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History of Math

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History of Math is a project in which students worked together in learning about historical development of mathematical ideas and theories. They were exploring about mathematical development from Sumer and Babylon till Modern age, and from Ancient Greek mathematicians till mathematicians of Modern age, and they wrote documents about their explorations. Also they had some activities in which they could work "together" (like writing a dictionary, taking part in the Eratosthenes experiment, measuring and calculating the height of each other schools, cooperating in given tasks) and activities that brought out their creativity and Math knowledge (making Christmas cards with mathematical details and motives and celebrating the PI day). Also they were able to visit Museum, exhibition "Volim matematiku" and to prepare (and lead) workshops for the Evening of mathematics (Večer matematike). At the end they have presented their work to other students and teachers.

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History of Math

  2. 2. History of Math  eTwinning project  Cooperation between and  We explored:  Development of mathematical thought from Sumer till Modern age  Distinguished mathematicians from Ancient Greece to Modern age
  3. 3. What did we do?  Presentations (.ppt) for introducing ourselves:  Personal  School  Hometown
  4. 4. What did we do?  We chose the LOGO of the project 13
  5. 5. What did we do?  We visited exhibition „I Love Math” („Volim matematiku”)/CRO
  6. 6. What did we do?  We organized „The Evening of Mathematics” /CRO
  7. 7. What did we do?  We exchanged Christmas cards Croatian in Greece Greek in Croatia
  8. 8. What did we do?  We celebrated the 𝜋 Day
  9. 9. What did we do?  We visited Technical Museum of Ancient Greece in Thessaloniki /GR
  10. 10. What did we do?  We did measures for calculating the Earth circumference – Eratosthenes experiment  We calculated the Earth circumference  We measured shadow length of schools  We calculated the height of schools
  11. 11. ... and our outcome for Novska school is 8.322 m !!! Greek Team
  12. 12. … and our outcome for Edessa school is 𝟏𝟏, 𝟗𝟖 ± 𝟎, 𝟏𝟒 𝒎 Croatian Team
  13. 13. What did we do?  We wrote documents (.doc i .ppt) about given tasks
  14. 14. What did we do?  We presented the project to mathematic teachers of Sisak- Moslavina County / CRO
  15. 15. What did we do?  We presented the project to teachers and students of our schools
  16. 16. What did we do?  We set up an exhibition of posters in school hallway / CRO
  17. 17.  On TwinSpace Forum we wrote a dictionary of mathematical words that have Greek root in:  English  Greek  Croatian What did we do?
  18. 18. What did we do?  We wrote seminars about given topics and merge them in a book /CRO  matematike-history-of-math  We edited our TwinSpace: 
  19. 19. Antonio Jakubek, 4.g
  20. 20.  Our first knowledge of math comes from Egypt and Babylon  Babylon math dates back to 4000 BC along with the Sumerians in Mesopotamia
  21. 21.  Little is known about Sumer  It was first inhabited 4500 and 4000 BC  Today, these people are called Ubaidiansi  Even less is known about their math
  22. 22.  They used cuneiform and wrote on clay tablets  They used over 2000 signs Picture 1. Sumerian cuneiform record
  23. 23.  They developed heksagezimal number system which was taken by the Babylonians  Babylonians, Assyrians and Hiti inherited Sumerian law and literature and more importantly their way of writing  What we have kept from the Sumerians today is the division of weeks to 7 days, days to 24 hours, 60 minutes in an hour and 60 seconds in one minute
  24. 24.  With the collapse of the Sumerian civilization in Mesopotamia Babylon is developed  They inherited from the Sumerians cuneiform and heksagezimal number system Picture 2. The digits of the Babylonian number system
  25. 25.  They used 2 basic forms for numbers:  They had no symbol for zero or a decimal point, so it was difficult to interpret the findings from this era Picture 3. Babylonian symbol for the number 10 Picture 4. Babylon symbol for number 1 or 60
  26. 26.  In 1940s German historians Otto Neugebauer and Abraham Sachs  Noticed how the verses on the board meet interesting propertyes  Decorated triplets of positive integers (a, b, c) that satisfy a2 + b² = c²
  27. 27.  Proof of the existence of Pythagorean triples thousands of years before the mathematicians of ancient Greece Picture 5. Plimpton 322
  28. 28.  The site in Nippur - found about 50 000 clay tablets  Witnessed considerable knowledge of mathematics Picture 7. The site in Nippur
  29. 29.  They were building up series of numbers that include triangular numbers (1, 3, 6, 10, 15 ...), square numbers (1, 4, 9, 16, 25 ...) and the pyramidal numbers (1, 5, 14, 30, 50 ...) Picture 8. i 9. Showing series of numbers
  30. 30.  An example of using a series of numbers is pyramid stacking of ammunition in Calcutta and easy calculation of the number of cannonballs Picture 10. Pile of ammo in Calcutta
  31. 31. Egypt Ella Cink, 4.g
  32. 32. Moscow papyrus - discovered in 1893 and the author is unknown - the greatest achievements of Egyptian geometry - length is about half a meter and width of less than 8 cm - kept in the Moscow Museum
  33. 33. Moscow papyrus
  34. 34. Rhinds papyrus • In 1858 he was discovered by Scottish Egyptologist Henry Rhind in Luxor • It was written by the scribe Ahmes around 1600 BC • It is 6 meters long, 30 cm wide, preserved in the British Museum in London
  35. 35. • A collection of tables and exercises with 87 math problems • It contains the oldest known written record number π Rhinds papyrus
  36. 36. Numbers The Egyptians used a number system with a base 10  number 1339
  37. 37. • Addition • Subtraction • Multiplication • Division because
  38. 38. Fractions • They only knew fractions • The exception was 2/3 • Fractions are formed by combining the individual parts of the symbol Horus eye the entire symbol of the eye has a value of 1 
  39. 39. Geometry • To build the pyramids and temples they were obliged to have a well-developed geometry and stereometry • They knew how to calculate the slope and volume of the pyramid, and the volume of a truncated pyramid
  40. 40. Algebra • Egyptian algebra was rhetorical • Problems and solutions are given by words • They used seven-digit numbers, and their calculations were a mixture of simplicity and complexity
  41. 41. Mathematics of Ancient Greece Doroteja Lukić, 3.g
  42. 42.  based on Greek texts  developed from the 7th century BC to the 4th century AD  along the eastern shores of the Mediterranean  mathematics - Greek Mathematica - Science  use general mathematical proofs and theories
  43. 43.  presided crucial and most dramatic revolution in mathematics ever  main goal: the understanding of man's place in the universe  mathematics has reached the highest level of development  began to use papyrus  Greek contribution to mathematics in three phases:  1. Thales and Pythagoras to Democritus  2. Euclidean system  3. phase of Alexandria
  44. 44.  Tales - founder of Greek mathematics  no documentary evidence  classical philosophy helped to reconstruct texts a closer period  editions of Euclid, Archimedes, Apollonius, etc.  difficult to follow the course of historical development  on Greek mathematics concludes: smaller components and observations of philosophers and other authors
  45. 45. Greek number system ( About 900 BC - 200 AD)  The first was based on the initial letters of the names of numbers
  46. 46.  the second used all the letters from  Greek alphabet and three from the Phoenician  Base - 10
  47. 47.  the idea of evidence and a deductive method of using logical steps to confirm or refute the theory  gave the mathematics force  ensures that the proven theories are true  laid the foundation for a systematic approach to mathematics The most important contribution of the Greeks
  48. 48. PYTHAGORA Petra Kalanja, 2.g
  49. 49. General…  the first "true" mathematician  born on the Greek island of Samos  Tales interested him in mathematics  traveled to Egypt around 535 BC  founded the Pythagorean school  Today he is known for the Pythagorean theorem
  50. 50. Through life ...  philosopher in Egypt  temple priest in Diospolisu  captive in Babylon  married at age 60  starved to death  the most perfect number 10  number - being in philosophy
  51. 51. Pythagorean school  established in Crotona  emphasis on secrecy and fellowship  Pythagorean theorem  the discovery of irrational numbers  five regular solids
  52. 52. Pythagorean theorem  Surface of the square on the hypotenuse of a right triangle is equal to the sum of the squares of the cathetus
  53. 53. Pythagorean triples  3, 4, 5 9+16=25  Egyptian triangle  We can get another infinite number of Pythagorean triples by making the numbers 3, 4 and 5 reproduce the same number 6, 8, 10 36+64=100
  54. 54. PLATO (428 - 347 BC) -lived and worked in Athens -387 BC founded the philosophical school ACADEMY where mathematics, arithmetic, trigonometry and planimetry was taught Picture 1.Plato Marija Kožarić, 4.g
  55. 55.  "No entrance for those who do not know geometry!" Picture 2.The inscription at the entrance to the Academy
  56. 56. Platonic solids  Described in the work Timaeus  5 regular polyhedrons: Picture 3. regular polyhedrons
  57. 57. TETRAHEDRON  4 peaks  6 edges  4 sides  equilateral triangles Picture 4. tetrahedron
  58. 58. HEXAHEDRON  cube  8 peaks  12 edges  6 sides  squares Picture 5. hexahedron
  59. 59. OCTAHEDRON  6 peaks  12 edges  8 sides equilateral triangles Picture 6. octahedron
  60. 60. DODECAHEDRON  20 peaks  30 edges  12 sides  regular pentagon Picture 7. dodecahedron
  61. 61. ICOSAHEDRON  12 peaks  30 edges  20 sides  equilateral triangles Picture 8. icosahedron
  62. 62. Chinese mathematics
  63. 63. • in the second millennium BC China had symbols for numbers • they counted with sticks until abacus appeared in the 16th century Picture 1. chinese numbers Picture 2. abacus • not much is known about the mathematics of ancient China, but it is fairly certain that the origins of astronomy and mathematics of ancient China date back to at least the second millennium BC, at that time the Chinese have already had an elaborate calendar • oldest surviving mathematical texts originate from the time around 200 BC
  64. 64. • Contributions of Chinese mathematicians: • The Holy Book of arithmetic (2nd - 12th century) - indirect talks about the Pythagorean theorem • Arithmetic in nine books (about 150 BC) - the process of calculating the area of a triangle, rectangle, circle, circular section and clip, the volume of prism, pyramid, cylinder, cone, deprived (truncated) cone and pyramid • The book of phases (I Ching) - one of the oldest surviving book - used for fortune telling and divination, contains elements of the binary notation of numbers
  65. 65. • The famous mathematicians: • Zhang Qiu Jian (5th c.) - Gave the formula for the sum of the arithmetic series • Tsu Chung - chih (430 to 500) - the value of the number π takes a precise six decimal places • Quin Jiu - Shao (1202 -1261) - Sought the solution of equations method that is called Horner (William Horner, 1819), although it was known in China 500 years earlier • Chu Shih - kieh (1270 -1330) - wrote two important texts that are the pinnacle of Chinese mathematics texts which contain Pascal's triangle binomial coefficients, which is known in China for four centuries before it was discovered by Pascal.
  67. 67. • in ancient Indian mathematics there is no great works exclusively devoted to mathematics; mathematics is present only as part of, as a separate chapter in astronomical or astrological works • the oldest known mathematical texts are Sulvasutre, accessories to religious texts  in them are the rules for measuring and building temples and altars at the level of elementary geometry • characteristics of Indian mathematical texts is that they are generally written in verse Picture 1. Indian numbers
  68. 68. Ancient India mathematics: • Aryabhatta (476 to 550) - he knew how to take out the second and third root of the division into groups of radikands • gave the correct formula for the area of a triangle and a circle, writes about quadratic equations and potencies • Brahmagupta (598 – about 670) • Brahmaguptas formula: a generalization of Heron's formula in the cyclic quadrilateral;
  69. 69. • Mahavira (9th century) - dealt with elementary mathematics and the first Indian mathematician who wrote the only math dedicated text • Bhaskara (1114 – 1185) - the most famous Indian mathematician to the 12th century, has contributed to the understanding of numerical systems and solving equations and proved the Pythagorean theorem • his main mathematical works of Lilavati and Bijaganita, dealt with plane and spherical trigonometry
  70. 70. Arab mathematics
  71. 71. • Today's western style math is much more similar to what we find in the Arab mathematics than that of ancient Greeks, many of the ideas that have been attributed to the Europeans proved to be actually Arabic Picture 1. Arabic numbers Al-Khwarizmi (780 – 850) first great Arab mathematician
  72. 72. Arabic mathematicians: • Al-Karaji (953 - 1029) - is considered the first person who completely freed algebra from geometrical operations and replaced them with arithmetic • founded the influential algebraic school that will work successfully for centuries •Al-Khwarizmi (780 - 850 ) first great Arab mathematician • he wrote about algebra, geometry, astronomy and he introduced Arabic numerals in mathematics • he dealt with linear and square equations. He built tables for sinus and tangens functions. He gave a general method for finding two roots of quadratic equations : 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑥1,2 = −𝑏 ± 𝑏2 − 4𝑎𝑐 2𝑎
  73. 73. • Al-Haytham (965 - 1040) is probably the first who attempted to classify the even perfect numbers • also was the first person that has imposed Wilson's theorem (if p is prime, then p divides 1 + (p - 1)!), it is unclear if he knew how to prove it • Omar Khayyam (1048 – 1131) with mathematics also dealt with astronomy, philosophy and poetry • gave a complete classification of cubic equations (14 types) and the first to notice that you do not need to have a unique solution • Nasir al-Din al-Tusi (1201 - 1274) wrote important works on logic, ethics, philosophy, mathematics and astronomy • The most important contribution was his creation of trigonometry as a mathematical discipline, not a means of astronomical calculations, and gave the first complete account of plane and spherical trigonometry. • This work gave the theorem of the sinuses for planar triangles: 𝑎 sin 𝛼 = 𝑏 sin 𝛽 = 𝑐 sin 𝛾
  74. 74. (12th – 13th century) Picture 1. Fibonacci Barbara Mašunjac, 4.g
  75. 75.  Italian mathematician  spent his youth in Arabia  the foundation of his mathematics - number  left behind a series of discoveries
  76. 76. FIBONACCI SERIES Picture 2. Fibonacci series
  77. 77. FIBONACCI SERIES IN NATURE Picture 3. Fibonacci series in sunflowe Picture 4. Fibonacci series in snail Nautilus shell Picture 5. Fibonacci series in human body
  78. 78. FIBONACCI SERIES IN ART Picture 6. Fibonacci series in Mona Lisa portrait Picture 7. Fibonacci series in Partenon
  79. 79.  ”Divine ratio” or the ratio of the golden section 𝜑 = 1 + 5 2 ≈ 1.618033989
  80. 80. The ratio of the golden section
  81. 81. The ratio of the golden section
  82. 82. The ratio of the golden section
  83. 83. The ratio of the golden section
  84. 84. The ratio of the golden section
  85. 85. The ratio of the golden section
  86. 86. LIBER ABACI  the most famous work of arithmetic  one of the first Western book that described the Arabic numerals  four parts Picture 8. Liber Abaci
  87. 87. JOHN NAPIER Laura Iličić, 3.g
  88. 88. GENERAL: • Born in Edinburgh 1550, died April 4th 1617 • He enrolled at the University of St. Andrews • He graduated in Paris, and then stayed in the Netherlands and Italy • He is known in mathematical and engineering circles • He is best known as the inventor of logarithms, Napier's bones, and the popularization of the decimal point • He worked in the fields of mathematics, physics, astronomy and astrology
  89. 89. Napier’s bones Decimal point Logarithms
  90. 90. MOST FAMOUS WORKS • Plaine Discovery of the Whole Revelation of St. John, 1593 • Statistical Account • Mirifici logarithmorum canonis descriptio, 1614 • Construction of Logarithms, 1619
  91. 91. • Mirifici logarithmorum canonis descriptio, Statistical Account i Construction of Logarithms
  92. 92. Henry Briggs English mathematician Professor of geometry at Oxford born in Warleywoodu in Yorkshire 1561 He studied at St. John's College, Cambridge Patricia Kujundžić, 3.g NO PICTURE
  93. 93. as a professor at Oxford he learned about Napier 1615 travels to him in Edinburgh Napier agrees with the proposal for the Briggs logarithms with base 10 After Napier's death continues his work 1624 publishes logarithmic table Arithmetica He died in Oxford 1630
  94. 94. Blaise Pascal Antonio Horaček, 4.g
  95. 95. Biography  Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a 'child prodigy' and was educated by his father.  Pascal’s earliest jobs were in applied and natural sciences, where he contributed to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of EvangelistaTorricelli.
  96. 96.  Pascal's first calculating machine
  97. 97. Pascal’s contribution to math  The first significant work, Blaise wrote at age sixteen, and it was a basic draft of his famous debate on the sections of the cone.
  98. 98.  Blaise Pascal, also created his famous  mystical hexagram (Pascal's theorem), which has not survived.  In his 'Treatise on the arithmetical triangle' '(Traité du triangle arithmétique), described the convenient, practical tabulation of binomial coefficients, now called "Pascal's Triangle'.
  99. 99. Pascal’s contribution to physics  His work in the field of hydrodynamics and hydrostatics has focused on the principles of hydraulic fluids. His inventions include the hydraulic press (using hydraulic pressure to multiply force) and the syringe.  Hydrostatic pressure increases the depth, acts equally in all directions and is equal in all places at the same depth.
  100. 100.  Pascal’s law  The fundamental law of hydrostatics:  The fluid contained in a closed vessel outer pressure p expands equally in all directions, that is, particles of the liquid pressure is transmitted equally in all directions.
  101. 101. History of infinitesimal calculus Stjepan Marijan, 4.g
  102. 102. Gottfried Wilhelm Leibniz • Leipzig 1st July 1646 • Philosopher, mathematician, physicist and diplomat • The forerunner of George Boole and symbolic logic • "Differential" and "integral" • 1559 French Academy of Science • The first model of the computer machine Picture 1.1.: Gottfried Wilhelm Leibniz Picture 1.2.: Leibniz’s mechanical computer
  103. 103. Isaac Newton • Woolsthorpe-by-Colsterworth 4th January 1643 • Astronomer, mathematician and physicist • methods of elimination • The general law of gravity • mirror telescope • The Royal Academy Picture 2.1.: Isaac Newton •Picture 2.2.: Mirror telescope
  104. 104. Infinitesimal calculus • Functions, derivation, integral limits and limit functions • differential calculus • integral calculus Picture 3.1.: Integral Picture 3.2.: Derivation
  105. 105. Newton – Leibniz’s formula • If 𝐹 is selected primitive functions of function 𝑓 on the interval 𝑎, 𝑏 , following applies: 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
  106. 106. History of infinitesimal calculus • 5th century BC- Zenon • 4th century BC- Eudokso • 225 years BC- Arhimed • 17th century - Bonaventura Francesco Cavalieri Picture 4.1.: Method of ekshaution Picture 4.2.: Geometria indivisibilibus continuorum nova quadam ratione promota
  107. 107. Conflict of Newton and Leibniz • Isaac Newton- 1671 - De Methodis Serierum et Fluxionum (published 60 years later), physical access • Gottfried Wilhelm Leibniz- 1684 - the first published results, the geometric approach Picture 5.2.: De Methodis Serierum et Fluxionum Picture: 5.3.: Transactions of the Royal Society of London
  108. 108. ABRAHAM DE MOIVRE Iva Ciprijanović, 4.g
  109. 109. DE MOIVRES BEGINIGS:  was born in Vitry, France, May 26, 1667  French mathematician famous for the formula which links complex numbers and trigonometry  He was a Protestant, and for a while after the Edict of Nantes (1685) he was in prison, after which he moved to England, where he lived the rest of his life.
  110. 110.  He was earning money as a private tutor of mathematics and he taught the students in their homes, but also in London bars.  He hoped to one day become a professor of mathematics, but in every country for some reason been discriminated against
  111. 111. DE MOIVRES ANECTODE:  His well-known anecdote is that he predicted the day of his death by determinating that he sleeps every day 15 minutes longer and summarizing the corresponding arithmetic progression, calculated that he would die on the day that he will sleep for 24 hours and he was right.
  112. 112. THE DOCTRINE OF CHANCE: A METHOD OF CALCULATING THE PROBABILITIES OF EVENTS IN PLAY  Main De Moivre’s work  In this book we can find the definition of statistical independence of events and a number of tasks related to various games. Picture1. De Moivre work: The Doctrine of Chance: A method of calculating the probabilities of events in play
  113. 113. DE MOIVRE’S FORMULAS: Picture 2. Formula for binomial coefficients Picture 3. The formula which could prove all the integer numbers n Picture 4. Famous DE MOIVRE’s formula
  114. 114. Picture 1. Johann Carl Friedrich Gauss Marta Ćurić, 3.g
  115. 115.  German mathematician (1777-1855)  except for mathematics, worked in astronomy, physics, geodesy and topography  designed "non-Euclidean geometry” at the age of sixteen  with twenty-four years he published a masterpiece Disquisitiones Arithmeticae  In 1801, according to his calculations discovered planetoid Ceres  discovered Kirchhoff's laws  made primitive telegraph  created his own newspaper - Magnetischer Verein
  116. 116. Picture 2. Disquisitiones Arithmeticae Picture 3. Magnetischer Verein
  117. 117.  devised a faster way of solving tasks of adding numbers from 1 to 100:  (100 + 1) + (2 + 99) + ... + (50 + 51) = 50 * 101 = 5050  realized the criteria of constructing proper heptadecagon  proved the basic theorem of algebra  created the Gaussian plane  created a Gaussian curve that is used in many sciences, especially in psychology
  118. 118. Picture 4. Gaussian plane Picture 5. Proper heptadecagon Picture 6. Gaussian curve
  119. 119. John Nash Lana Matičević, 3.g
  120. 120. John Nash (1928) is an economist and mathematician. He has published several theories that are used and who have contributed to the economy. He won the 1994 Nobel Prize for economics. His most famous theory: Nash Equilibrium (game theory)
  121. 121. What is the Nash Equilibrium?  The concept, which was initially designed as a tactic for simple games  It is not the best strategy that can be used, but it is the best tactic to not use other players in order to reach the goal
  122. 122. Interesting facts  He was suffering from schizophrenia (up to 1990)  Movie Beautiful Mind is based on his life.
  123. 123. From a scientific rationality to the illusory chaos
  124. 124. Equation N.E.
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History of Math is a project in which students worked together in learning about historical development of mathematical ideas and theories. They were exploring about mathematical development from Sumer and Babylon till Modern age, and from Ancient Greek mathematicians till mathematicians of Modern age, and they wrote documents about their explorations. Also they had some activities in which they could work "together" (like writing a dictionary, taking part in the Eratosthenes experiment, measuring and calculating the height of each other schools, cooperating in given tasks) and activities that brought out their creativity and Math knowledge (making Christmas cards with mathematical details and motives and celebrating the PI day). Also they were able to visit Museum, exhibition "Volim matematiku" and to prepare (and lead) workshops for the Evening of mathematics (Večer matematike). At the end they have presented their work to other students and teachers.


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