Problem 16 to nd three numbers such that the sum of any two are given 20,30,40. For example, in problem 14, book i of the arithmetica, he chose a given ratio as well as a second value for x, thus creating a rather simple problem to solve gow 120. Diophantus died 4 years after the death of his son. Other problems seek a value for x such that particular types of polynomials in x up to degree 6 are squares. We can use his method to find solutions to the ops case, a 1. Diophantusanddiophantine equations diophantus diophantus of alexandria, about 200 284, was a greek mathematician. He lived in alexandria, egypt, during the roman era, probably from between ad 200 and 214 to 284 or 298. The remaining 7 cannot even be traced to even arab times. If a problem leads to an equation in which certain terms are equal to terms of the same species but with different coefficients, it will be necessary to subtract like from like on both sides, until one term is found equal to one term. As diophantus writes in the concluding section of the introduction, the problems are roughly ordered from the simpler to the more in volved, p. So, the cube root or side 517 57, the cube 573 125343 and the. The side of 16 is the square root of 16 think of 16 as the area of square, so in modern terms his statement means. Much later, fermat 5 notes that the product of any two of 1, 3, 8, and 120 increased by 1 is the square of an integer.
The solution diophantus writes we use modern notation. The problem in the very first problem in the very first book of arithmetica diophantus asks his readers to divide a given number into two numbers that have a given difference. I feel i am sufficiently knowledgeable about the properties of quadratic relations. Books iv to vii of diophantus arithmetica springerlink. It was at first found that diophantus lived between ad 250350 by analysing the price of wine used in many of his mathematical texts and finding out the period during which wine was sold at that price. Diophantus main claim to fame rests on his book arithmetika, which. One of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. On the other hand theon of alexandria, the father of hypatia, quotes one of diophantuss definitions. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so. The distinctive features of diophantus s problems appear in the later books. Diophantus takes the square to be 16 and solves the problem as follows.
In book iii, diophantus solves problems of finding values which make two linear. Two interpretations of problem 16 from book i of the arithmetica by diophantus. Solve problems, which are from the arithmetica of diophantus. Diophantus gives the sum as 20 and the product as 96. Diophantus solution is quite clear and can be followed easily. Of course our modern decimal numbers have been used where diophantus would use the greek numbers on page 14.
In book iii, diophantus solves problems of finding values which make two linear expressions simultaneously into squares. The problems of book i are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. In other words, for the given numbers a and b, to find x and y such that x y a and x3 y3 b. Answer to solve problems, which are from the arithmetica of diophantus. Diophantus life span problem diophantus youth lasted 16 of his life. Apr 30, 2009 this wonderful book may be one of the most important arithmetic books ever translated into the english language. Diophantus of alexandria, arithmetica and diophantine equations. Diophantus lived in alexandria in times of roman domination ca 250 a.
Find three numbers such that when any two of them are added, the sum is one of three given numbers. I feel as if, however, the wikipedia page, which states this contains both indeterminate and determinate equations might be slightly misleading, because i never encountered a definitively determinate equation. Book iii problem 9 to nd three squares at equal intervals. Diophantus is sometimes called the father of algebra, partly because he introduced symbolism into the subject. The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares. The following is problem 7 of the first book of arithmetica. The meaning of plasmatikon in diophantus arithmetica. He had his first beard in the next 112 of his life. Since diophantus method produces rational solutions, we have to clear denominators to get. For example, the first seven problems of the second book fit. Diophantus of alexandria had a great impact in the world of mathematics. Once again the problem is to divide 16 into two squares. An example of this is found in problem 16, book i of the arithmetica, and it reads as.
Find two numbers such that their sum and product are given. Diophantuss main achievement was the arithmetica, a collection of arithmetical problems involving the solution of determinate and indeterminate equations. For example to find a square between 54 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 25 16 to the original problem. The conclusion of knorr as to diophantuss dates is 16.
Find two square numbers whose di erence is a given number, say 60. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions. Page 3 his hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed. Assignment 1 due a week thursday again watch for a. Diophantus of alexandria university of connecticut. Mar 10, 2009 to find two numbers such that their sum and the sum of their squares are given numbers. Book v contains sixteen problems, covering pages 7397 of the manuscript.
For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. The eighth problem of the second book of diophantus s arithmetica is to divide a square into a sum of two squares. Alexandrian algebra according to diophantus mathematics. The arithmetica as written by diophantus originally contained thirteen books. Even remarkable translators like heath and many of the most famous mathematicians who have read or studied diophantus s book were not convinced that diophantus d. Long ago diophantus of alexandria 4 noted that the numbers 116, 3316, 6816, and 10516 all have the property that the product of any two increased by 1 is the square of a rational number. A presentday mathematician, when faced with an equation, would. Derive the necessary condition on a and b that ensures a rational solution. Intersection of the line cb and the circle gives a rational point x 0, y 0. Because little is known on the life of diophantus, historians have approximated his birth to be at about 200 ad in alexandria, egypt and his death at 284 ad in alexandria as well. For simplicity, modern notation is used, but the method is due to 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. Five years after his marriage, was born a son who died 4 years before his father, at 1 2 log on.
In section 1 we differentiate between numerical problem solving techniques and that of. Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. However, the necessity of his necessary condition must be explored. A similar problem involves decomposing a given integer into the sum of three squares. And if diophantus states a necessary condition for dividing a number into two or three squares as in the previous case of v. For example to find a square between 54 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 2516 to the original problem. Diophantus passed 1 6 of his life in childhood, 1 12 in youth and 1 7 more as a bachelor. Nov 18, 2003 another type of problem which diophantus studies, this time in book iv, is to find powers between given limits. Taking a 3 r x 4 and m 2 x 4, m 2 x 4 m n, we obtain, how to solve the general problem.
At the end of the following 17 of his life diophantus got married. To divide a given square into a sum of two squares. Let the first number be n and the second an arbitrary multiple of n diminished by the root of 16. Problem 15 to nd two numbers such that when a given number 30 is transferred from the second to the rst, and when another number 50 is transferred from the rst to the second, the resulting pairs have given ratios 2. Immediately preceding book i, diophantus gives the following definitions to solve these simple problems. A contribution of diophantus to mathematics the following is a statement of arithmetica book ii, problem 28 and its solution.
This wonderful book may be one of the most important arithmetic books ever translated into the english language. The number he gives his readers is 100 and the given difference is 40. Book iv problem 21 to nd four numbers such that the product of any two added to one gives a square. This book features a host of problems, the most significant of. Problem to nd a number whose di erences from two given numbers 9,21 are both squares.
Then in problem 20, book iv, he treated the problem of finding four numbers such that all six pairwise products are 1 less than a square. Diophantus married at the age of 33 and had a son who later died at 42, only 4 years before diophantus death at 84. Diophantus is aware of the fact that his method produces many more solutions. Two works have come upon us under the name of diophantus of alexandria. This book features a host of problems, the most significant of which have come to be called diophantine equations. Diophantus about 200 about 284 mactutor history of mathematics. To find three numbers such that the sums of pairs are given numbers. Ix reaches the same solution by an even quicker route which is very similar to the generalized solution above. Another type of problem which diophantus studies, this time in book iv, is to find powers between given limits. 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.
Five years after his marriage, was born a son who died 4 years before his father, at 12 log on. Even remarkable translators like heath and many of the most famous mathematicians who have read or studied diophantuss book were not convinced that diophantus d. Algebra customizable word problem solvers age solution. To divide a given number into two num bers with given di. Step 1 of 4 consider the problem statement provided in the textbook. Two interpretations of problem 16 from book i of the arithmetica by. Diophantus occasionally resumes, in the middle of a proof, what is sought for in a problem. Diophantus passed 16 of his life in childhood, 112 in youth and 17 more as a bachelor. One of the most famous problems that diophantus treated was writing a. Just as we see in medieval arabic and italian algebra, diophantus worked out the operations. Problem 8 to split a given square 16 in two squares.