Before I talk about the history of Pi I want to explain what Pi is. Webster's Collegiate Dictionary defines Pi 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" 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 define a number's behavior in different situations. In order to understand where Pi fits in to the world of mathematics, one must understand several of its properties pi is irrational and pi is transcendental. A rational number is one that can be expressed as the fraction of two integers. Rational numbers converted into decimal notation always repeat themselves somewhere in their digits. For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0.142857142857…, a repetition of six digits. However, the square root of 2 cannot be written as the fraction of two integers and is therefore irrational. For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number. The first attempt at a proof was by Johaan Heinrich Lambert in 1761. Through a complex method he proved 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. Many people saw Lambert's proof as too simplified an answer for such a complex and long-lived problem. In 1794, however, A. M. Legendre found another proof which backed Lambert up. This new proof also went as far as to prove that Pi^2 was also irrational. In the long history of the...

...The History of Pi
The History of Pi
Most individuals who have a general mathematical education that touches on the basics of geometry commonly know pi. The definition of pi is the ratio of the circumference to the diameter of the circle (Smoller, 2001). The majority of the population, however, does not realize the history associated with the symbol, which not only spans throughout the centuries but throughout the millenniums. The Babylonians, Egyptians, Archimedes of Syracuse, Leonardo of Pisa, Francois Viete, Leonhard Euler, Asian mathematicians such as Liu Hiu, Tsu Ch’ung-Chih, Arya Bhatta, Gottfried Leibniz, Isaac Newton, William Jones, John Machin. George Buffon and Srinivasa Ramanujan, have all played a role in the enriched past of this important mathematical symbol.
The ancient Babylonians dates back to the 18th century BCE and reigned in Mesopotamia. The Babylonia, even though it declined drastically in the 17th century, existed until 539 when the Persians consumed Babylonia (Kjeilen, 2009). During this time, they made magnificentstructures with archways that held religious emphasis. The Babylonians used a developed mathematical system, which included six as the root number as opposed to 10 which are commonly used today (Kjeilen, 2009). Even though the Babylonians has a variation on their mathematical system, they calculated the area of a circle by taking three...

...What is π?
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"
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 define a number's behavior 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.
The History of π
In the long history of the number π, there have been many twists and turns, many inconsistencies that reflect the condition of the human race as a whole. Through each major period of world history and in each regional area, the state of intellectual thought, the state of mathematics, and hence the state of π, has been dictated by the same socio-economic and geographic forces as every other aspect of civilization. The following is a brief history, organized by period and region, of the development of our understanding of the number π.
In ancient times, π was discovered independently by the first civilizations to begin agriculture. Their new sedentary life style first freed up time for mathematical pondering,...

...History of Pi
There are many people who have discovered and proved what pi is. As time goes on people discover more and more of the seemingly random numbers. Four of the people who proved pi are the Liu Hui, Archimedes of Syracuse, James Gregory, and the Bible.
The first proof I will be talking about is Liu Hui’s. Liu Hui was a Chinese mathematician whose method for proving pi was to find the area of a polygon inscribed in a circle. When the number of sides on the inscribed polygon increased, its area became closer to the circumference of a circle and pi. For finding the side length of an inscribed polygon Liu Hui used a simple formula. (13Ma3)
To find the side length of an inscribed polygon of 2n sides, if the side length of a polygon with n sides is known he used the following formula:
In this formula k stands for a temporary variable, and Sn stands for the side length of an inscribed polygon with n sides. (13Ma3)
We will start with a hexagon inside of a circle. The radius of the circle is one, the area is pi. The side length of the hexagon is 1. To calculate the next k value, all we need to do is do an addition and a square root like in the following:
The area of a regular polygon is A=1/2nsa. The n stands for number of sides, s stands for side length, and a stands for apothem. As the number of sides increases, the apothem becomes closer and closer to...

...Paper - I
1. Sources: Archaeological sources:Exploration, excavation, epigraphy, numismatics, monuments Literary sources: Indigenous: Primary and secondary; poetry, scientific literature, literature, literature in regional languages, religious literature. Foreign accounts: Greek, Chinese and Arab writers.
2. Pre-history and Proto-history: Geographical factors; hunting and gathering (paleolithic and mesolithic); Beginning of agriculture (neolithic and chalcolithic).
3. Indus Valley Civilization: Origin, date, extent, characteristics, decline, survival and significance, art and architecture.
4. Megalithic Cultures: Distribution of pastoral and farming cultures outside the Indus, Development of community life, Settlements, Development of agriculture, Crafts, Pottery, and Iron industry.
5. Aryans and Vedic Period: Expansions of Aryans in India. Vedic Period: Religious and philosophic literature; Transformation from Rig Vedic period to the later Vedic period; Political, social and economical life; Significance of the Vedic Age; Evolution of Monarchy and Varna system.
6. Period of Mahajanapadas: Formation of States (Mahajanapada): Republics and monarchies; Rise of urban centres; Trade routes; Economic growth; Introduction of coinage; Spread of Jainism and Buddhism; Rise of Magadha and Nandas. Iranian and Macedonian invasions and their impact.
7. Mauryan Empire: Foundation of the Mauryan Empire, Chandragupta, Kautilya and Arthashastra; Ashoka;...

...The History of Pi
Throughout the history of mathematics, one of the most enduring challenges has been the calculation of the ratio between a circle's circumference and diameter, which has come to be known by the Greek letter pi. From ancient Babylonia to the Middle Ages in Europe to the present day of supercomputers, mathematicians have been striving to calculate the mysterious number. They have searched for exact fractions, formulas, and, more recently, patterns in the long string of numbers starting with 3.14159 2653..., which is generally shortened to 3.14. William L. Schaaf once said, "Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi" (Blatner, 1). We will probably never know who first discovered that the ratio between a circle's circumference and diameter is constant, nor will we ever know who first tried to calculate this ratio. The people who initiated the hunt for pi were the Babylonians and Egyptians, nearly 4000 years ago. It is not clear how they found their approximation for pi, but one source (Beckman) makes the claim that they simply made a big circle, and then measured the circumference and diameter with a piece of rope. They used this method to find that pi was slightly greater than 3, and came up with the value 3 1/8 or 3.125 (Beckmann, 11). However, this theory is...

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...
Today one of the most cherished ideologies of America is the fact that everyone is and should be created equal. With this cherished ideology bringing a sense of pride and diversity to America we must keep in mind that this cherished ideology did not always exist. Since 1865 various individuals and groups have not been able to receive and express their rights to full equal status in the United States. These different individuals and groups have seemingly fought for their rights in equality and have become pioneers in the fight for evolution for equality.
In 1865 African Americans in the United States under the 13th amendment were freed from the terrible burden of slavery. Through the 14th amendment they were given the right to citizenship and the right to equal protection. The 15th amendment gave them the right to vote regardless of their skin color race or any other type of servitude. These amendments were meant to be enforced and make a serious change in the everyday life of the average American.
With these amendments passing in 1865 they were meant to make a serious change towards the evolution of equality. These changes did not seem to happen right away and African Americans were still not being treated with equality. The average African American at this time were being denied there newly given rights every day making life extremely hard to stay...

...Name:
Date:
Graded Assignment
Final Exam Part 2
I. Map
On this world map, indicate the following features:
Amazon Rainforest
Panama Canal
The Himalayas
The Ring of Fire
The Mississippi River
The Gobi Desert
(10 points)
II. Graphic Organizer
Fill in the table below about these five major world religions. Do not fill in the shaded boxes.
(10 points)
Religion
Name at least
one Holy Text
How do you achieve
enlightment?
Describe their view about the afterlife.
Hinduism
Bhagvada Gata
Do good deeds to get good karma until you break the samsara or cycle of reincarnation and reach enlightenment
Buddhism
Believe the Four Truths are true and real, follow the Eightfold Path, meditation is one of the major steps to reach enlightenment
Judaism
Old Testament
God promised the Jews, people of Israel, paradise and those who hate the Jews and mistreat them are going to go to Hell
Christianity
New Testament
Islam
Quran
People who believe in all the five pillars and do them and do righteous deeds go to heaven while the disbelievers and those who sin are punished and go to Hell
III. Short Answer
1. Explain the role of river valleys in the development of civilizations. Name at
least two river valleys as examples. (10 points)
Rivier valleys first and foremost provided water, a basic necessity for humans. It also provided fertile soil for agriculture, which led to settlements and brought hunting and gathering to an end....

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