A beautiful presentation describing the history of pi and its use and application in real life situations. It also covers calculating pi and world records about the number of digits of pi that have been calculated. Hope you enjoy and use it!!
Pi has been studied for millennia. Ancient texts like the Bible contained approximations of pi. Archimedes was the first to calculate pi theoretically, determining that 223/71 is less than pi and pi is less than 22/7. It has since been calculated to trillions of digits. Pi is vital because the circumference of any circle is equal to its diameter multiplied by pi, so without pi we could not calculate properties of spherical objects like the Earth. Mathematicians continue striving to more precisely calculate pi.
π (pi) is the ratio of a circle's circumference to its diameter. It is an irrational and transcendental number represented by the Greek letter π. William Jones is believed to have first used π in its modern sense in 1706 to represent the constant ratio, rather than a varying circumference. Important formulas using π include the circumference of a circle (2πr), area of a circle (πr^2), and volume/surface area of a sphere. While π cannot be expressed exactly as a fraction, it is commonly approximated as 22/7 or 3.14159.
The document discusses the history and properties of pi (π). It describes how various ancient civilizations like the Babylonians, Hebrews, and Egyptians approximated pi. The Greeks studied pi's relationship to circles, cones, and cylinders. Over centuries, mathematicians like Machin, Euler, and Lambert improved approximations of pi and proved its irrationality. With modern computers, pi has been calculated to extreme accuracy. The document also notes how pi is
This document provides an overview of coordinate geometry. It defines key concepts like the Cartesian coordinate system, quadrants, and using an ordered pair (x,y) to locate points on a plane. It then explains how to calculate the distance between two points using the distance formula. Other topics covered include finding the area of a triangle using the coordinates of its vertices, using the section formula to divide a line segment internally, and finding the midpoint of two points.
This document provides information about pi (π), including:
- Pi is a mathematical constant that is the ratio of a circle's circumference to its diameter. It is an irrational and non-terminating number with a value of approximately 3.14159.
- The history of pi discusses how European mathematicians developed formulas to calculate pi more accurately over time.
- Pi is used in geometric formulas to calculate areas and volumes of shapes like circles, spheres, and cones. Pi day is celebrated on March 14th in honor of the first three digits of pi.
This document discusses the history of Indian mathematics through several prominent mathematicians such as Aryabhata, Bhaskaracharya, Varaha Mihira, and Srinivasa Ramanujan. It notes that while attitudes are slowly changing, Indian mathematical contributions remain neglected or attributed to other cultures. The document aims to address this neglect by discussing several influential Indian mathematicians and their achievements, as well as examining why Indian works were neglected and why this represents an injustice.
Pi has been studied for millennia. Ancient texts like the Bible contained approximations of pi. Archimedes was the first to calculate pi theoretically, determining that 223/71 is less than pi and pi is less than 22/7. It has since been calculated to trillions of digits. Pi is vital because the circumference of any circle is equal to its diameter multiplied by pi, so without pi we could not calculate properties of spherical objects like the Earth. Mathematicians continue striving to more precisely calculate pi.
π (pi) is the ratio of a circle's circumference to its diameter. It is an irrational and transcendental number represented by the Greek letter π. William Jones is believed to have first used π in its modern sense in 1706 to represent the constant ratio, rather than a varying circumference. Important formulas using π include the circumference of a circle (2πr), area of a circle (πr^2), and volume/surface area of a sphere. While π cannot be expressed exactly as a fraction, it is commonly approximated as 22/7 or 3.14159.
The document discusses the history and properties of pi (π). It describes how various ancient civilizations like the Babylonians, Hebrews, and Egyptians approximated pi. The Greeks studied pi's relationship to circles, cones, and cylinders. Over centuries, mathematicians like Machin, Euler, and Lambert improved approximations of pi and proved its irrationality. With modern computers, pi has been calculated to extreme accuracy. The document also notes how pi is
This document provides an overview of coordinate geometry. It defines key concepts like the Cartesian coordinate system, quadrants, and using an ordered pair (x,y) to locate points on a plane. It then explains how to calculate the distance between two points using the distance formula. Other topics covered include finding the area of a triangle using the coordinates of its vertices, using the section formula to divide a line segment internally, and finding the midpoint of two points.
This document provides information about pi (π), including:
- Pi is a mathematical constant that is the ratio of a circle's circumference to its diameter. It is an irrational and non-terminating number with a value of approximately 3.14159.
- The history of pi discusses how European mathematicians developed formulas to calculate pi more accurately over time.
- Pi is used in geometric formulas to calculate areas and volumes of shapes like circles, spheres, and cones. Pi day is celebrated on March 14th in honor of the first three digits of pi.
This document discusses the history of Indian mathematics through several prominent mathematicians such as Aryabhata, Bhaskaracharya, Varaha Mihira, and Srinivasa Ramanujan. It notes that while attitudes are slowly changing, Indian mathematical contributions remain neglected or attributed to other cultures. The document aims to address this neglect by discussing several influential Indian mathematicians and their achievements, as well as examining why Indian works were neglected and why this represents an injustice.
The document provides information about pi (π) in 3 paragraphs. It states that pi is the ratio of a circle's circumference to its diameter, is an irrational number, and has been calculated to over 206 billion digits. The history of pi is discussed, noting it was proven irrational in 1761 and transcendental in 1882. Methods of calculating pi are also summarized, including a supercomputer calculating over 206 billion digits in 1999.
Presentation on famous mathematicians in indiaFabeenaKMP
(1) Aryabhata was a famous Indian mathematician from the classical age who lived in the 5th century AD. Some of his key contributions included a place value numeral system, approximations of pi, and trigonometric formulas.
(2) Brahmagupta was a 7th century Indian mathematician who is known for being the first to use zero as a number and introduce basic algebraic rules and formulas.
(3) Bhaskara was a 12th century mathematician whose main work Siddhanta Shiromani covered topics in arithmetic, algebra, astronomy and advanced mathematics. He made contributions in calculus, arithmetic progressions, and solving indeterminate equations.
Euclid (325-265 BCE) is considered the father of geometry. He organized geometry into a logical system using definitions, axioms, and postulates in his work Elements. Some key ideas are:
- Euclid defined basic geometric terms like points, lines, and planes. He also stated basic axioms about equality and properties of wholes and parts.
- Euclid proposed five postulates, including ones about drawing straight lines and circles. The fifth postulate about parallel lines was controversial and spurred development of non-Euclidean geometries.
- Euclid proved 465 theorems in Elements through deductive reasoning based on the definitions, axioms, and postulates
This document discusses the history and properties of the mathematical constant pi (π). It describes how pi has been calculated and approximated throughout history using different methods, from the ancient Greeks to modern computers. The document also discusses how pi is an irrational number that cannot be expressed as a fraction, and how computing pi to increasing numbers of decimal places has helped test and develop computing technology over time.
Pi is the ratio of a circle's circumference to its diameter. It has been known and studied since ancient times by cultures like the Egyptians and Babylonians, though its precise value was unknown. Pi is represented by the Greek letter π because in Greek, "p" stands for perimeter. The decimal representation of pi goes on indefinitely without repeating in a pattern. It is celebrated on March 14th (3.14) as Pi Day.
Indian mathematicians and their contribution to the field of mathematicsBalabhaskar Ashok Kumar
- Mathematics originated in India as early as 200 BC during the Shulba period, where the Sulba Sutras were developed as part of the Indus Valley civilization.
- During the "golden age" of Indian mathematics between 500-1000 AD, great mathematicians like Aryabhata, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II made significant contributions and advances in many areas of mathematics. Their work spread throughout Asia and influenced mathematics in the Middle East and Europe.
- Aryabhata, in particular, made early approximations of pi and proposed that it is irrational. He also discussed sine, verses, and solutions to indeterminate equations in his
Pi is the ratio of a circle's circumference to its diameter. It is an irrational number that cannot be expressed as a fraction and its value never repeats. Throughout history, mathematicians have worked to calculate Pi to greater levels of precision, advancing from the ancient Babylonians' approximation of 3 1/8 to modern computers calculating Pi to billions of decimal places. Analytic geometry and calculus in the 17th century allowed Pi to be applied to shapes beyond just circles.
This document discusses the history and evolution of geometry. It begins by defining geometry as the measurement of earth and outlines its origins in ancient Egypt and the Indus Valley civilization where it was used to measure land and construct buildings. It then covers important early Greek mathematicians like Thales and Pythagoras and their theorems. Most of the document focuses on Euclid's Elements, outlining his definitions, postulates, axioms and use of deductive reasoning to prove 465 theorems. It also discusses criticisms of Euclid's definitions and the development of non-Euclidean geometry on curved surfaces.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
Aryabhat was an Indian mathematician and astronomer from the 5th century who made early contributions to trigonometry and algebra. He was the first to calculate pi accurately to four decimal places and devised formulas for calculating areas of triangles and circles. Brahmagupta was a 7th century mathematician who wrote the Brahmasphutasiddhanta, covering mathematics such as arithmetic, algebra and trigonometry. Brahmagupta's formula gives the exact area of a cyclic quadrilateral. Bhaskara was an 11th century mathematician who made contributions representing numbers in the Hindu decimal system with a circle for zero and providing rational approximations of sine functions.
Zero originated in ancient India, Babylon, and the Mayan civilization. The concept of zero as a number was first attributed to India in the 9th century CE, where it was treated as any other number in calculations. The symbol and rules for using zero in arithmetic operations were further developed by Indian mathematicians like Aryabhata and Brahmagupta. Their work influenced Arabic mathematicians who helped spread the concept of zero to Europe. While other ancient cultures used placeholder symbols, it was in India that zero was first understood and used as a true number.
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
The document discusses key concepts related to polynomials including:
1) Algebraic expressions, polynomials, monomials, binomials, trinomials, and their definitions.
2) Constants, variables, and the degree of a polynomial.
3) How to classify polynomials based on degree and number of terms.
4) How to rewrite a polynomial in standard form by arranging terms by descending degree.
5) Remainder and factor theorems related to polynomials.
6) Common identities used to factorize polynomials.
Indian mathematics emerged in the Indian subcontinent from 1200 BC until the 18th century. Key classical Indian mathematicians included Aryabhata, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the decimal number system, zero, negative numbers, and trigonometry. These concepts were transmitted to other parts of the world and helped develop mathematics further. Some notable later Indian mathematicians included Bhaskara II, who introduced concepts related to calculus, and Ramanujan, who had a natural genius for mathematics despite a lack of formal education.
The number π is a mathematical constant. Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in the month/day date format) since 3, 1, and 4 are the first three significant digits of 휋. In 2009, the United States House of Representatives supported the designation of Pi Day.
Using pi, it can measure things like ocean wave, light waves, sound waves, river bends, radioactive particle distribution and probability like the distribution of pennies, the grid of nails and mountains by using a series of circles.
This document is a student project on consumer awareness that was compiled by Salahudin habibullah. It includes an acknowledgements section thanking the economic teacher for guidance. The content sections outlines topics covered like a spreadsheet of a consumer survey, analysis of the survey results, the rights and duties of consumers, and recommendations. The survey analyzed consumer behaviors related to checking expiration dates, product ingredients, complaining to stores, awareness of consumer courts, and responses to marketing. It recommends consumers more carefully check product quality and standardization and utilize consumer courts when needed.
1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Some properties of triangles discussed are: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
1. The document discusses properties and congruence of triangles. It defines congruence as two triangles being the same shape and size with corresponding angles and sides equal.
2. There are five criteria for congruence: side-angle-side, angle-side-angle, angle-angle-side, side-side-side, and right angle-hypotenuse-side.
3. Additional properties discussed include isosceles triangles having equal angles opposite equal sides, and relationships between sides and opposite angles/angles and opposite sides in all triangles.
This document discusses Pi Day activities that will take place, including singing Pi Day songs, participating in Pi trivia games and contests, making Pi-themed crafts like bracelets revealing digits of Pi, and learning interesting facts about the number Pi. Students are encouraged to celebrate Pi Day on March 14th (3/14) through interactive games and songs that involve reciting digits of Pi.
This document provides a historical overview of calculations of the mathematical constant pi. It describes how ancient cultures like the Egyptians, Babylonians, and Bible approximated pi as 3. It then discusses how Archimedes was the first to theoretically calculate pi between 223/71 and 22/7. The document traces how mathematicians and scientists from cultures like Greek, Islamic, Indian, and European progressively calculated pi to more decimal places over centuries, with modern computers allowing for thousands of decimal places. It emphasizes how calculating pi more precisely has been an ongoing goal of the scientific community to enable more accurate measurements.
The document provides information about pi (π) in 3 paragraphs. It states that pi is the ratio of a circle's circumference to its diameter, is an irrational number, and has been calculated to over 206 billion digits. The history of pi is discussed, noting it was proven irrational in 1761 and transcendental in 1882. Methods of calculating pi are also summarized, including a supercomputer calculating over 206 billion digits in 1999.
Presentation on famous mathematicians in indiaFabeenaKMP
(1) Aryabhata was a famous Indian mathematician from the classical age who lived in the 5th century AD. Some of his key contributions included a place value numeral system, approximations of pi, and trigonometric formulas.
(2) Brahmagupta was a 7th century Indian mathematician who is known for being the first to use zero as a number and introduce basic algebraic rules and formulas.
(3) Bhaskara was a 12th century mathematician whose main work Siddhanta Shiromani covered topics in arithmetic, algebra, astronomy and advanced mathematics. He made contributions in calculus, arithmetic progressions, and solving indeterminate equations.
Euclid (325-265 BCE) is considered the father of geometry. He organized geometry into a logical system using definitions, axioms, and postulates in his work Elements. Some key ideas are:
- Euclid defined basic geometric terms like points, lines, and planes. He also stated basic axioms about equality and properties of wholes and parts.
- Euclid proposed five postulates, including ones about drawing straight lines and circles. The fifth postulate about parallel lines was controversial and spurred development of non-Euclidean geometries.
- Euclid proved 465 theorems in Elements through deductive reasoning based on the definitions, axioms, and postulates
This document discusses the history and properties of the mathematical constant pi (π). It describes how pi has been calculated and approximated throughout history using different methods, from the ancient Greeks to modern computers. The document also discusses how pi is an irrational number that cannot be expressed as a fraction, and how computing pi to increasing numbers of decimal places has helped test and develop computing technology over time.
Pi is the ratio of a circle's circumference to its diameter. It has been known and studied since ancient times by cultures like the Egyptians and Babylonians, though its precise value was unknown. Pi is represented by the Greek letter π because in Greek, "p" stands for perimeter. The decimal representation of pi goes on indefinitely without repeating in a pattern. It is celebrated on March 14th (3.14) as Pi Day.
Indian mathematicians and their contribution to the field of mathematicsBalabhaskar Ashok Kumar
- Mathematics originated in India as early as 200 BC during the Shulba period, where the Sulba Sutras were developed as part of the Indus Valley civilization.
- During the "golden age" of Indian mathematics between 500-1000 AD, great mathematicians like Aryabhata, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II made significant contributions and advances in many areas of mathematics. Their work spread throughout Asia and influenced mathematics in the Middle East and Europe.
- Aryabhata, in particular, made early approximations of pi and proposed that it is irrational. He also discussed sine, verses, and solutions to indeterminate equations in his
Pi is the ratio of a circle's circumference to its diameter. It is an irrational number that cannot be expressed as a fraction and its value never repeats. Throughout history, mathematicians have worked to calculate Pi to greater levels of precision, advancing from the ancient Babylonians' approximation of 3 1/8 to modern computers calculating Pi to billions of decimal places. Analytic geometry and calculus in the 17th century allowed Pi to be applied to shapes beyond just circles.
This document discusses the history and evolution of geometry. It begins by defining geometry as the measurement of earth and outlines its origins in ancient Egypt and the Indus Valley civilization where it was used to measure land and construct buildings. It then covers important early Greek mathematicians like Thales and Pythagoras and their theorems. Most of the document focuses on Euclid's Elements, outlining his definitions, postulates, axioms and use of deductive reasoning to prove 465 theorems. It also discusses criticisms of Euclid's definitions and the development of non-Euclidean geometry on curved surfaces.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
Aryabhat was an Indian mathematician and astronomer from the 5th century who made early contributions to trigonometry and algebra. He was the first to calculate pi accurately to four decimal places and devised formulas for calculating areas of triangles and circles. Brahmagupta was a 7th century mathematician who wrote the Brahmasphutasiddhanta, covering mathematics such as arithmetic, algebra and trigonometry. Brahmagupta's formula gives the exact area of a cyclic quadrilateral. Bhaskara was an 11th century mathematician who made contributions representing numbers in the Hindu decimal system with a circle for zero and providing rational approximations of sine functions.
Zero originated in ancient India, Babylon, and the Mayan civilization. The concept of zero as a number was first attributed to India in the 9th century CE, where it was treated as any other number in calculations. The symbol and rules for using zero in arithmetic operations were further developed by Indian mathematicians like Aryabhata and Brahmagupta. Their work influenced Arabic mathematicians who helped spread the concept of zero to Europe. While other ancient cultures used placeholder symbols, it was in India that zero was first understood and used as a true number.
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
The document discusses key concepts related to polynomials including:
1) Algebraic expressions, polynomials, monomials, binomials, trinomials, and their definitions.
2) Constants, variables, and the degree of a polynomial.
3) How to classify polynomials based on degree and number of terms.
4) How to rewrite a polynomial in standard form by arranging terms by descending degree.
5) Remainder and factor theorems related to polynomials.
6) Common identities used to factorize polynomials.
Indian mathematics emerged in the Indian subcontinent from 1200 BC until the 18th century. Key classical Indian mathematicians included Aryabhata, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the decimal number system, zero, negative numbers, and trigonometry. These concepts were transmitted to other parts of the world and helped develop mathematics further. Some notable later Indian mathematicians included Bhaskara II, who introduced concepts related to calculus, and Ramanujan, who had a natural genius for mathematics despite a lack of formal education.
The number π is a mathematical constant. Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in the month/day date format) since 3, 1, and 4 are the first three significant digits of 휋. In 2009, the United States House of Representatives supported the designation of Pi Day.
Using pi, it can measure things like ocean wave, light waves, sound waves, river bends, radioactive particle distribution and probability like the distribution of pennies, the grid of nails and mountains by using a series of circles.
This document is a student project on consumer awareness that was compiled by Salahudin habibullah. It includes an acknowledgements section thanking the economic teacher for guidance. The content sections outlines topics covered like a spreadsheet of a consumer survey, analysis of the survey results, the rights and duties of consumers, and recommendations. The survey analyzed consumer behaviors related to checking expiration dates, product ingredients, complaining to stores, awareness of consumer courts, and responses to marketing. It recommends consumers more carefully check product quality and standardization and utilize consumer courts when needed.
1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Some properties of triangles discussed are: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
1. The document discusses properties and congruence of triangles. It defines congruence as two triangles being the same shape and size with corresponding angles and sides equal.
2. There are five criteria for congruence: side-angle-side, angle-side-angle, angle-angle-side, side-side-side, and right angle-hypotenuse-side.
3. Additional properties discussed include isosceles triangles having equal angles opposite equal sides, and relationships between sides and opposite angles/angles and opposite sides in all triangles.
This document discusses Pi Day activities that will take place, including singing Pi Day songs, participating in Pi trivia games and contests, making Pi-themed crafts like bracelets revealing digits of Pi, and learning interesting facts about the number Pi. Students are encouraged to celebrate Pi Day on March 14th (3/14) through interactive games and songs that involve reciting digits of Pi.
This document provides a historical overview of calculations of the mathematical constant pi. It describes how ancient cultures like the Egyptians, Babylonians, and Bible approximated pi as 3. It then discusses how Archimedes was the first to theoretically calculate pi between 223/71 and 22/7. The document traces how mathematicians and scientists from cultures like Greek, Islamic, Indian, and European progressively calculated pi to more decimal places over centuries, with modern computers allowing for thousands of decimal places. It emphasizes how calculating pi more precisely has been an ongoing goal of the scientific community to enable more accurate measurements.
This document discusses the many applications of pi (π) in mathematics and other fields. It is defined as the ratio of a circle's circumference to its diameter. Pi appears in formulas for areas and volumes of geometric shapes like circles, spheres, ellipses and cones. It also appears in trigonometric functions, complex analysis, probability, statistics, physics equations for mechanics, electromagnetism, and more. Pi is an irrational number that goes on forever without repeating, and understanding its applications has expanded over time across multiple disciplines.
Pi (π) is a mathematical constant that is approximately equal to 3.14159. It is the ratio of a circle's circumference to its diameter. Over time, many mathematicians developed different formulas to calculate Pi, including formulas developed by Leibniz, Wallis, Machin, Sharp, and Euler. Pi is an irrational number that never repeats and has been computed to over 10,000 places. The early history of calculating Pi involved inscribing and circumscribing polygons around circles. While Pi seems like a simple concept, it has been studied and calculated for centuries and continues to be relevant in mathematics, science, and computing.
The Origin And History Of Pi By Nikitha ReddyJohn Williams
Pi is the ratio of a circle's circumference to its diameter. It has been estimated and calculated since ancient times but was more accurately defined over thousands of years of mathematical development. Pi is used in formulas to calculate the area, circumference, and volume of circles and spheres, and it has applications in many fields like engineering, agriculture, and construction.
Este documento describe el número pi (π), la relación entre la circunferencia y el diámetro de un círculo. Explica que pi es un número irracional y una de las constantes matemáticas más importantes. También resume brevemente cómo varios matemáticos a través de la historia han calculado aproximaciones de pi, incluyendo Arquímedes que fue capaz de determinar su valor real.
The Raspberry Pi is a credit-card sized computer that can connect to keyboards, monitors and TVs to function similarly to a desktop computer. It was developed by the Raspberry Pi Foundation in the UK to inspire teaching of basic computer science in schools and develop interest in programming. While low in cost at $25-35, the Raspberry Pi runs Linux and can be used for a variety of applications including robotics, programming practice and basic computing tasks.
Pi is an irrational number that represents the ratio of a circle's circumference to its diameter. It is approximately 3.14159 and used to calculate the circumference and area of circles. Pi has been studied for thousands of years by various ancient cultures but its exact value cannot be known as it is an irrational number with random, non-repeating digits. Pi is important in fields like science, architecture, and engineering as circles are present in many applications.
A jeopardy game I built in PP. I got the template offline. Depending on the version you open it up with it may not operate properly but you can tweak it and fix it in any version.
This document is a math worksheet for 7th grade on converting measurements. It contains 10 multiple choice questions about converting between grams, kilograms, and milligrams. The document also provides the class name, subject, date, and homework assignment.
The document provides an overview of the objectives and content to be covered in a math 7 tutoring session, including order of operations, solving equations, and calculating perimeter, area, and circumference. Key topics are powers, order of operations using PEMDAS, solving equations using mental math and word problems, finding perimeter of rectangles and squares, and calculating circumference using pi and diameter/radius. The conclusion restates that after the session, students will be able to solve problems involving equations, perimeters, and circumferences.
1) Various Germanic tribes started moving from their homelands in the early centuries AD for reasons like fascination with the Roman civilization, need for more land and resources, and employment opportunities along the Roman frontier.
2) Major Germanic tribes that migrated included the Visigoths, Vandals, Lombards, Ostrogoths, Burgundians, Angles, Saxons, and later the Franks of the Merovingian dynasty.
3) The migrations of the Huns also contributed to displacement of Germanic tribes from their lands.
Pi is the ratio of a circle's circumference to its diameter. It was first discovered around 250 BC by Archimedes, who calculated upper and lower bounds for pi. Over centuries, many mathematicians have calculated pi to increasing levels of accuracy using polygon and series methods. In the modern era, computers have been used to calculate pi to trillions of digits. Pi is an irrational and transcendental number that is ubiquitous in mathematics and physics, appearing in formulas relating to circles, spheres, trigonometry, and more.
This document provides an overview of some major figures and developments in the growth of Christianity from the 4th to 7th centuries CE. It discusses several influential Church Fathers like St. Jerome, St. Ambrose, and St. Augustine. It also outlines the contributions of St. Benedict in establishing the first monastic order in the West and rules of monastic life. The document further notes the development of celebrating saints' days, use of relics, and the work of missionaries like St. Columba and St. Patrick in spreading Christianity.
Pi is the ratio of a circle's circumference to its diameter. The document traces the history of pi from ancient Egyptians and Babylonians through Archimedes and the development of calculus. Pi is now known to over 6 billion places due to modern computers. Pi has many applications and is used in formulas by engineers, architects, agriculturists, and other professionals for calculating areas, circumferences, and volumes of circles and spheres.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Pi (π) represents the ratio of a circle's circumference to its diameter. It is an irrational number that cannot be written as a fraction. Archimedes first calculated pi to approximate its value around 250 BC. Since then, mathematicians have developed various methods and formulas to calculate pi to greater decimal places, aided by advances like calculus and digital computers. Pi is widely used in fields like engineering, science, and architecture for calculating areas and volumes of circles.
The document provides an overview of the historical development of methods for calculating pi. It discusses ancient approximations of pi from cultures like Egypt, Babylon, and India. It describes how Archimedes developed the first mathematical analysis and algorithm to approximate pi by calculating the circumference of polygons with increasingly more sides. Later developments include Machin's fast-converging formula using arctangents and Euler's new method for calculating arctangent values. The document traces the history of pi calculations through various mathematicians and cultures over thousands of years.
The document discusses the history and development of pi. Pi is the mathematical constant that represents the ratio of a circle's circumference to its diameter. Ancient civilizations like Egypt and Greece were aware of pi as a constant ratio slightly higher than 3. Archimedes calculated pi more accurately in ancient Greece. Over centuries, mathematicians have calculated pi to increasing numbers of decimal places. Pi is ubiquitous in formulas used by many fields like engineering and has no discernible pattern in its decimal representation.
This presentation discusses the mathematical constant Pi (π). It begins by introducing Pi as the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. Pi has been represented by the Greek letter π since the mid-18th century. The document then defines Pi more precisely as the ratio of a circle's circumference to its diameter. It notes that the ancient Greek mathematician Archimedes discovered Pi and was able to approximate it to high precision using polygons. The presentation concludes by discussing the importance of Pi in mathematics, science, and engineering and some historical methods used to calculate Pi more accurately such as measuring circles and using polygons.
[1] The document presents a method to derive the exact diagonal length of an inscribed square from the circumference of the enclosing circle in order to validate a new proposed value of Pi.
[2] Using the current Pi value of 3.14159265358, the derived diagonal length does not match the expected value of 2 times the side length.
[3] However, when the calculation is repeated with the proposed new Pi value of 14^2/4 - 1, the derived diagonal length exactly equals the expected value, validating this as the true Pi value.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Pi is the ratio of a circle's circumference to its diameter. Ancient Egyptian and Babylonian mathematicians approximated pi as fractions like 256/81 and 25/8. Later, Indian, Greek, and Chinese mathematicians calculated more accurate approximations, with Archimedes proving pi is between 3 1/7 and 3 10/71. Over centuries, mathematicians have computed pi to increasing decimal places of accuracy using new calculation methods. Pi is now known to at least one trillion decimal places and has many uses in mathematics, science, and engineering.
Pi is the ratio of a circle's circumference to its diameter. It was first discovered around 250 BC by Archimedes, who calculated upper and lower bounds for pi. Over centuries, many mathematicians have calculated pi to increasing levels of accuracy using polygon approximations of circles. In the modern era, computers have been used to calculate pi to trillions of digits. Pi is an irrational and transcendental number that is ubiquitous in formulas relating to circles, spheres, and trigonometry.
The document discusses the history and properties of the mathematical constant pi (π). It covers:
- Pi is the ratio of a circle's circumference to its diameter.
- Pi is an irrational number that repeats endlessly. Early cultures estimated pi to around 3 or 3.14. Archimedes first approximated it as between 223/71 and 22/7.
- Pi is useful for calculating the circumference and area of circles. Many fields of science rely on pi in their equations.
This document provides instructions for constructing and using different types of scales for technical drawing. It includes:
1. An overview of plain, diagonal, comparative, Vernier and cord scales. Plain scales can measure dimensions to one decimal place, while diagonal and Vernier scales provide more precision up to two decimals.
2. Examples of how to construct plain scales to specified ratios to measure distances and time intervals on technical drawings. This includes dividing lines into units, labeling scales, and indicating values.
3. Three practice problems for students to construct plain scales showing distances, calculate representative factors, and indicate values on the scales. This helps students learn how to appropriately scale measurements for technical drawings.
The document provides an overview of the history of mathematics from counting in 50,000 BC to modern unsolved problems. It discusses important mathematicians like Euclid and Descartes and theorems like Fermat's Last Theorem and the Four Color Theorem. The document also outlines different levels of mathematics taught at various grade levels and provides external resources on math games, jokes, and videos.
This document provides a high-level overview of the history of mathematics, levels of mathematics taught at different grades, famous mathematicians, unsolved math problems, and resources for math games and jokes. It discusses important developments like the first evidence of counting 50,000 BC, the definition of the 360 degree circle in 180 BC, the first trigonometry in 140 BC, and the proofs of Fermat's Last Theorem in 1994 and the Four Color Theorem in the 1970s using computers.
This document discusses perimeter and area. It defines perimeter as the sum of all sides of a shape and area as the region enclosed by a shape's boundary. It provides examples of calculating perimeter and area for rectangles, squares, triangles, circles, and parallelograms. Formulas are given for finding the perimeter and area of these shapes. Sample problems are worked through, such as finding the area of a rectangular land that is 60m by 45m and finding the area of a square park with a perimeter of 48m.
This document provides information about Pythagoras' theorem and how it can be used to solve problems involving right triangles. It begins with a brief history of Pythagoras and the times he lived in. It then explains Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Examples are provided to demonstrate how to use the theorem to calculate unknown sides of right triangles. The document also notes that rearranging the theorem allows calculating the lengths of the shorter sides given the hypotenuse.
Pythagoras was an ancient Greek philosopher and mathematician born on the island of Samos in around 570 BC. He is best known for the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. While Pythagoras likely did not discover this theorem himself, he is credited as being the first to prove why it is true. The Pythagorean theorem is one of the earliest and most important theorems in mathematics.
This document discusses the mathematical constant Pi. Pi is the ratio of a circle's circumference to its diameter and is approximately 3.14. It is an irrational and non-terminating number. The document outlines Pi's history, uses in formulas for areas and volumes of shapes, facts about Pi such as the longest memorization of its digits, and Pi Day which is celebrated on March 14th in honor of the first three digits of Pi.
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
3. 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
5. Pi is the ratio of the circumference of
a circle to the diameter.
6. 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”:
𝐴 = 𝜋𝑟2
7. 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 pi
Babylonians
3
1
8
= 3.125
Egyptians
4 ×
8
9
2
≈ 3.160484
› These approximations were slightly less accurate and much
harder to work with.
8. 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
1 to
16
5
−
4
239
−
1
3
16
53 −
4
2393 + ⋯ = 3.14159 … = 𝜋
› 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’.
9. 2 Pi in radians form is 360 degrees. Therefore Pi
radians is 180 degrees and 1/2 Pi radians is 90
degrees.
10. History of Calculating Pi
Archimedes
250 B.C.
Newtonand
Leibnitz
Late 17th
Century
Machin’s
Formula
Early 18th
Century
Ramanujan
20th Century
Electronic
Computers
Till Date
11. Archimedes (250 B.C.)
› The first mathematician to calculate pi with reasonable accuracy
was Archimedes, around 250 B.C., using the formula:
𝐴 = 𝜋𝑟2
› 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.
12. › Archimedes showed in this way that pi
is between 3
1
7
and 3
10
71
.
› 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…
13. 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:
arctan 𝑥 = 𝑥 −
𝑥3
3
+
𝑥5
5
−
𝑥7
7
+ ⋯
If you substitute 𝑥 = 1 and notice that arctan(1) is
𝜋
4
, you get a
formula for pi.
14. 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:
𝜋
4
= 4 arctan
1
5
− arctan
1
239
› 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!)
15. 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:
1
𝜋
=
2 2
9801
𝑛=0
∞
4𝑛 ! (1103 + 26390𝑛)
(𝑛!)43964𝑛
› In 1985, William Gosper used this formula to calculate the first
17 million digits of pi.
16. 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.
17. 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:
1
𝜋
= 12
𝑘=0
∞
−1 𝑘
6𝑘 ! (13591409 + 545140134𝑘)
3𝑘 ! (𝑘!)3640320
3𝑘+3
2
› 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.
18. 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.
19. 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.
20. 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?
21. 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?
22. The Other Cases
There 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.
23. RECORD
APPROXIMATIONS OF
PI
Given 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).
24. The 'famous five' equation connects the five most
important numbers in mathematics, viz., 0, 1, e, π,
and i: 𝒆𝒊𝝅 + 𝟏 = 𝟎.
25. 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).
26. Irrationality of Pi
Here's a proof of the irrationality of Pi from Robert Simms:
Theorem: Pi is irrational.
Proof: Suppose 𝜋 =
𝑝
𝑞
, where p and q are integers. Consider the
functions 𝑓𝑛(𝑥) defined on [0, π] by
𝑓𝑛 𝑥 =
𝑞 𝑛
𝑥 𝑛
(𝜋 − 𝑥) 𝑛
𝑛!
=
𝑥 𝑛
(𝑝 − 𝑞𝑥) 𝑛
𝑛!
Clearly 𝑓𝑛 0 = 𝑓𝑛 𝜋 = 0 for all n. Let 𝑓𝑛[𝑚](𝑥) denote the mth
derivative of 𝑓𝑛(𝑥). Note that 𝑓𝑛 𝑚 0 = −𝑓𝑛 𝑚 𝜋 = 0 for
𝑚 ≤ 𝑛 or for 𝑚 > 2𝑛; otherwise some integer
27. Continued…
max 𝑓𝑛(𝑥) = 𝑓𝑛
𝜋
2
=
𝑞 𝑛 𝜋
2
2𝑛
𝑛!
By repeatedly applying integration by parts, the definite integrals
of the functions 𝑓𝑛 𝑥 sin 𝑥 can be seen to have integer values. But
𝑓𝑛 𝑥 sin 𝑥 are strictly positive, except for the two points 0 and pi,
and these functions are bounded above by
1
𝜋
for all sufficiently
large n. Thus for a large value of n, the definite integral of
𝑓𝑛 𝑥 sin 𝑥 is some value strictly between 0 and 1, a contradiction.
28. Pi day is celebrated on March 14 at the
Exploratorium in San Francisco (March 14 is 3/14).
29. 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 (𝑑 = 2𝑟).
› We use the Greek letter π to represent
the ratio of the circumference of a circle
to the diameter. For simplicity, we use
𝜋 = 3.14.
› The formula for circumference of a
circle is: 𝐶 = 𝜋𝑑, i.e., 𝐶 = 2𝜋𝑟.
Parts of a circle.
30. 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:
𝐴 = 𝜋𝑟2
or 𝐴 = 𝜋. 𝑟. 𝑟
where 𝐴 is the area, and 𝑟 is the radius.
31. 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.
32. 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.