Chapter 5 Born about 370 AD. She was the first woman to make a - PDF document

The Saga of Mathematics A Brief History Hypatia of Alexandria Chapter 5 � Born about 370 AD. � She was the first woman to make a substantial contribution to the development of mathematics. Must All Good Things � She taught the philosophical ideas of Come to an End? Neoplatonism with a greater scientific emphasis. Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics 2 Neoplatonism Neoplatonism � The founder of Neoplatonism was Plotinus. � Plotinus stressed that people did not have the mental capacity to fully understand both the � Iamblichus was a developer of Neoplatonism ultimate reality itself or the consequences of its around 300 AD. existence. � Plotinus taught that there is an ultimate reality � Iamblichus distinguished further levels of reality which is beyond the reach of thought or language. in a hierarchy of levels beneath the ultimate � The object of life was to aim at this ultimate reality. reality which could never be precisely described. � There was a level of reality corresponding to every distinct thought of which the human mind was capable. The Saga of Mathematics 3 The Saga of Mathematics 4 Lewinter & Widulski Lewinter & Widulski Hypatia of Alexandria Hypatia of Alexandria � She is described by all commentators as a � She was murdered in March of 415 AD by charismatic teacher. Christians who felt threatened by her scholarship, learning, and depth of scientific knowledge. � Hypatia came to symbolize learning and science which the early Christians identified � Many scholars departed soon after marking the with paganism. beginning of the decline of Alexandria as a major center of ancient learning. � This led to Hypatia becoming the focal � There is no evidence that Hypatia undertook point of riots between Christians and non- original mathematical research. Christians. The Saga of Mathematics 5 The Saga of Mathematics 6 Lewinter & Widulski Lewinter & Widulski Lewinter & Widulski 1

The Saga of Mathematics A Brief History Hypatia of Alexandria Diophantus of Alexandria � Often known as the “ Father of Algebra ”. � However she assisted her father Theon of � Best known for his Arithmetica , a work on the Alexandria in writing his eleven part commentary solution of algebraic equations and on the theory on Ptolemy's Almagest. of numbers. � She also assisted her father in producing a new � However, essentially nothing is known of his life version of Euclid's Elements which became the and there has been much debate regarding the date at which he lived. basis for all later editions. � The Arithmetica is a collection of 130 problems � Hypatia wrote commentaries on Diophantus's giving numerical solutions of determinate Arithmetica and on Apollonius's On Conics . equations (those with a unique solution), and indeterminate equations. Lewinter & Widulski The Saga of Mathematics 7 Lewinter & Widulski The Saga of Mathematics 8 Diophantus of Alexandria Diophantus of Alexandria � Diophantus looked at three types of quadratic equations � A Diophantine equation is one which is to be ax 2 + bx = c , ax 2 = bx + c and ax 2 + c = bx . solved for integer solutions only. � He solved problems such as pairs of simultaneous � The work considers the solution of many problems quadratic equations. concerning linear and quadratic equations, but � For example, consider y + z = 10, yz = 9. considers only positive rational solutions to these – Diophantus would solve this by creating a single quadratic equation in x . problems. – Put 2 x = y - z so, adding y + z = 10 and y - z = 2 x , we have y = 5 + � There is no evidence to suggest that Diophantus x , then subtracting them gives z = 5 - x . – Now 9 = yz = (5 + x )(5 - x ) = 25 - x 2 , so x 2 = 16, x = 4 realized that a quadratic equation could have two – leading to y = 9, z = 1. solutions. The Saga of Mathematics 9 The Saga of Mathematics 10 Lewinter & Widulski Lewinter & Widulski Diophantus of Alexandria Diophantus of Alexandria � Another type of problem is to find powers � Diophantus solves problems of finding values between given limits. which make two linear expressions simultaneously � For example, to find a square between 5/4 and 2 into squares. he multiplies both by 64, spots the square 100 � For example, he shows how to find x to make 10 x between 80 and 128, so obtaining the solution 25/16 to the original problem. + 9 and 5 x + 4 both squares (he finds x = 28). � Diophantus also stated number theory results like: � He solves problems such as finding x such that – no number of the form 4n + 3 or 4n - 1 can be the sum simultaneously 4 x + 2 is a cube and 2 x + 1 is a of two squares; square (for which he easily finds the answer x = – a number of the form 24n + 7 cannot be the sum of three squares. 3/2). The Saga of Mathematics 11 The Saga of Mathematics 12 Lewinter & Widulski Lewinter & Widulski Lewinter & Widulski 2

The Saga of Mathematics A Brief History Arabic/Islamic Mathematics Arabic/Islamic Mathematics � Research shows the debt that we owe to � A remarkable period of mathematical progress Arabic/Islamic mathematics. began with al-Khwarizmi's (ca. 780-850 AD) work and the translations of Greek texts. � The mathematics studied today is far closer in � In the 9 th century, Caliph al-Ma'mun set up the style to that of the Arabic/Islamic contribution than to that of the Greeks. House of Wisdom ( Bayt al-Hikma ) in Baghdad which became the center for both the work of � In addition to advancing mathematics, Arabic translating and of research. translations of Greek texts were made which preserved the Greek learning so that it was � The most significant advances made by Arabic available to the Europeans at the beginning of the mathematics, namely the beginnings of algebra, sixteenth century. began with al-Khwarizmi. Lewinter & Widulski The Saga of Mathematics 13 Lewinter & Widulski The Saga of Mathematics 14 Arabic/Islamic Mathematics Geometric Constructions � It is important to understand just how significant � Euclid represented this new idea was. numbers as line segments. � It was a revolutionary move away from the Greek concept of mathematics which was essentially � From two segments a , geometric. b , and a unit length, it is possible to construct � Algebra was a unifying theory which allowed a + b , a – b , a × b , a ÷ rational numbers, irrational numbers, geometrical b , a 2 , and the square magnitudes, etc., to all be treated as “algebraic root of a . objects”. The Saga of Mathematics 15 The Saga of Mathematics 16 Lewinter & Widulski Lewinter & Widulski Geometric Construction of ab Arabic/Islamic Mathematics � Algebra gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. � Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. � All of this was done despite not using symbols. The Saga of Mathematics 17 The Saga of Mathematics 18 Lewinter & Widulski Lewinter & Widulski Lewinter & Widulski 3