The Origin and History of Pi

By Pritish Avilash NayakClass-IX Section-A Roll no.-32

Objectives

Study the history of pi.Learn the meaning of pi.Examine the development of pi along with

the development of mathematics.Evaluate some of the different formulas

that pi is used in.Explore professions that use pi.

Intro to pi

Ever wondered what that symbol means on your calculator. It is the first Greek letter of the Greek work meaning “perimeter”. It turns out that it is a very important symbol.

Pi = 3.1459.

Some of you may already know that pi is used in a lot of different mathematical formulas.

As you study pi keep in mind that for thousands of years people had to calculate pi by hand. This took an incrediably long time.

What is Pi?• Let's look at what

Pi really is. Some declare that Pi is an edible dessert, usually circular, consisting of something sweet enclosed within a baked crust.

About Pi

• Pi is defined as the ratio of the circumference to the diameter of a circle

• Pi is usually denoted with this greek symbol:

ππ =

circumferencediameter

Pi=3.14159

Pi is the mathematical constant whose value is the ratio of a circle’s circumference to its diameter.

Circumference = The distance around a circle

Diameter = The width of a circle.

Some people became famous by discovering ways to calculate Pi

LEIBNITZ (1671) Pi= 4(1/1-1/3+1/5-1/7+1/9-1/11+1/13+...) WALLIS Pi= 2(2/1*2/3*4/3*4/5*6/5*6/7*...) MACHIN (1706) Pi=16(1/5- 1/(3+5^3) +1/(5+5^5) -1/(7+5^7)+...) -4(1/239 -1/(3*239^3) + 1/(5*239^5)-...) SHARP (1717) Pi= 2*Sq.Rt(3)(1-1/3*3 + 1/5*3^2 - 1/7*3^5...) EULER (1736) Pi= Sq.Rt(6(1+1/1^2+1/2^2+ 1/3^2...)) BOUNCKER Pi= 4 --- 1+1 --- 2+9 --- 2+25 +...

Pi and “e” are irrational Pi is one of the longest numbers ever computed,

second only to “e” another IRRATIONAL number with a value of 2.718281828459045 --

It never repeats like the decimal values of 1/3=.33333… or 5/7=.7142857142857…

Visit http://www.sciboard.louisville.edu to see if you can find a computer program that calculates Pi or e.

History of Pi

• The hunt for pi began in Egypt and in Babylon about two thousand years before Christ. The Egyptians obtained the value (4/3)4 and the Babylonians the value 3 1/8 for pi. About the same time, the Indians used the square root of 10 for pi. These approximations to pi had an error only as from the second decimal place.

Pi slept for nearly

1500 years

Analytical Geometry and Calculus

During the 17th century, analytic geometry and calculus were developed. They had a

immediate effect on Pi. Pi was freed from the circle! An ellipse has a formula for its area which involves Pi (a fact known by the Greeks); but this is also true of the sphere, cycloid arc, hypoclycloid, the witch, and many other curves.

RESEARCH ON YOUR OWN

Look in the Mathematics Teacher for articles on Pi. Learn to calculate Pi using

the methods described.

Using license plates

Buffoon's Needle and Monte Carlo Techniques

Calculations of Pi continued

One of the major chapters in calculus deals with infinite series. Several such series have been discovered which approximate Pi. These discoveries and the conjuncture that Pi is a transcendental number led others to compute Pi to more places Vega 1794 137 digits Dase 1844 201 Rutherford 1853 441 Shanbis 1873 707 (only 527 were correct)

Who Discovered Pi?

• Archimedes was the first to theoretically approximate pi

• He calculated that pi was “trapped” between 223/71 and 22/7, or roughly 3.1428

• Today we use better approximations, most of which are derived by computers

Hi, I’m Archimedes… Pi = Between 223/71 and 22/7

The Death of ArchimedesArchimedes was born in 287 BC in a Greek state called Syracuse, Sicily.

The city of Syracuse was taken over by the Romans and Archimedes was killed.

It is said that he was busy drawing circles in the dust and writing mathematical equations at the time of his death

THAT’S HOW IMPORTANT PI WAS TO ARCHIMEDES!!!!!

THE DISCOVERY OF PIWho? When? Discovery Equivalence

Egyptians 2000 BC (4/3)4 3.160493827...

Babylonians 2000 BC 3 1/8 3.125

Indians 2000 BC square root of 10

3.16227766...

Archimedes 250 BC 22/7 3.14128…

Computers Today Pi 3.1415926535…Currently the value of pi is known to 6.4 billion places!

How Pi was developed.

The history of pi is complicated. No one “invented” pi. It was discovered rather than made. The use of pi developed as the use of mathematics developed.

Ancient Egyptians and Greeks knew the ratio of the circumference to the diameter of a circle is the same for all circles and is slightly higher than 3.

Though the estimate of ancient civilizations was close to pi it was not until Archimedes, a Greek mathematician and physicist, who more accurately calculated pi.

How Pi was developed. (con’t) Archimedes found that if he drew circles in polygons and

calculated the inner and outer polygon’s perimeters and calculated the values that the estimate of pi was 3.1419.

The second major advancement in understanding pi came with the development of calculus.

Polygon=geometric figure with three or more sides or angles.

How Pi was made. (con’t)Isaac Newton was able to calculate 15 decimals in the pi series, it is said that Newton spoke about his work with pi saying he was ashamed how long it took him just to calculate the 15 digits without working on anything else.

How Pi was made. (con’t) In 1706 John Machin developed a converging series for pi that

calculated up to 100 decimal places.

By the year 1949 digital computers were invented and could calculate 100’s of decimal places of pi in hours. In 1949 John von Nuemann calculated 2037 digits of pi in 70 hours.

The Usefulness of Pi

• Pi is extremely useful in calculating the area and circumference of a sphere: A = πr2 and C = 2πr.

• Many disciplines of science use π in their equations to describe the world

• In fact DNA, rainbows, the human eye, music, color, and ripples all have some natural roots in pi.

Using Pi

Pi has mathematical applications in most professions.

Can you think of situations where using pi might be important to you?

As you probably already know you use pi for finding the Area and Circumference of a circle.

Using Pi (con’t)

Other applications of pi are finding the volume of something.

Volume = The amount of 3-D space an object takes up.

Everyone in the world at some time has used pi.

Agriculturalist, engineers, architects, and construction workers are just a few of the professions that use pi extensively.

AREA Area=pi x the radius squared.

The radius of a circles is the distance from the center most point to the edge of that circle .

For example. Tim the architect has to build a circular column that has a radius of 5 feet. What would be the area of the entire column?

To calculate this Tim would take pi x 5 squared

3.14 x 5 squared = 78.5

Therefore the area of the circular column is 78.5 feet.

<= radius

Circumference Circumference is the distance around the outside of the circle.

To find distance we would use the formula circumference= pi x diameter

As you know diameter is the width of a circle.

For example Jim the engineer needs to design a metal tube that has a diameter of 15 feet.

Jim needs to find the circumference of the circle in order to make a metal tube to fit around it. Jim would use the formula c=pi x d

Circumference = 3.14 x 8 = 25.12

The circumference Jim is looking for is 25.12 feet.

Volume To find volume we would use the formula volume = pi x radius

squared x height.

Tom the agriculturist needs to know the volume of a dipping vat for his cows to safely and comfortably fit in.

Before he can calculate he needs to know the radius and the height of the structure before he can begin building.

Tom measures the radius at 5 feet and the height at 9 feet.

Therefore Tom’s formula would look like Volume=3.14 x 5 squared x 9

3.14 x 5 squared x 9 = 706.5 feet cubed

DIAMETER AND RADIUS

• The distance across a circle through its center is called its diameter, D.

• The radius, R of a circle is the distance from the center of a circle to a point on the edge of the circle.

• So a circle's diameter is twice as long as its radius: D = 2 × R.

diameter, D

radius, R

radius

diameter

•The distance around a circle is its circumference

diameter ncecircumfere

π = circumference

diameter

circumference2 x radiusπ =

Pi = circumference divided by diameter

Every time you use these formulas, you will come up with

3.14159265…

area, A

2 area r

•The space inside a circle is called the Area

APPROXIMATELY

~=The symbol means

When speaking of pi we always use ~=Example: π ~= 3.14

LET’S REVIEW THE FORMULAS!

2 area r

π =

circumferencediameter

diameter ncecircumfere

SOME NAMES FOR PI

3.14~= ~= 227

___

pi π

Hope you enjoyedlearning about Pi!!!

What is the record today? Around 1950, Borel noted that numbers like the Square Roots of 2, 3, etc. appear to be a mere jumble of digits, but on the average each digit appears a fixed fraction of the time. (Some people say this is characteristic of a random set of numbers. Do the digits of Pi occur randomly?)

Such number are called 'normal.' With computers widely available the race was on again!

1950 Eniac in 70 hours produced 2,036 digits 1954 More in 13 minutes produced 3,093 digits 1959 IBM 708 in 1 hr 40 min produced 10,000 digits 1959 Pegasus produced > 100,000 digits

PI IN THE PROFESSIONS.

Agricultural professionals may use pi to determine the area covered by a pivot irrigation system or storage facility. The would use the formula

Architects and construction works would both use the formula for Area extensively. They also use the formula for volume extensively to fill columns of concrete and to know the space a building would take up.

PI IN THE PROFESSIONS. (CON’T)

Engineers use advance formulas that include pi.

These are just some of the formulas an engineer would use.

Moving around structures such as landmasses and buildings would require the use of some of these formulas.

Engineers would probably use pi more than other professions.

SUMMARY

Pi is the mathematical constant whose value is the ratio of a circle’s circumference to its diameter.

The use of pi dates back to 1900 BC with ancient Egypt and Greece.

It has taken 100’s of different algorithms to help estimate pi’s number because it is a repeating decimal.

Pi has been developed along with the development of mathematics.

There are many different applications for pi.

Many different professionals such as engineers, agriculturalist, and construction workers use pi.

Pi day is celebrated on 3/14 of every year because pi = 3.14