MathematicsA Project on Pi

Certificate of Accomplishment

This is to certify that Manul Goyal, a student of class X Sarabhai has successfully completed the project on the topic A Project on Pi under the guidance of Amit Sir (Subject Teacher) during the year 2015-16 in partial fulfillment of the CCE Formative Assessment.

Signature of Mathematics Teacher

Contents

Introduction

Whatis

Pi?History

Calculating Pi

Historyof

Calculating

Pi

Buffon’s Needle Method

Properties of Pi

DigitsofPi

IrrationalityofPi

Facts about Pi

Piand

the Areaof Circles

Piin

Real Life

IntroductionPi is the ratio of the circumference of a circle to the diameter.

What is Pi (π)?› 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.

› For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159, but if you want a better approximation you can use a computer to get it. Here’s pi to many more digits: 3.14159265358979323846.

› The area of a circle is pi times the square of the length of the radius, or “pi r squared”:

History› Pi is a very old number. 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.

Community Approximation of piBabylonians

Egyptians

› These approximations were slightly less accurate and much harder to work with.

Continued…› The modern symbol for pi [π] was first used in our modern sense

in 1706 by William Jones, who wrote:There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as to

› Pi (rather than some other Greek letter like Alpha or Omega) was chosen as the letter to represent the number 3.141592… because the letter π in Greek pronounced like our letter ‘p’ stands for ‘perimeter’.

Calculating Pi2 Pi in radians form is 360 degrees. Therefore Pi radians is 180 degrees and 1/2 Pi radians is 90 degrees.

History of Calculating PiA

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Archimedes (250 B.C.)› The first mathematician to calculate pi with reasonable accuracy

was Archimedes, around 250 B.C., using the formula:

› Using the area of a circle, he approximated pi by considering regular polygons with many sides inscribed in and circumscribed around a circle. Since the area of the circle is between the areas of the inscribed and circumscribed polygons, you can use the areas of the polygons (which can be computed just using the Pythagorean Theorem) to get upper and lower bounds for the area of the circle.

› Archimedes showed in this way that pi is between and .

› The same method was used by the early 17th century with polygons with more and more sides to compute pi to 35 decimal places.

› Van Ceulen did the biggest 674 calculations.

The area of the circle lies between the areas of the

inscribed and the circumscribed octagons.

CONTINUED…

Newton and Leibnitz (late 17th century)› When Newton and Leibnitz developed calculus in the late 17th

century, more formulas were discovered that could be used to compute pi.

› For example, there is a formula for the arctangent function:

If you substitute and notice that is , you get a formula for pi.

Machin’s Formula (early 18th century)› Using the arctangent function to calculate pi is not useful because

it takes too many terms to get any accuracy, but there are some related formulas that are very useful.

› The most famous of this is Machin’s formula:

› This formula and similar ones were used to push the accuracy of approximations of pi to over 500 decimal places by the early 18th century (this was all hand calculation!)

The 20th Century› In the 20th century, there have been two important developments:

– the invention of electronic computers– the discovery of much more powerful formulas for pi

› In 1910, the great Indian mathematician Ramanujan discovered the following formula for pi:

› In 1985, William Gosper used this formula to calculate the first 17 million digits of pi.

Electronic Computers (since the 20th century)› From the mid-20th century onwards, all calculations of pi have

been done with the help of calculators or computers.› George Reitwiesner was the first person who used an electronic

computer to calculate pi, in 1949, on an ENIAC, a very early computer. He computed 2037 decimal places of pi.

› In the early years of the computer, an expansion of pi to 100,000 decimal places was computed by Maryland mathematician Daniel Shanks and his team at the United States Naval Research Laboratory in Washington, D.C.

› The first 100,265 digits of pi were published in 1962.

Continued…› In 1989, the Chudnovsky brothers correctly computed pi to over

1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan’s infinite series of pi:

› In 1999, Yasumasa Kanada and his team at the Univesity of Tokyo correctly computed pi to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP using another variation of Ramanujan’s infinite series of pi. In October 2005 they claimed to have calculated it to 1.24 trillion places.

The 21st Century – Current Claimed World Record› In August 2009, a Japanese supercomputer called the T2K Open

Supercomputer was claimed to have calculated pi to 2.6 trillion digits in approximately 73 hours and 36 minutes.

› In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of pi in a total of 131 days.

› In August 2010, Shigeru Kondo used Alexander Yee’s y-cruncher to calculate 5 trillion digits of pi. The calculation was done between 4 May and 3 August. In October 2011, they broke their own record by computing ten trillion (1013) and fifty digits using the same method but with better hardware.

› In December 2013 they broke their own record again when they computed 12.1 trillion digits of pi.

Deriving Pi: Buffon’s Needle Method› Buffon's Needle is one of the oldest problems in the field of

geometrical probability. It was first stated in 1777.› It involves dropping a needle on a lined sheet of paper and

determining the probability of the needle crossing one of the lines on the page.

› The remarkable result is that the probability is directly related to the value of pi.

The following pages will present an analytical solution to the problem.

THE SIMPLEST CASE

Let's take the simple case first. In this case, the length of the needle is one unit and the distance between the lines is also one unit. There are two variables, the angle at which the needle falls (θ) and the distance from the center of the needle to the closest line (D). Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the center to the closest line can never be more that half the distance between the lines. The graph alongside depicts this situation.

The needle in the picture misses the line. The needle

will hit the line if the closest distance to a line (D) is less

than or equal to 1/2 times the sine of theta. That is, D ≤

(1/2)sin(θ). How often will this occur?

CONTINUED…

The shaded portion is found with using the definite integral of (1/2)sin(θ) evaluated from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(π) or π/2. So, the probability of a hit is 1/(π/2) or 2/π. That's approximately .6366197.To calculate pi from the needle drops, simply take the number of drops and multiply it by two, then divide by the number of hits, or 2(total drops)/(number of hits) = π (approximately).

In the graph above, we plot D along the ordinate and

(1/2)sin(θ) along the abscissa. The values on or below the curve represent a hit (D ≤

(1/2)sin(θ)). Thus, the probability of a success it the ratio shaded area to the entire

rectangle. What is this to value?

The Other CasesThere are two other possibilities for the relationship between the length of the needles and the distance between the lines. A good discussion of these can be found in Schroeder, 1974. The situation in which the distance between the lines is greater than the length of the needle is an extension of the above explanation and the probability of a hit is 2(L)/(K)π where L is the length of the needle and K is the distance between the lines. The situation in which the needle is longer than the distance between the lines leads to a more complicated result.

RECORD APPROXIMATIONS OF PIGiven alongside is a graph showing the historical evolution of the record precision of numerical approximations to pi, measured in decimal places (depicted on a logarithmic scale; time before 1400 is not shown to scale).

Properties of PiThe 'famous five' equation connects the five most important numbers in mathematics, viz., 0, 1, e, π, and i: .

Digits of Pi› Pi is an infinite decimal, i.e., pi has infinitely many numbers to

the right of the decimal point. If you write pi down in decimal form, the numbers to the right of the 0 never repeat in a pattern. Although many mathematicians have tried to find it, no repeating pattern for pi has been discovered - in fact, in 1768 Johann Lambert proved that there cannot be any such repeating pattern.

› As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) pi is irrational: it cannot be written as a fraction (the ratio of two integers).

Irrationality of PiHere's a proof of the irrationality of Pi from Robert Simms:Theorem: Pi is irrational.Proof: Suppose , where p and q are integers. Consider the functions defined on [0, π] by

Clearly for all n. Let denote the mth derivative of . Note that for or for ; otherwise some integer

Continued…

By repeatedly applying integration by parts, the definite integrals of the functions can be seen to have integer values. But are strictly positive, except for the two points 0 and pi, and these functions are bounded above by for all sufficiently large n. Thus for a large value of n, the definite integral of is some value strictly between 0 and 1, a contradiction.

Facts about PiPi day is celebrated on March 14 at the Exploratorium in San Francisco (March 14 is 3/14).

PI AND THE AREA OF CIRCLES

› The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter.

› The diameter of a circle is twice as long as the radius ().

› We use the Greek letter π to represent the ratio of the circumference of a circle to the diameter. For simplicity, we use .

› The formula for circumference of a circle is: , i.e., .Parts of a circle.

CONTINUED…

Square units with area 1 cm2 each inside a circle.

› The area of a circle is the number of square units inside that circle.

› If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2.

› However, it is easier to use one of the following formulas:

or where is the area, and is the radius.

Pi in Real Life

› Pi is used in areas ranging from geometry to probability to navigation. Common real-world application problems involve finding measurements of circles, cylinders, or spheres, such as circumference (one-dimensional), area (two-dimensional), or volume (three-dimensional). Students can look for real-life applications of knowing measurements of objects in these shapes.

CONTINUED…

› For example, if a student would decide to build a robot, he might want to be able to program the robot to move a certain distance. If the robot is made with wheels that have a three-inch diameter, how far would the robot move with each complete rotation of the wheels? This would be a circumference problem, as the robot would move the length of the circumference with each rotation of its wheels – in this case, 5π, or approximately 15.7 inches.

Applications of Pi.

A Vote of Thanks

Submitted By:πManul GoyalπRoll No. 20πClass X

17th June’15 MLZS