Unformatted text preview: Greek Mathematics Greek Mathematics
600 B.C. – 450 A.D. Birth of Demonstrative Mathematics Birth of Demonstrative Mathematics
• Beginning of Greek civilization around 1100 B.C. • Developed into a prosperous, independent and sophisticated thinkers. Everything was open to debate and analysis with the development of Greek democracy. • By 400 B.C., this remarkable civilization had a rich intellectual history. Demonstrative Mathematics Demonstrative Mathematics
• The Egyptians and Babylonians never asked the question of “why”. They only worried about “how”. • Deductive features of mathematics came into prominence – fundamental characteristic of mathematics. • Beginning of modern mathematics. Outstanding Greek Mathematicians Outstanding Greek Mathematicians
• • • • • •
Thales of Miletus (640 B.C. – 546 B.C.) Pythagoras (572 B.C. – 497 B.C.) Eudoxus (408 B.C. – 355 B.C.) Euclid (300 B.C.) Archimedes (287 – 212 B.C.) Apollonius (250 – 175 B.C.) Thales of Miletus Thales of Miletus
• First to prove the following geometric 1. 2. 3.
results: A circle is bisected by any diameter. The base angles of an isosceles triangle are equal. The vertical angles formed by two intersecting lines are equal. Theorems of Thales Theorems of Thales
1. Two triangles are congruent if they have two angles and one side in each respectively equal. 2. An angle inscribed in a semicircle is a right angle. • Thales proved these theorems by some logical reasoning instead of intuition and experiment. Pythagoras and the Pythagoreans Pythagoras and the Pythagoreans
• Pythagoras was born in Samos around • • • •
572 B.C. Studied with Thales and travel to Egypt. Fled to the greek town of Crotona in Italy. Founded a society known as the Pythagoraen Brotherhood. Originally an academy of philosophy, mathematics and natural sciences. Pythagoreans Pythagoreans
• Later developed into a brotherhood with secret • •
rites and observances. Pythagorean philosophy rested on the assumption that whole numbers formed the basic organizing principle of all. Pythagorean program of study consists of the study of number properties, arithmetic, geometry, music and spherics (astronomy). Pythagoras’ Teaching Pythagoras’ Teaching
• Its difficult to know which mathematical findings should be credited to Pythagoras and which to his followers as his teaching was oral. • Ancient Greeks made a distinction between the abstract relationship of numbers (arithmetic) and the practical art of computing with numbers (logistic). Number Theory Number Theory
• Discovery of amicable numbers. Two numbers are amicable if each is the sum of the proper divisors of the other. • Example: 220 and 284 since the proper divisors of 220 are 1,2,4,5,10,11,20,22,44, 55, 110 and their sum is 284. The proper divisors of 282 are 1,2,4,71,142 and their sum equals 220. Amicable numbers Amicable numbers
• 220 and 284 was the first pair of amicable numbers discovered by Pythagoras. These two numbers came to play an important role in magic, astrology and horoscopes. No new pair was discovered until 1636 by Pierre de Fermat – 17,296 and 18,416. Recently announced that this was a rediscovery. These numbers were found previously by Arab alBanna in the late 13th century. • • Perfect, deficient and abundant Perfect, deficient and abundant numbers
• Other numbers from originated from the Pythagorean brotherhood are: • Perfect – if it equals the sum of its proper divisors. • Deficient if it exceeds the sum of its proper divisors. • Abundant if t is less than the sum of its proper divisors. Examples Examples
• • • •
Perfect – 6=1+2+3 , 28=1+2+4+7+14 Deficient – 8>1+2+4, 32>1+2+4+8+16 Abundant – 40 < 1+2+4+5+8+10+20 Until 1952, there were only 12 known perfect numbers, all even. First three are 6, 28 and 496. • Existence or nonexistence of odd perfect numbers – one of the unsolved problems. Prime numbers Prime numbers
• A prime number is a positive integer greater than 1 that has no positive integral divisors other than 1 and itself. An integer greater than 1 that is not a prime number is called a composite number. Last proposition of the 9th Book of Euclid’s Elements proves that if is a prime 2n − 1 1 2 n − (2 n − 1) number, then is a perfect number. • Perfect numbers Perfect numbers
• In 1952, with the help of a digital computer, five more perfect numbers were found for n=521, 607, 1279, 2203, and 2281 in Euclid’s formula. In 1957, the next perfect number was found for n=3217. In 1961, two more were found for n=4253 and n=4423. • 30 known perfect numbers. Figurate numbers Figurate numbers
• These numbers represent a link between geometry and arithmetic. Number of dots in certain geometrical configurations. • Example: Triangular (1,3,6,10,…), Square (1,4,9,16,…), Pentagonal (1, 5, 12, 22,…). • Many interesting theorems concerning figurate numbers can be established in a purely geometrical fashion. Pythagorean Theorem Pythagorean Theorem
• “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs”. • This theorem was known to the Babylonians a thousand years earlier but the first general proof was given by Pythagoras. • Over 370 proofs since then. Pythagorean Triples Pythagorean Triples
• Closely related to the Pythagorean theorem is the problem of finding integers a, b, c that can represent the legs and hypotenuse of a right triangle – Pythagorean triple. Pythagoreans credited with the formulas n2 − 1 n2 + 1 , n, , n − odd number 2 2 m 2 m 2 m, − 1, + 1 , m − even number 2 2 Irrational numbers Irrational numbers
• A rational number is the quotient of two integers •
p/q, q not equal to 0. Two line segments AB and CD are said to be commensurable if there exists a smaller segment EF that goes evenly into both AB and CD, that is, uuu r uuu r AB = pEF , uuu r uuu r CD = qEF Commensurability Commensurability
• Pythagoreans felt that any two magnitudes are commensurable. Later Hippasus discovered that the side and its diagonal are not commensurable. 2 • To show this, it suffices to show that is not rational. • This discovery had different impacts on the Pythagorean philosophy and proofs. Discovery of irrationals Discovery of irrationals
• On the supremacy of whole numbers – this discovery was a blow to their philosophy that all depends upon the whole numbers. • On their proofs that rested upon the assumed commensurability of all segments. Notion of commensurability Notion of commensurability
• A break from the Babylonian and Egyptian concepts of calculation with numbers. • No question that one can assign a numerical value to the length of the diagonal of a square of side one unit (as the Babylonians did), but the notion that no ‘exact’ value can be found is first formally recognized by the Greeks. Irrational numbers Irrational numbers
2 • For some time, was the only known irrational. • Later, according to Plato, Theodorus of Cyrene (425 B.C.) showed that are 3, 5, 6, 7, 8,... also irrational. About 370 B.C., the ‘scandal’ was resolved by the brilliant Eudoxus, a pupil of Plato and the Pythagorean Archytas who put forth a new definition of proportion. Coincides with treatment of irrationals by Richard Dedekind in 1872. Algebraic Identities Algebraic Identities
• Due to lack of proper algebraic notation, the early Greeks use geometrical processes for carrying out algebraic operations. Much of this geometrical algebra can be attributed to the Pythagoreans and can be found in several of the earlier books of Euclid’s Elements. Algebraic Identities Algebraic Identities
• Examples: Proposition 4 of Book II shows geometrically the identity (a + 2 = + b) a2 2 ab + b2 • “If a straight line is divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the two parts”. Algebraic Identities Algebraic Identities
• Proposition 4 a
b b b a a a a b Algebraic Identities Algebraic Identities
• Proposition 5 of Book II is “If a straight line is divided equally and also unequally, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line”. Geometric Solution of Quadratic Geometric Solution of Quadratic Equations
• Example; Special case of Proposition 28. “To divide a given line segment so that the rectangle contained by its parts will equal a given square, the square not exceeding the square on half the given line segment”. ...
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 Spring '10
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 Proper divisors, Identities Algebraic Identities, Algebraic Identities Algebraic Identities, Pythagoreans Pythagoreans

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