Diophantusanddiophantine equations diophantus diophantus of alexandria, about 200 284, was a greek mathematician. From aristarchus to diophantus dover books on mathematics book 2. Thus the problem has been reduced to a linear equation, which. But i dont think the current state is the desired one. Again, the 1 dynamis, 1 unit, being the result of the addition, is a collection of two objects of different kinds. Easily share your publications and get them in front of issuus. Such a triple is commonly written a, b, c, and a wellknown example is 3, 4, 5. Each of these lowc points forms one of the more easily recognizable radiating lines in the scatter plot.
It comes from a fifth century greek anthology of number games and puzzles created by metrodorus. Following is a sample of problems in the other books. After him, any problem where only integer solutions are allowed came to be known as diophantine equations. The symbolic and mathematical influence of diophantus s arithmetica. A similar problem involves decomposing a given integer into the sum of three squares. Though fermat admitted the extreme difficulty of the problems, he assured mersenne, and through him his two correspondents, that the problems had solutions.
Some clarifications on diophan tus method of solution. You can use cents instead of pence, and i thought of changing the chops and cutlets to hot dogs and hamburgers. The emphasis in this book will be on problem solving, with problems about general graphs and applied graph models. Read the question carefully and determine the appropriate operation. Find two square numbers whose di erence is a given number, say 60. This problem became important when fermat, in his copy of diophantus arithmetica edited by bachet, noted that he had this wonderful proof that cubes cant. He was interested in problems that had whole number solutions. Books iv to vii of diophantus arithmetica springerlink.
Diophantus and pappus ca 300 represent a shortlived revival of greek mathematics in a society that did not value math as the greeks had done 500750 years earlier. It is about the life of diophantus, the father of algebra, who lived in the second century. The symbolic and mathematical influence of diophantuss. Diophantus of alexandria arithmetica book i joseph. Diophantus s main achievement was the arithmetica, a collection of arithmetical problems involving the solution of determinate and indeterminate equations. This book features a host of problems, the most significant of which have come to.
Applied combinatorics 6th edition by alan tucker 2012 by. Butterfly in the quantum world morgan claypool publishers. The problemsolving of diophantus of alexandria article pdf available in historia mathematica 402. You come home, and find a book called fermats last theorem by amir aczel. On intersections of two quadrics in p3 in the arithmetica 18 5. His birth until death may be determined from an epitaph revealing the fact that he passed a sixth of his life in childhood, a twelfth in adolescence and a seventh more as a bachelor.
Liked by einstien, x, y, tarleton, alexandria, gauss. We know little about this greek mathematician from alexandria, called the father of algebra, except that he lived around 3rd century a. His major contribution to mathematics is a collection of books called arithmetica, in which only 6 survived through the centuries, and exhibit a high degree of math skills and ingenuity. They accused fermat of having posed impossible problems. Given two numbers not prime to one another, to find their greatest common measure. The problem solving of diophantus of alexandria article pdf available in historia mathematica 402. Diophantus studied at the university of alexandria in egypt. The history of algebra is very intriguing because of the many cultures that contributed to its origins. What is your reading rate for the first half of the book. To divide a given square into a sum of two squares. The name of diophantus is associated with problems whose solutions are whole numbers, such as the number of mutton chops and cutlets in the above puzzle. What douglas hofstadters godel, escher, bach did for godels incompleteness theorema crucial discovery that was poorly understood outside of the domain of professional mathematicianspetzolds book does for turings universal computer.
The date of diophantus death is the date of his sons birth plus his sons life plus 4, so. On the other hand im not entirely convinced that we shouldnt just merge all three articles. Oct 20, 2015 euclids algorithm appears as proposition ii in book vii elementary number theory of his elements. As part of this theory, stormer also investigated divisibility relations among solutions to pells equation. Here we look at other polygons of dots such as triangles, pentagon and so on the polygonal numbers. Full text of diophantus of alexandria a study in the. Hypatias work on diophantus appalachian state university. This, as the 27 th, is also worked out methodically via the 2 nd of the 1st book of diophantus. This edition of books iv to vii of diophantus arithmetica, which are extant only in a recently discovered arabic translation, is the outgrowth of a doctoral dissertation submitted to the brown university department of the history of mathematics in may 1975. In both cases, there are about five times more scholarly sources for the versions without diacritics.
Diophantus wrote a seminal series of books called the arithmetica. The following is problem 7 of the first book of arithmetica. He defines a number to be a multitude composed of units. The son died four years before diophantis at half the diophantis was when he himself died. Polygonal and figurate numbers or numbers as shapes we call some numbers square numbers because they can be arranged into a square shape.
Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. Some manuscripts divide the six books into seven and others list the separate work on polygonal numbers as book vii 6. One of these poems relates to the life, and the age at death, of a thirdcentury mathematician named diophantus, who lived in or around alexandria, egypt but was probably of greek heritage. But suppose i have a piece of meadow that is three times. In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. Arithmetica is the major work of diophantus and the most prominent work on algebra in greek mathematics. Table of contents for introduction to number theory martin erickson, anthony vazzana, available from the library of congress. Polygonal and figurate numbers or numbers as shapes. The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares. One must divide the 12 in sesquitertan ratio, and the number becomes 12 7 ths.
Structures, examples and problems will appeal to senior high school and undergraduate students, their instructors, as well as to all who would like to expand their mathematical horizons. As lenstra 2002 describes, pells equation can also be used to solve archimedes cattle problem. So the the day that has passed will be 36 7 ths, while the remaining, 48 7 ths. The words used to express addition reflect this interpretation, too. A contribution of diophantus to mathematics the following is a statement of arithmetica book ii, problem 28 and its solution. It is the purpose of book iv to present a survey of the fragmentary data from the early stages of greek astronomy. Diophantus of alexandria, a third century mathmatician, lived onesixth of his life in childhood, onetwelfth in his youth, and oneseventh as a bachelor.
Diophantus on fakebook fakebook create a fictional social profile at. Pisa, after 1240 mathematics leonardo fibonacci, the first great mathematician of the christian west, was a member of a family named bonacci, whose presence in pisa since the eleventh century is documented. Find a number whose subtraction from two given numbers say 9 and 21 allows both remainders. For example, the first seven problems of the second book fit much better with the problems of the first, as do problems ii, 17, and ii, 18. In other words, for the given numbers a and b, to find x and y such that x y a and x3 y3 b. Accordingly, equations of this type are called diophantine equations. It captures the spirit of an important mathematical literature and distills the essence of a rich problemsolving culture. If a, b, c is a pythagorean triple, then so is ka, kb, kc for any positive integer k.
If you have any interest whatsoever in the theory of computing, make this the first book you read. Another unresolved question about diophantus is that concerning the relationship. Derive the necessary condition on a and b that ensures a rational solution. Diophantus passed 16 of his life in childhood, 112 in youth and 17 more as a bachelor. Mar 30, 2007 diophantus back to the cool math games. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. One of the problems sometimes called his epitaph is the riddle you see above. Diophantus was the first greek mathematician who recognized fractions as numbers, thus allowed positive rational numbers for the coefficients and solutions. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. About six centuries have to be covered by such an attempt, beginning with the calendaric cycles of meton and his school in the fifth century b. The problem of computing probabilities of results of coin tosses for coins weighted in a speci. Find two numbers such that the square of either added to the sum of both gives a square. Issuu is a digital publishing platform that makes it simple to publish magazines, catalogs, newspapers, books, and more online.
The riddle can be written as an equation where \x\ is the age diophantus died. Intersection of the line cb and the circle gives a rational point x 0,y 0. Answer to solve problems, which are from the arithmetica of diophantus. The text used is the edition of tannery 1893, but i have also consulted the translation of ver eecke 1959 and the paraphrase of heath 1910.
A primitive pythagorean triple is one in which a, b and c are coprime that is, they have no common divisor larger than 1. There was some controversy last year related to this, see wikipedia talk. Diophantus passed 16 of his life in childhood, 1 12 in youth and 1 7 more as a bachelor. After finishing half the book, you realize that you must read 12 pages more per day to finish on time. This book features a host of problems, the most significant of which have come to be called diophantine equations. There you see that diophantus was a greek philosopher from approximately 250 ad, who wrote several books of problems, all of them requiring integer solutions. He writes, if 1 unit is added to it, it makes 1 dynamis, 1 unit 126. Diophantus of alexandria, arithmetica and diophantine equations. Diophantus is one of the brilliant greek mathematicians born around 250 ad. The problems in book i of the arithmetica are determinate ie, having a unique solution or a.
The eighth problem of the second book of diophantus s arithmetica is to divide a square into a sum of two squares. In paris in 1770, etienne bezout wrote a 474page book on a. From the same number to subtract two given numbers so that their remainders have a. Solve problems, which are from the arithmetica of diophantus. Other readers will always be interested in your opinion of the books youve read. Immediately preceding book i, diophantus gives the following definitions to solve these simple problems. Multivariable calculus with analytic geometry, fifth edition. Thanks to an admirer of his, who described his life by means of an algebraic riddle, we know at least something about his life. A particular attention should be given to the content of book v, beginning with the seventh problem. Then write a mathematical expression that will help to solve the problem. Of the original thirteen books of which arithmetica consisted only six have survived, though there are some who believe that four arabic books discovered in 1968 are also by diophantus.
Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day. Five years after his marriage, was born a son who died 4 years before his father, at 12 log on. Algebra customizable word problem solvers age solution. Table of contents for introduction to number theory. Full text of diophantus of alexandria a study in the history. Reading rate in pages per day number of pages read number of days first half second half 20. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions.
The problems one of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. Use a table like the following to help you solve the problem. Not every heronian triple is a pythagorean triple, however, as the example 4, 15 with area 24 shows. This, as the 27 th, is also worked out methodically via the 2 nd problem of the 1st book of diophantus. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. Diophantus wrote a thirteenvolume set of books called arithmetica of which only six have survived. Diophantus lived in alexandria in times of roman domination ca 250 a. For simplicity, modern notation is used, but the method is due to diophantus. Note, for example, that 6, 8, 10 is not a primitive pythagorean triple, as it is a multiple of 3, 4, 5. Clearly, any pythagorean triple is a heronian triple, since in a pythagorean triple at least one of the legs a, b must be even, so that the area ab2 is an integer. If a problem leads to an equation in which certain terms are equal to terms of the same species. The author thanks benjamin braun, for whose history of mathematics course this paper was originally written, and an anonymous referee for their guidance and suggestions.
542 810 1427 57 602 479 1163 1227 824 1244 841 1411 1366 1610 133 617 23 1030 847 1416 28 1397 798 1373 1417 82 1646 1398 94 513 1389 816 1053 1499 1049 384 478