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MATH 2306 History of Mathematics

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1 MATH 2306 History of Mathematics
Instructor: Dr. Alexandre Karassev

2 } } 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 (17th – 18th century CE)

3 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

4 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 a2+b2=c2 “Let no one unversed in geometry enter here” was written over the door of Plato’s Academy (≈ 387 BCE) c2 a2 a b c b2

5 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

6 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 ( BCE) Greeks: between Euclid (300 BCE) and Diophantus (250 CE)

7 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.

8 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)

9 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 X O x y 1 P

10 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-t2) / (1+t2), y = 2t / (1+t2) where t = p/q Y X O 1 Q -1 R

11 1.4 Right-angled Triangles
Proof of Pythagoras’ Theorem c2 a2 a b c b2

12 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

13 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)

14 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 Y X O P R Y X O P ? x y

15 Alternative approach: Definition A point is an ordered pair (x,y)
P (x2 , y2 ) Pythagoras’ theorem: y2-y1 x2-x1 R (x1 , y1 ) X O 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

16 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)


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