on, when we
use pi in a formula, they know what it stands for.
Lesson 2
Because pi is 3.14159…, some schools hold their big Pi Day celebrations on 3/14 at exactly 1:59 p.m.
Pi Day Lessons: http://ck022.k12.sd.us/specialevents/piday.htm
Pre-Calculus
Geometry
Algebra I
Trigonometry
Algebra 1/2
Algebra II
Why I chose this question: I was interested in the history of π, I was taught that it was a “magic” number and
simply to memorize it because it shows up all the time.
References
MacTutor: “A History of Pi”, Article by: J J O'Connor and E F Robertson, Retrieved June 12, 2010:
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_through_the_ages.html
Pi Day http://ck022.k12.sd.us/specialevents/piday.htm
The Math Forum @ Drexel: “About Pi”, Retrieved June 10, 2010: http://mathforum.org/
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, ed. by Victor J. Katz.
Princeton University Press, 2007
Your piece of the π, Retrieved June 13, 2010: http://library.thinkquest.org/C0110195/uses/uses.html
Weaver, Douglas: The History of Mathematical Symbols, Retrieved on June 15, 2010, from:
http://www.roma.unisa.edu.au/07305/symbols.htm
What Is Pi? “A Cool Lesson” From: Dr. Mavis Kelley, [email protected]. Retrieved June
15, from: http://www.middleweb.com/INCASEpi.html