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- 1. HISTORY OF MATH CROATIA-GREECE 2014-15
- 2. They developed as the life became more complicated (calculations,product exchange) They drived in development of civilization of the Ancient Egyptians (pyramids,tombs) They were written with three ways: the hieroglyphic,the hieratic and the demotic.
- 3. The system of the Egyptian arithmetic was based at the simple iterative beginning according to differently symbols repeated for the successive forces of ten to form the requested number. With this seven symbols the Egyptians could wrote any integer number from 1 until 9.999.999
- 4. Addition: was a simple process that all you needed was the replacing ten similar symbols with one symbol for the next category. Deduction: they faced her as otherwise process of addition. Multiplication: was the base of the whole Egyptian arithmetic and it made with the method of doubling, halving and with the addition. Division: they weren’t consider different process from the multiplication.(namely 168:12 «multiplied with 12 until you find 168»)
- 5. At first Egyptians ignored the metric relations of rectangles triangle and moreover they weren’t deal with theorems and evidences. The content of geometry was the computation of areas and tumors of different shapes based on rules(some right and some not).The most remarkable results are the tumor calculation of a bun pyramid and the calculation of circle areas based on a rule that correspond to formula Ε= [(1 -1/9)d]2 (d the diameter). The formula drive us to the approximate value of p= 256/81 = 3,16.., (better than the value p= 3 That used by Babylonians).
- 6. Papyrus Rhind, is a collection of 84 problems that copied approximately 1650 BC Papyrus of Moscow , (1850BC ) is a collection of 25 problems. The leather roller,(1650 BC) include 26 sum of unit fractons. At the end, papyrus the Kahun and the papyrus Of Berlin, (1850 BC) they include mathematical operations and problems. The sources of Egyptian Mathematics
- 7. He was born in AD 90 and died in AD 168 He was a Greco-Egyptian writer of Alexandria, known as: • a mathematician • an astronomer • a geographer • an astrologer • and a poet
- 8. He wrote in Greek. Three of his most important scientific treatises are: • the Almagest which is the only surviving comprehensive ancient treatise on astronomy • the Geography • the Tetrabiblos or the Apotelesmatika.
- 9. He was an Egyptian Muslim mathematician during the Islamic Golden Age. He was the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations. His mathematical techniques were later adopted by Fibonacci. He made important contributions to algebra and geometry.
- 10. He was born in Egypt between 950 and 952 and died in 1009. His full name was Abu al-Hasan 'Ali ibn 'Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri. He was an important Egyptian Muslim astronomer and mathematician whose works are noted for being ahead of their time. He was also known as an eccentric and a poet.
- 11. He was born in Alexandria, Egypt, on November 11, 1911 In 1944 he began publishing books and articles in scientific and other journals. By 1988, he had written about 120 books and 500 articles. In 1947 he began running seminars for international groups which he continued until his death. He died in Paris in 1988.
- 12. Founded the International Commission for the Study and Improvement of Mathematics Education. Founded The Association of Teachers of Mathematics. Founded The Cuisenaire Company in England.
- 13. He was born in Cairo in 1936. He is a: mathematician, philosopher and historian of science. His work: • focuses on mathematics and physics of medieval Arab world. •explores and illuminates the unrecognized Arab scientific tradition.
- 14. Ahmes Ismail Mustafa al-Falaki Ahmad ibn Yusuf Mohammed Reda Madwar Menelaus of Alexandria Serenus
- 15. Al Khwarizmi (780-850) was Persian: Mathematician Geographer Astronomer He was born in Khwarezm ( Uzbekistan ) Al Khwarizmi was translating Greek and Sanskrit scientific manuscripts .
- 16. He created the foundation in algebra and trigonometry, for what is considered the "father" of Algebra ( with Diophantus ). The reason of spread of Indian number system was his book written around 825, Calculating with Indian numbers. His book "The image of the earth" had to corrected values, the coordinates of Asian Mediterranean and Africa. He wrote for devices like the sundial and the astrolabe. In the 12th century introduced the Latin West the Arabic numerals, which was based on the decimal system developed by Indian sources.
- 17. "The Compendious Book on Calculation by Completion and Balancing".(830 CE) al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala AL-TZAMPR والمق الجبر حساب في المختصر الكتاب The word ALGEBRA from the book’s title "The Book of Addition and Subtraction According to the Hindu Calculation" which refers to Arithmetic. He design tables for the trigonometric functions of sines and cosine and tangent in Zīj al-Sindhind which includes116 tables with calendrical, astronomical and astrological data and 37 chapters on calendrical and astronomical calculations. "Book of the Description of the Earth“. Refers to the coordinates of 2402 cities and other geographical features. (880 CE)
- 18. •Fibonacci (September1170-1240)was Italian Mathematician who was born in Piza and lived in Bejaia(Algeria).He gained popularity with the Fibonacci sequence .He was taught accounting and he traveled together with his father. Fibonacci Numbers constitute the following sequence 0 1 1 2 3 5 8 13 21 34 55 89 144….or −21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21 By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.
- 19. Golden Ratio The number F is related to golden mean. The golden ratio also is called the golden mean or golden section .Other names include ,extreme and mean ratio ,medial section, divine proportion, divine section ,golden proportion, golden cut, and golden number. The Fibonacci sequence is named after Fibonacci. His book Liber Abaci introduced the sequence to Western European mathematics .The book was widely known after the introduction of typography.
- 20. The most characteristic experiment that Fibonacci made was the Rabbit Problem. The question was (if we know that once a pair is two months old, it bears another pair and from then bears one pair every month. ) ”Starting with a newborn pair at the beginning of a year, how many pairs of rabbits will there be at the end of the year?” In nature
- 21. Spirals are patterns that are related to the Fibonacci sequence. By using quarter-circle arcs inscribed in squares of integer Fibonacci-number side, shown for square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34. The most famous and beautiful examples of the occurrence of the Fibonacci sequence in nature are found in a variety of trees and flowers, generally associated with some kind of spiral structure. For instance, leaves on the stem of a flower or a branch of a tree often grow in a helical pattern, spiraling around the branch as new leaves form further out. A completely understandable example is the one about the daisy. A daisy has a central core consisting of tiny florets arranged in opposing spirals. There are usually 21 going to the left and 34 to the right.
- 22. Birth: on 1540, in Vante which was the kingdom of France. Death: in Paris on 1603. Occupation: Mathematician Point of interest: law, mathematics and especially algebra. Known for: the establishment of the algebraic symbolism.
- 23. Initially he deciphered the Spanish code of correspondence. Ηe expressed the number π with the assistance of the infinity product and he calculated with accuracy of nine decimal figures, improving the Archimedes’ result. 2 2 2 2 2 2 2 2 2 2
- 24. He used the vowels to express the unknown variable and consonants for the known ones. Lastly the formulas known as “Vietas’ formulas” which give many terms between roots and the quotients of a polynomial. Especially for the trinomial: 2 0, 0ax bx c a b cS Pa a
- 25. René Descartes
- 26. • French philosopher and mathematician. • He was born in La Haye on 1596 from a wealthy family. • He was educated at a boarding school and after finishing his studies in Law, it came his military service. • In 1628 he moved to the Netherlands, where he devoted himself to his scientific studies and especially in mathematics. • He died on 11 February 1650 in Stockholm.
- 27. He developed the analytic geometry. He was the first to introduce the concept of variable size and variable function (The dual form of the variable determined the unity of geometry and algebra). He invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c and the way of writing the powers. He made the "Cartesian rule" for the determination of the number of positive and negative roots. Showed that the cubic equation is solved by squaring and with the assistance of diabetes and rule. He was involved with the study of the theory of sets and set the principles of physics and biological determinism. His works:
- 28. “ Discourse on the Method ”, which was published in 1637, “Meditations on First Philosophy”, published in 1641 in Latin, "Principles of Philosophy", published in 1644 also in Latin, "Passions of the Soul" (1649) and his endless work "Rules for the Direction of the Mind“ (1626–1628). All of his works published in Paris from 1897 to 1910 in 12 volumes. Other works:
- 29. The Geometry of Descartes with the method of coordinate system that he created and with the construction of the vertical tangents and the flat curves, he helped remarkably the work of Newton, Leibniz, Euler and all of the other important people after him and he also had an enormous influence on the development of mathematics.
- 30. Quotes - Testimonials - Sayings: «Reading good books is like conversation with the most perfect people in the past.» «I think, therefore I live.» «In the effort of finding the truth it is necessary we doubt for everything, as much as we can.» Foundation of his philosophical thoughts is doubt because senses often mislead people. Since he disputes, he is forced to think, so he justifies his existence as a human being.
- 31. Born: 17 August 1601 Died: 12 January 1665 Profession: Lawyer Origin: Basque Occupied with: Mathematics as amateur from 1629 Knowledgeable: French, Latin, Ancient Greek, Spanish, Italian and perhaps Basque dialect
- 32. Project: He composed various texts about the maximum and the minimum of functions, which later gave to Etienne d’Espagnet. His project on the Analytic Geometry was available for the public in manuscript in 1636. In one text of his work, he developed a determination method of the minimum, maximum and tangent in curves of different functions, equivalent with the one of the differential calculus. He also worked with Pascal. In 1664 they established the basic foundations of the probability theory. That is the reason why they are considered to be co-authors of the probability theory.
- 33. He invented: A technique for the positioning of the center of gravity of many levels and solids, which has led to several analyzes on the integrals. A type of calculation of the integral attributed to a sum of geometric progress terms. Terms that were useful later for many scientists, like Newton. The factorization method and the technique of infinite descent, a special case of the proof of contradiction, which he used in order to prove the Last Theorem for the case n=4.
- 34. He studied: A special case of Diophantine equation which was called ‘the equation of Pell’. The perfect numbers, the friendly numbers and the numbers later known as the Fermat numbers. where the n is a non negative integer. 2 2 1x n y 2 2 1 n nF
- 35. Born: 19th of June1623 in Clermont-Ferrand of France Died: 19th of August 1662, Considered: one of the most charismatic mathematics as his contribution was great, especially in the sections of Probability and Fluid. Occupied with: mathematics, philosophy, religion, physics. Despite the removal of all geometry books from home by his father , Pascal began at age 12 to read geometry alone
- 36. “Pascalina” (invention of his) contained pinions, which was marked by numbers 1 to 10 and the sum or removing mapped to rotation angles. When a gear made a full turn, swept the immediate left of the gear located and thus conveys the “prisoner”.
- 37. Formulated those theories: 1. The principle of communicating vessels , particularly when in communicating vessels balances a liquid , all points of the liquid having the same pressure and the free surface of all containers located on the same horizontal plane. 2. The Pascal's law is one of the basic laws of Hydrostatic and determines that any pressure that may be exerted on the surface of a liquid spread evenly throughout, in all directions and throughout the depth of it. 3. The third theory of Pascal , according to which the points of intersection of the opposite sides of the hexagon inscribed in a circle ( and generally recorded on a conic section ) is collinear ( on a line called a straight Pascal 's hexagon ( the red line in the figure).
- 38. The arithmetic triangle, known as ‘Pascal’s Triangle’ is a figure where every number from third line and under, except from the units, is the sum of the numbers from the nearest line before Features: The first line constitutes from one number, the second line of two numbers, the third one of three numbers, etc. The n -th line has n numbers. The numbers of the n -th line are coefficients of the blank (a + b ) n The sum of every line’s numbers equals 2. In case the figure is coloured, the multiples of two form equilateral triangles.
- 39. Pascal’s Quotes Do not try to add years to your life. You should better add life to your years. The story of mankind would be different if Cleopatra’s nose had different shape. There are two kinds of people: the fair ones, who consider themselves as sinful, and the sinful ones, who consider themselves as fair. The greatness of a man is located behind his ability to think.
- 40. A logician A philosopher A mathematician
- 41. He was born on the 4th of August of 1834. He died on the 4th of April of 1934 of unknown causes. He was educated by private lesson until 1853 when he went to Gonville and Caius College of Cambridge. In 1857 he graduated and became a member of the college staff. In 1862 he returned to Cambridge as a lecturer of Ethics while he studied and taught Logic and the Theory of Probability.
- 42. Venn diagrams were invented around 1880. They are used in many fields, such as: Set theory Probability Logic Statistics Information technology
- 43. Venn Diagrams are an illustration of sets. In every Venn Diagram there are: A rectangle which symbolizes the biggest set there could be, depending on what we want to show, which is usually symbolized by U. Closed lines, usually curves and circles. The surface they cover symbolizes the set itself.
- 44. In a Venn diagram every surface which is defined by any combination of lines symbolizes a set.
- 45. He was born in 630 in Militos He died at the age of 78 around 543 b.C at the Olympic games due to the heat and thirst. He is known as the first philosopher of the seven in the ancient world Except of philosophy, he was also interested in mathematics, physics, astronomy, engineering, meteorology He set the Milicius school up He loved travelling and he made a lot of trips all over the world
- 46. He thought that The shape of earth was circular disk and Water is the starting point of the whole life. He invented electricity and the magnetic fields.
- 47. Some of Thales’ theories are: When parallel lines are being crossed by two other lines then the parts between the parallel lines are analogous. Ιf A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved as part of the 31st proposition, third book of Euclid's Elements
- 48. Τhales’ quotes: Σοφώτατον χρόνος· ἀνευρίσκει γὰρ πάντα. ◦ Time is the wisest of all things that are; for it brings everything to light. Οὔ τι τὰ πολλὰ ἔπη φρονίμην ἀπεφήνατο δόξαν ◦ A multitude of words is no proof of a prudent mind. Ἐὰν ἃ τοῖς ἄλλοις ἐπιτιμῶμεν, αὐτοὶ μὴ δρῶμεν ◦ Avoid doing what you would blame others for doing. Μέγιστον τόπος· ἅπαντα γὰρ χωρεῖ ◦ Place is the greatest thing, as it contains all things. Γνῶθι σαυτόν ◦ Know themselves.
- 49. Pythagoras was a very important Greek : Philosopher Mathematician Geometer Theoretical in music He was born between 592 B.C and 572 B.C . . He died in old age in Metapontis ,Italy. . He was an intelligent personality and with his fluency he gained admiration and respect of all his fellow citizens of that time .He travelled to Egypt and Samos and he tried to teach other people . He was so intelligent that his fame reached as far as Miletus and Priene to two of the seven wise men of antiquity(Thales and Vianta) and in many places people admired the young Pythagoras.
- 50. Pythagorean theorem Pythagorean triads Asymmetric sizes Study regular pentagonal Musical scale construction Pythagorean school
- 51. "The square of the hypotenuse (the side opposite the right angle) of a right-angled triangle is equal to the sum of the squares of the two vertical sides."
- 52. Pythagoras studied and created at least three (dodecahedron, tetrahedron, cube) of the five regular polyhedral.
- 53. Pythagorean school was in Kroton, Italy, whose founder was Pythagoras. The Pythagorean school had religious, political and scientific nature. On the school’s entrance the following quote was carved by the Pythagoreans: «MΗΔΕΙΣ ΑΓΕΩΜΕΤΡΗΤΟΣ ΕΙΣΗΤΟ» That is no one, who cannot count every object using human measures, is allowed to enter or participate in this fraternity.
- 54. “Words are the winds of soul”. “Education is a golden wreath ,not only because of its great value but also of its benefit that it offers”. “ As it seems justice resembles a square all parts are equal and identical”.
- 55. • Ancient Greek philosopher (427 b.C.-347b.C) who died in the age of 80 • Born in Athens • Socrates is his most known student • Aristotle’s teacher
- 56. Plato established Academia in Athens, around 387 b.C., to organize his educational plans. Academia was named after the location where it was established; the gardens of Academus. Ioustinianos, the emperor of Byzantium shut it down in 529 b.C.
- 57. Tetrahedron Cube Octahedron Dodecahedron Icosahedron In mathematics he is widely known for the Platonic Solids which were named like this because they were studied by Plato's Academia Platonic solid is a regular regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex
- 58. According to Plato, geometry relates the world of ideas with the natural world. The natural world doesn’t consist perfect cycles, straight lines or points and the geometric objects don’t exist as eternal and unstoppable ones. The geometrical knowledge isn’t conquered with clear thought or with memory of the soul.
- 59. Ancient Greek mathematician Geographer Astronomer Poet Historian Music theorist Born in Cyrene in 276 b.C. (current Libya) He became the chief librarian at the Library of Alexandria in 236 b.C. And he stayed there in charge for 40 years teaching in its museum. In194 b.C. he went blind and a year later he stopped eating and then he died in Alexandria
- 60. His most important achievements were the sieve of Eratosthenes and the measurement of the Earth's circumference. In mathematics the sieve of Eratosthenes’ is a simple algorithm for finding prime numbers.
- 61. In the 3rd century b.C. Eratosthenes was informed that in Swenet during the noon on the summer solstice, the sun appears directly overhead and its reflection heads towards the bottom of a well. Simultaneously in Alexandria the sun rays form an angle 7ο with the zenith of the location. Then he measured the distance between Alexandria and Swenet and calculated, as it is shown in the picture below, with great accuracy the Earth’s circumference.
- 62. He lived from : 325 B.C. – 265 B.C He born– live-died : In Alexandria, Egupt. He was: Mathematician He called : “Father’’ of Geometry He contributed to: History of maths logic. He innovated at: The production of a formal cohesive total with propositions
- 63. Euclid’s contributions at the sector of Geometry and consequently in Hellenic science is enormous. He wrote one of the best tasks that named ''elements'‘
- 64. ‘’ELEMENTS’’ Had gather all the geometric knowledge until that time and he (Euclid) had add his own knowledge in a marvelous way separated at 13 books and conclude 382 theories mention Pythagoras an Theoklitos discoveries. 1-6mention plan meter 7th to 10th arithmetic and the last three stereometry
- 65. Based on one of the most important theorems. Ex. Let’s suppose that there is a straight line ε and an exterior point Α, there is only one parallel straight line which can pass from the ε. His simple perception of the space supports Euclidean geometry.
- 66. However, he had written another important tasks: 1.“dedomena” 2. “Peri diairesewn” 3. “Psedaria” 4.“Kwnika” 5. “Optica” 6. “Phenomenika” 7.“Katoptrika” 8. “Katanomi Kanonwn” 9. “Topoi of Epifanion”
- 67. Nothing is totally sure for his life . Also we know that: he related with Alexandria’s library. he probably studies at Platoons academy. What is more he acquired his reputation in Pallada for his achievements at Math’s due to the fact that Ptolemy invited him in Alexandria and called him "savior" . Additionally, his work appreciated by Pharaoh. For his character we only know he was gentle and contagious.
- 68. He lived from 287 B.C – 212 B.C and died at the age of 75 He was at the same time a physician, mathematician and mechanic He is considered as the greatest Mathematician of his age
- 69. A weapon that threw rocks and arrows against enemies) Street meter ( means that could calculate the distance that traversed a moving object) Wreckers (a machine that could capture the opponents’ boats and cause them damage, while they were sieging his town) Planetarium (a device capable of calculating the exact position of the sun, the moon and other planets) A device that could define the concentration and salinity of fluids) A screw (that was used for pulling water and utilize it in irrigation)
- 70. On the measurement of circle On the Sphere and the Cylinder On spirals On floating bodies (o) Stomachion These and many others are only a few of the Writings of Archimedes that were saved, while the ones that are gone are not a few. Furthermore he did a lot of research based on the area of circle, ellipse,parabola and spiral . He also studied the area and volumes of cylinders, cones and especially the volume of the sphere. Thus he did he calculated satisfactorily the decimals of ‘π’ and that’s why it is called “constant of Archimedes”.
- 71. In Euclid Geometry the number π is a mathematical constant, the ratio of a circle's circumference to its diameter.
- 72. Other scientists argue that ‘π’ is a fully-made rotation of a circle, diameter 1, on a horizontal floor.
- 73. One day, the king Ieronas called Archimedes to give him a task. Earlier, Ieronas had ordered a crown to be made for him. So, he wanted Archimedes to find out whether the crown was purely made out of gold or not. Later,while Archimedes was at his bathroom he realised that as he was sinking in it, the surface of the water was emerging! He had finally found the solution of that task! He immediately jumped out of the bathroom and started running on the streets claiming “Eureka”. He thought that if he sank the crown the surface of the water would emerge too! So he tried that experiment comparing the crown with pure gold, and discovered that the crown was fake!
- 74. His life Diophantus is a Greek mathematician who lived in Alexandria, Egypt in the 3rd century AD and he died in old age He worked on the solution of problems that had the form of equation and this helped the evolution of Αlgebra .Some people call him "father" of Algebra. The Epigram is a known mathematical riddle. From the solution we learn that Diophantus passed away 84 years.
- 75. The Numerically is the best known and oldest Greek textbook. It includes 130 problems Also studied and developed the undefined or Diophantine equations, namely the equations with multiple solutions. To these problems required only positive solutions The most famous Diophantus problem is the problem 8 of the II book of numerical: “Analyze a given square number into two square numbers”
- 76. Diophantus 1/6 his life he was a child. 1/12 was young and then spent 1/7 his life until he married. Five years later, his son was born. The life of his son, was the half life of Diophantus. After the death of his son, he lived four years in deep sorrow and then he died. How many years has lived Diophantus? (The answer is 84 years)
- 77. She was a mathematician, philosopher ,engineer and astonomer. She was born in 370 B.C. and died in 416 B.C. Her father’s name was Theon She took part at the coursework in Neoplatoniki school from Plutarch the younger
- 78. She discovered that the movement of the earth is elliptical. Astronomical canon. The Apollonian cones. She wrote 13 books with comments about the arithmetic of Diophantus .
- 79. She had a big interest for engineering and practical technologies So she made several organs such as : One astrolabe used for measuring the positions of stars, planets and the sun. Developed a device for refining water And the hydrometer to measure a liquid for specific gravity
- 80. Hypatia was a woman who separated the community in two parts. • Those who considered it a miracle of light. • Those who was seeing it like a apostle of darkness. Her action considered dangerous for the spread of Christianity, gradually has cultured a climate which was versus her and has led to her violent murder from a mob or from fanatic monks teams.
- 81. C R O A T I A - G R E E C E

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