History_of_Mathematics_Part_2 (3).pdf

History of Mathematics: Early Ages Ancient Greece Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 IŞIK UNIVERSITY Persians dominated the Middle-East around 600 BC. They invaded Athens between 500-480 BC. Greeks reclaimed Athens at 479 BC. This date is assumed to...

History of Mathematics: Early Ages Ancient Greece Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 IŞIK UNIVERSITY Persians dominated the Middle-East around 600 BC. They invaded Athens between 500-480 BC. Greeks reclaimed Athens at 479 BC. This date is assumed to be the beginning of Ancient Greek Civilization. ”History of Mathematics: Early Ages” by Deniz Karlı 2 0 IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı Thales (624-547 BC) Two ancestors of Greek Mathematics are Thales & Pythagoras. Thales was born in Miletus (which is in Aydın Turkey today). He traveled to Egypt and learnt Geometry. It is told that he measured the height of pyramids by using ratio between his and pyramid’s shadows. After returning Miletus, he formed a group (a school) to teach Geometry. It is assumed that that he introduced theoretical proofs (which is based on logic and not the experimental deductions) to mathematics first time. 3 IŞIK UNIVERSITY Pythagoras (569-475 BC) ”History of Mathematics: Early Ages” by Deniz Karlı 4 Pythagoras was born in Samos Island (which is in Greece near Miletus). He stayed with Thales and then traveled to Egypt by Thales’ advice. He learnt Geometry in Egypt until captured by Persians. He was sent to Babylon. There, he studied Mathematics, Music and Theology for 5 years. After 5 years, Pythagoras returned to Samos and formed his school. In 518 BC, he moved to Crotone (Southern Italy) and started a new school. IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 5 Pythagors’ School The School was a cult-like community consisting of 2 groups: matematikoi and students. Matematikoi was the upper class of this community. They lived together and are bounded with a oath to each other. Ideology was built on numbers. They claim that there is a divine harmony between numbers and everything can be explained by reducing to numbers. Their idea of numbers was built on rational number. (Whole numbers like 1,2,3,... and fractions like 1/2,3/7...) Pythagoras’ Theorem led to a big crisis. C b c 9762 IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı Pythagoras’ Theorem a Pythagoras’ Theorem: white area 0462 left c b Obtaining an irrational (not rational) number by Pythagoras’ Theorem: C 1,1C b a Visual proof: the on c 6 C 1712 2 √ 2 IŞIK UNIVERSITY Why isn’t √ ”History of Mathematics: Early Ages” by Deniz Karlı 2 a rational number? 7 1 5 All rational numbers can be written as a ratio of two integers whose greatest common divisor is 1. That is, all rational numbers are of the form EE a b where both a and b are integers and no number other than 1 can divide both. Assume √ a 2=. b a seven 9 even b E az where b k is 2 2 on b integer b EYE is even 4.42 2k 2 b 2 fbELEitte.ba m a 2 k BEN This is a contradiction f.db.Myassumptienwaswrang IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 8 Schools of Mathematics in Athens & Alexandria There are many great names and famous schools in Ancient Athens & Alexandria. We will discuss Academius by Plato (387 BC - 529 AD) and Museum by Ptolemy (312 BC - 421 AD). IŞIK UNIVERSITY Plato (427-347 BC) ”History of Mathematics: Early Ages” by Deniz Karlı 9 Plato was a student of Socrates. After Socrates was killed, he traveled and studied in Egypt, Sicilia, Italy for 10 years. There he learnt Geometry from Pythagoreans. There, he understood how important Mathematics was for critical thinking. He returned Athens in 387 BC, and established his school called Academius. This is the first university in history. In this school, Philosophy, Geometry, Music and Gymnastics were main courses. At this age the word ”Geometry” was used instead of ”Mathematics”. IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 10 IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı Academius The first great mathematician, who studied in Academius, was Euclid (325-265 BC) and the last one was Proclus (411-485 BC). Proclus wrote many books which became our main source about Mathematics of this era. Another mathematician from Academius is Eudoxus (395–390 BC). He developed the method of ”Exhaustion”. Exhaustion: Measuring a non-regular area or volume by filling it with regular shapes. This led to the idea of integration more than 1000 years later. 11 IIIEĞiniple Fill areas in the remaining with smaller shapes Then fill area in regular further with even the remaining smaller regular shapes so on IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 12 The Museum (Alexandria) One of the most important Mathematical centers of the age was The Museum in Alexandria which was established by Ptolemy (312 BC - 421 AD). It was both a research and teaching institute for advanced sciences. More than 100 scientist worked here where all the expanses were compensated by the state. There was a very rich library, botanical gardens and observatory. IŞIK UNIVERSITY Euclid (325-265 BC) One of the great mathematicians who used to teach in The Museum was Euclid. He is referred to as ”the father of Geometry”. His book ”Elements” is one of the most influential books of the human history. Elements was printed over 1000 times, translated to all languages and used as a textbook by all civilizations throughout the human history. ”History of Mathematics: Early Ages” by Deniz Karlı 13 IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 14 Elements The book, Elements, consists of 13 volumes written on papyrus.(In this era, one papyrus has an average length of 20-50 pages of a book of nowadays.) Euclid gather arguments of Geometry developed up to his age in Elements. What makes Elements so special is the way it is presented by Euclid. All arguments are based on a systematic formation. The Elements contain many well-known mathematical results such as Pythagoras’s Theorem, Thales Theorem, Sum of Angles in a Triangle, Prime Numbers, Greatest Common Divisor and Least Common Multiple, Euclidean Algorithm, Areas and Volumes, Tangents to a point and Algebra. IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 15 Elements’ Systematic A. First, Euclid gives 5 axioms which are universally valid. B. Then he makes definition of terminologies such as point, line, plane etc. C. Then he lists his 5 postulates: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 1 1 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. D. Finally, Euclid proves his statements (theorems) using only axioms and logical deduction without introducing anything else than these axioms and postulates. This is the fundamental systematic approach of modern Mathematics and science. IŞIK UNIVERSITY Contents of Elements ”History of Mathematics: Early Ages” by Deniz Karlı 16 Euclidean Geometry There are All ideas other geometries A 1 7 works on planes IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı 17 Archimedes (287-212 BC) Archimedes is one of the greatest minds of all history. He was was a mathematician, physicist, engineer, astronomer, and inventor It is believed that he was studied or at least visited the Museum. He used the ”exhaustion” method to compute area of many shapes, which later became fundamental idea leading to modern integration. Archimedes proved many theorem including the ones about the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. IŞIK UNIVERSITY ”History of Mathematics: Early Ages” by Deniz Karlı Archimedes (287-212 BC) Archimedes showed that the number π is in the interval 10 10 3