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) Number Theory in Asia (Ch. 5) Polynomial Equations (Ch. 6) Calculus (Ch. 9) Infinite Series (Ch. 10) The Number Theory Revival (Ch. 11) Complex Numbers in Algebra (Ch. 14) Greek Mathematics (≈ 300 BCE – 250 CE) } } China and India (≈ CE) Europe (17 th – 18 th 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 a 2 +b 2 =c 2 a b c c2c2 b2b2 a2a2 “Let no one unversed in geometry enter here” was written over the door of Plato’s Academy (≈ 387 BCE )

Remarks Converse statement: if a,b and c satisfy a 2 +b 2 =c 2 then there exists a right-angled triangle with corresponding sides. One can consider a 2 +b 2 =c 2 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 a 2 +b 2 =c 2 are called Pythagorean triples a 2 +b 2 =c 2 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 a 2 +b 2 =c 2 China (200 BCE -220 CE ), India ( 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 = (p 2 -q 2 )r, b = 2qpr, c = (p 2 +q 2 )r where p, q and r are arbitrary integers. Special case: a = p 2 -q 2, b = 2qp, c = p 2 +q 2 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 x 2 + y 2 = 1 → P (x,y) is a rational point on the unit circle. Y X O x y 1 P

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 all rational points on the circle which have the form x = (1-t 2 ) / (1+t 2 ), y = 2t / (1+t 2 ) where t = p/q Y X O 1 Q R

1.4 Right-angled Triangles Proof of Pythagoras’ Theorem a b c c2c2 b2b2 a2a2

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 1 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 (17 th CE ) Notion of distance Y X O P x y Y X O P R ?

Y X O R (x 1, y 1 ) P (x 2, y2 ) x 2 -x 1 y 2 -y 1 Pythagoras’ theorem: 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

1.7 Biographical Notes: Pythagoras Born on island Samos Learned mathematics from Thales ( 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 )