GREEK MATHEMATICS

1. GREEK MATHEMATICS

2. INTRODUCTION • The beginnings of Greek mathematics originated from the 6th century BC to the 5th century AD • The word mathematics comes from the Greek word μάθημα (mathema), meaning "subject of instruction“

3. PERIODS IN GREEK MATHEMATICS • FIRST – influenced by Pythagoras • SECOND – Plato and his school • THIRD – Alexandrian School flourished in Grecian Egypt and extended its influence to Sicily and Palestine

4. GREEK NUMBERS • Greeks had a variety of different ways of writing down numbers • Some Greeks used a system based on writing the first letter of the word for that number • For number ten “Deka”, they would draw a D to mean 10. (a delta, in the Greek alphabet)

5. Some other numbers in greek symbols

6. Because the Greeks had very clumsy ways of writing down numbers, they didn't like algebra • They were more focused on geometry, and used geometric methods to solve problems that you might use algebra for • They found it very hard to write down equations or number problems.

7. Greek mathematicians were very interested in proving that certain mathematical ideas were true. • They spent a lot of time using geometry to prove that things were always true,even thoughpeople like Egyptians and Babylonians already knew that they were true most of the time away.

8. Because the Greeks had very clumsy ways of writing down numbers, they didn't like algebra • They were more focused on geometry, and used geometric methods to solve problems that you might use algebra for • They found it very hard to write down equations or number problems.

9. MOST FAMOUS GREEK MATHEMATICIANS • Thales • Pythagoras • Anaxagoras • Democritus • Aristotle • Hipocrates • Euclid • Archimedes

11. THALES(grč.Θαλής) • Born 624. BC in Miletus • the first of the Greeks who took any scientific interest in mathematics in general • Improved Egyptian mathematics

12. THALES • He knew many number relations • In his work is the foundation of deductive geometry • He is credited with a few of the simplest propositions relating to the plane figures • His great contribution lay in suggesting a geometry of lines and in making the subject abstract • He gave the idea of a logical proof as applied to geometry

13. PROPOSITION RELATING PLANE FIGURES • a circle is bisected by its diameter, • the angles at the bases of any isosceles triangle are equal • if two straight lines cut one another, the opposite angles are equal. • if two triangles have two angles and a side in common, the triangles are identical.

14. INTERCEPT THEOREM • The ratios of any 2 segments on the first line equals the ratios of the according segments on the second line

15. THALES THEOREM • If AC is a diameter, then the angle at B is a right angle

16. PHYTAGORAS (grč.Πυθαγόρας) • Born 570. BC in Samos • Died 495. BC • worked with abstract geometric objects and numbers • gathered his school as a sort of mathematician secret brotherhood

17. PHYTAGORAS THEOREM • in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse

18. TV screen size is measured diagonally across the screen. A widescreen TV has an aspect ratio of 16:9, meaning the ratio of its width to its height is 16/9. Suppose that a TV has a one inch boundary one each side of the screen. If Joe has a cabinet that is 34 inches wide, what is the largest size wide screen TV that he can fit in the cabinet?

19. SQUARE NUMBERS • These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots

20. PYTHAGORAS AND MUSIC • musical notes could be translated into mathematical equations

21. DEMOCRITUS(grč.Δημόκριτος) • Born 460. BC, died 370.BC • Famous atomist • introduced the idea of an infinite number of points that make up the line

22. He observed that a cone or pyramid has one-third the volume of a cylinder or prism respectively with the same base and height

23. Plato (428 BC – 348 BC), Philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western World.

24. Plato’s Cave Analogy

26. In Plato’s Divided Line, Mathematics falls under the following category: • Highest form of true knowledge • Second highest form of true knowledge • A form of belief, but not true knowledge • A form of perception

27. ARISTOTLE (grč.Ἀριστοτέλης) • Born 384. BC, died 322. BC • Greek philosopher, a student of Plato and teacher of Alexander the Great

28. For him the base of mathematics is logic, but the nature of mathematical relations is completely specified by postulates that dictates the physical experience

29. HIPPOCRATES (grč. Ἱπποκράτης ) • Lived from 460. BC to 377. BC • an ancient Greek physician and was considered one of the most outstanding figures in the history of medicine

30. HIPPOCRATUS PROBLEM • He proved that the lune bounded by the arcs labeled E and F in the figure has the same area as does triangle ABO

31. EUCLID (grč.Εὐκλείδης) • Born 300. BC • pioneer of axiomatics in geometry • His work Elements fundamental work in the field of Greek mathematics • influenced the development of mathematics in the next 20 centuries

32. ELEMENTS • written about 300 B.C. • textbook that includes number theory • the Euclidean algorithm for finding the greatest common divisor of two numbers

33. the first edition of the translation from Arabic into Latin 1482.

34. The axiomatic method The Elements begins with definitions and five postulates. There are also axioms which Euclid calls 'common notions'. These are not specific geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science. For example: “Things which are equal to the same thing are equal to each other.””

36. Euclid's fifth postulate cannot be proven from others, though attempted by many people. Euclid used only 1—4 for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823,Bolyai and Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold.

37. Our world is non Euclidean Restate the fifth postulate:Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. Spherical geometry is just as real as Euclidean geometry, but the theorems and general results are very different. There are quite a few results from Euclidean geometry that are completely false in spherical geometry (and vice versa).

38. ARCHIMEDES (grč.Ἀρχιμήδης) • mathematician and inventor born 287. BC in Syracuse • founder of quantitative physics • as a mathematician, advocate of logical processes

39. He determined approximate values of some irrational numbers 1351/780> >265/153 28/7> π >223/71

40. A sphere has 2/3 the volume and surface area of its circumscribing cylinder • A sphere and cylinder were placed on the tomb of Archimedes at his request

41. LITERATURA • Vladimir Devide: “Na izvorima matematike” • Dadić Žarko: “Povijest ideja i metoda u matematici i fizici”; ŠK, 1992. • http//www.ibilio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html • http://www.historyforkids.org

42. Authors: Ivana Pušić Dajana Rudić Ines Malić