GROUP MEMBERS

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.

What is pi??

A number can be placed into several categories based on its properties. Is it prime or composite? Is it imaginary or real? Is it transcendental or algebraic? These questions help to define a number's behaviour in different situations. In order to understand where π fits in to the world of mathematics, one must understand several of its properties: π is irrational and π is transcendental. Another important concept to understand is that of how π is calculated and how the methods have changed over time.

π is:-

"1: the 16th letter of the Greek alphabet... 2 : the symbol pi denoting the ratio of the circumference of a circle to its diameter.”

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.

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.

~=

The early Babylonians andHebrews used “3”as a valuefor Pi.

Later, Ahmed, an Egyptianfound the area of a circle .

Down through the ages,countless people havepuzzled over this samequestion, “What is Pi?

The Greeks found Pi to berelated to cones, ellipses,cylinders and other geometricfigures.

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)+...) x- 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 +...

THE DISCOVERY OF PI

Who? When? Discovery Equivalence

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

Babylonians 2000 BC 3 1/8 3.125

Indians 2000 BCsquare

root of 103.16227766...

Archimedes 250 BC 22/7 3.14128…

Computers Today Pi 3.1415926535…

Currently the value of pi is known to 6.4 billion places!

Pi slept for nearly 1500 years

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 Archimedes

Archimedes 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

Pi in everyday life

We use it for Drawing, machining, plans, planes, buildings, bridges, geometry problems, radio, TV, radar, telephones, estimation, testing, simulation, global paths, global positioning, space science, orbit calculation, Space ships, satellites, Speedometers at vehicles… etc.

Pi is used by every career whether you are a electrical engineer, statistician, biochemist, or physicist. Pi is indeed a necessity for life.

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.

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 facts

”Pi Day” is celebrated on March 14. The official celebration begins at 1:59 p.m. to make an appropriate

3.14159 when combined with the date.

Albert Einstein was born on Pi Day (14/3/1879) in Ulm Wurttemberg, Germany.

Pi goes on for ever

Decimals have no pattern and don’t repeat.

.

Pi Pie at Delft University

HOW TO FIND PI

The circumference of the circle is smaller than the perimeter of the outer hexagon, and bigger than the perimeter of the inner hexagon.

The perimeter of the inner hexagon is exactly 6 radii.

The perimeter of the outer hexagon is harder to work out.

The length of each side is 2R Tan 30O

The length of each side is 2R Tan 30O

So, the perimeter of the outer hexagon is

6 x 2R Tan 30O = 12R Tan 30O

6R < Circumference < 12RTan 30O

C = 2 p R

3 < p < 3.4

sides p min p max6 3.000000 3.46410212 3.105829 3.21539024 3.132629 3.15966048 3.139350 3.14608696 3.141032 3.142715192 3.141452 3.141873384 3.141558 3.141663768 3.141584 3.1416101,536 3.141590 3.1415973,072 3.141592 3.1415946,144 3.141593 3.141593

In general, for an n-sided polygon inside the circle and another one outside of the circle,

n x Sin (180/n) < p < n x Tan (180/n)

Summary

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

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

The mathematical constant e is the base of the natural logarithm (ln(x) or loge(x)).

It is often called Euler's number, due to the related and extensive discoveries of Euler. The exact reasons why Euler himself started to use the letter e for the constant are unknown, but it may be because it is the first letter of the word "exponential" rather than of his name, because he was a very modest man.

Euler's numbere＝2,7182818284590452...

History

Jacob Bernoulli, another famous Basel mathematician, had defined the number e as that he estimated between 2 and 3.

He was conscious of the unproved equivalence with the 2 other formulas:

Mathematical properties

The exponential function f(x) = ex is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive:

Euler's formula of complex analysis

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "the most beautiful equation."

The comple x analysis equation eix = cos + i sinx is

called Euler's identity. The special case with x = π is the famous Euler's formula:

e in physics and engineering

Complex numbers are used in almost every field of physics and engineering.

Euler's formula is widely used in quantum mechanics, relativity.

In electrical engineering, signals that vary periodically over time are often described as a combination of sine and cosine functions, and are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula. The Fourier series are based on it.

In fluid dynamics, Euler's formula is used to describe potential flow. Concrete examples of problems include the calculation of optimal shapes of transport vehicles, of energy generators, of boat turbines, the simulation of chemical/physical processes, the weather forecast, etc.

In differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity is an easy consequence of Euler's formula.

e in finance

e is also the amount of money if you deposit 1 Swiss Frank at a continuously compounded interest rate of 100%: the terminal value of an interest compounded m times per annum is (1 + 100%/m)m , and it tends to e=2.71828... if m →∞.

e in operations research:the "optimal stopping problem"

e in probability and statistics

The normal distribution ,describing measurement errors and that is used daily in statistics, was developed based on certain discoveries of euler.

The function f(x) = ex

and its derivative

Euler made very important contributions to complex analysis, with the e number. He discovered what is now known as Euler's formula, i.e. that for any real number x, the complex exponential function satisfies

$2,718,281,828Google in 2004 announcedits intention to raise$2,718,281,828 stock.

eix = cos + i sinx

FORMULAS RELATED TO E

The natural log at (x-axis) e, ln(e), is equal to 1

The area between the x-axis and the graph y = 1/x, between x = 1and x = e is 1.

WAYS TO SHOW E:

E SONG

REFERENCES

https://en.wikipedia.orghttps://www.youtube.comPLANE GEOMETRY 10http://euler-2007.chhttps://en.wikipedia.org/wiki/Euler