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MATH 2306 History of Mathematics Instructor: Dr. Alexandre Karassev COURSE OUTLINE • • • • The Theorem of Pythagoras (Ch. 1) Greek Geometry (Ch. 2) Greek Number Theory (Ch. 3) Infinity in Greek Mathematics (Ch. 4) } Greek Mathematics (≈ 300 BCE – 250 CE) • Number Theory in Asia (Ch. 5) } China and India (≈ 300-1200 CE) • Polynomial Equations (Ch. 6) • • • Calculus (Ch. 9) Infinite Series (Ch. 10) The Number Theory Revival (Ch. 11) • Complex Numbers in Algebra (Ch. 14) } Europe (17th – 18th century CE) Chapter 1 The Theorem of Pythagoras • • • • • • • Arithmetic and Geometry Pythagorean Triples Rational Points on the Circle Right-angled Triangles Irrational Numbers The Definition of Distance Biographical Notes: Pythagoras 1.1 Arithmetic and Geometry The Theorem of Pythagoras If c is the hypotenuse of a right-angled triangle and a, b are two other sides then a2+b2=c2 “Let no one unversed in geometry enter here” was written over the door of Plato’s Academy (≈ 387 BCE) a2 a c b b2 Remarks • Converse statement: if a,b and c satisfy a2+b2=c2 then there exists a right-angled triangle with corresponding sides. • One can consider a2+b2=c2 as an equation • It has some simple solutions: (3,4,5), (5,12,13) etc. • Practical use - construction of right angles • Deep relationship between arithmetic and geometry • Discovery of irrational numbers 1.2 Pythagorean Triples • Definition Integer triples (a,b,c) satisfying a2+b2=c2 are called Pythagorean triples • Examples: (3,4,5), (5,12,13), (8,15,17) etc. • Pythagoras: around 500 BCE • Babylonia 1800 BCE: clay tablet “Plimpton 322” lists integer pairs (a,c) such that there is an integer b satisfying a2+b2=c2 • China (200 BCE -220 CE), India (500-200 BCE) • Greeks: between Euclid (300 BCE) and Diophantus (250 CE) • Diophantine equation (after Diophantus, 300 CE) - polynomial equation with integer coefficients to which integer solutions are sought • It was shown that there is no algorithm for deciding which polynomial equations have integer solutions. General Formula • Theorem Any Pythagorean triple can be obtained as follows: a = (p2-q2)r, b = 2qpr, c = (p2+q2)r where p, q and r are arbitrary integers. • Special case: a = p2-q2, b = 2qp, c = p2+q2 • Proof of general formula: Euclid’s “Elements” Book X (around 300 BCE) 1.3 Rational Points on the Circle • Pythagorean triple (a,b,c) • Triangle with rational sides x = a/c, y = b/c and hypotenuse c = 1 • x2 + y2 = 1 → P (x,y) is a rational point on the unit circle. Y P 1 O x y X Construction of rational points on the circle • Base point (trivial solution) Q(x,y) = (-1,0) • Line through Q with rational slope t y = t(x+1) intersects the circle at a second rational point R • As t varies we obtain allY rational points on the circle which have the form R x = (1-t2) / (1+t2), y = 2t / (1+t2) where t = p/q Q X -1 O 1 of Pythagoras’ Triangles Theorem 1.4 Proof Right-angled a2 a c b b2 1.5 Irrational Numbers • For Pythagoreans “a number” meant integer • The ratio between two such numbers is a rational number • According to the Pythagoras theorem, the diagonal of the unit square is not a rational number • Discovery of incommensurable lengths (not measurable as integer multiple of the same unit) • Irrational numbers (diagonal ) 2 12 12 2 1 diagonal 2 1 Consequences of this discovery • According to the legend, first Pythagorean to make the discovery public was drowned at sea • Split between Greek theories of number and space • Greek geometers developed techniques allowing to avoid the use of irrational numbers (theory of proportions and the method of exhaustion) 1.6 The Definition of Distance • Coordinates of a point on the plane: pair of numbers (x,y) • Development of analytic geometry (17th CE) • Notion of distance P R y O P Y Y x X ? X O Y Pythagoras’ theorem: P (x2 , y2 ) x2-x1 R (x1 , y1 ) O y2-y1 PR 2 ( x2 x1 ) 2 ( y2 y1 ) 2 X PR ( x2 x1 ) 2 ( y2 y1 ) 2 Alternative approach: Definition A point is an ordered pair (x,y) Distance between two points R (x1 , y1 ) and P (x2 , y2 ) is defined by formula ( x2 x1 ) 2 ( y2 y1 ) 2 1.7 Biographical Notes: Pythagoras • Born on island Samos • Learned mathematics from Thales (624 - 547 BCE) (Miletus) • Croton (around 540 BCE) • Founded a school (Pythagoreans) – “All is number” – strict code of conduct (secrecy, vegetarianism, taboo on eating beans etc.) – explanation of musical harmony in terms of whole-number ratios Pythagoras (580 BCE – 497 BCE)