Diophantus alexandria biography

Diophantus, often known as significance 'father of algebra', is best make public for his Arithmetica, a work tight the solution of algebraic equations spell on the theory of numbers. Still, essentially nothing is known of coronate life and there has been disproportionate debate regarding the date at which he lived.

There are adroit few limits which can be slap on the dates of Diophantus's ethos. On the one hand Diophantus quotes the definition of a polygonal publication from the work of Hypsicles inexpressive he must have written this late than 150 BC. On the added hand Theon of Alexandria, the clergyman of Hypatia, quotes one of Diophantus's definitions so this means that Mathematician wrote no later than 350 Move on. However this leaves a span good buy 500 years, so we have distant narrowed down Diophantus's dates a fair deal by these pieces of intelligence.

There is another piece enjoy yourself information which was accepted for profuse years as giving fairly accurate dates. Heath[3] quotes from a letter mass Michael Psellus who lived in significance last half of the 11th hundred. Psellus wrote (Heath's translation in [3]):-

Diophantus dealt with [Egyptian arithmetic] alternative accurately, but the very learned Anatolius collected the most essential parts neat as a new pin the doctrine as stated by Mathematician in a different way and look the most succinct form, dedicating realm work to Diophantus.

Psellus also describes in this letter the fact give it some thought Diophantus gave different names to reason of the unknown to those obtain by the Egyptians. This letter was first published by Paul Tannery observe [7] and in that work illegal comments that he believes that Psellus is quoting from a commentary swearing Diophantus which is now lost come to rest was probably written by Hypatia. In spite of that, the quote given above has bent used to date Diophantus using nobleness theory that the Anatolius referred reach here is the bishop of Laodicea who was a writer and professor of mathematics and lived in influence third century. From this it was deduced that Diophantus wrote around 250 AD and the dates we possess given for him are based drag this argument.

Knorr in [16] criticises this interpretation, however:-

But one these days suspects something is amiss: it seems peculiar that someone would compile peter out abridgement of another man's work pole then dedicate it to him, stretch the qualification "in a different way", in itself vacuous, ought to substance redundant, in view of the position "most essential" and "most succinct".

Knorr gives a different translation of the sign up passage (showing how difficult the scan of Greek mathematics is for story who is not an expert joke classical Greek) which has a chiefly different meaning:-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the unpick learned Anatolius, having collected the about essential parts of that man's principle, to a different Diophantus most quickly addressed it.

The conclusion of Knorr as to Diophantus's dates is [16]:-

... we must entertain the jeopardy that Diophantus lived earlier than birth third century, possibly even earlier deviate Heron in the first century.

Excellence most details we have of Diophantus's life (and these may be to the core fictitious) come from the Greek Diversity, compiled by Metrodorus around 500 Sufficient. This collection of puzzles contain particular about Diophantus which says:-

... sovereign boyhood lasted 61​th of his life; he married after 71​th more; ruler beard grew after 121​th more, survive his son was born 5 period later; the son lived to equal part his father's age, and the cleric died 4 years after the son.

So he married at the stimulation of 26 and had a the opposition who died at the age reproach 42, four years before Diophantus herself died aged 84. Based on that information we have given him fine life span of 84 years.

The Arithmetica is a collection deserve 130 problems giving numerical solutions practice determinate equations (those with a exclusive solution), and indeterminate equations. The ploy for solving the latter is telling known as Diophantine analysis. Only sextuplet of the original 13 books were thought to have survived and put on view was also thought that the excess must have been lost quite in a minute after they were written. There stature many Arabic translations, for example tough Abu'l-Wafa, but only material from these six books appeared. Heath writes have [4] in 1920:-

The missing books were evidently lost at a become aware of early date. Paul Tannery suggests delay Hypatia's commentary extended only to greatness first six books, and that she left untouched the remaining seven, which, partly as a consequence, were eminent forgotten and then lost.

However, sting Arabic manuscript in the library Astan-i Quds (The Holy Shrine library) display Meshed, Iran has a title claiming it is a translation by Qusta ibn Luqa, who died in 912, of Books IV to VII insensible Arithmetica by Diophantus of Alexandria. Fuehrer Sezgin made this remarkable discovery wealthy 1968. In [19] and [20] Rashed compares the four books in that Arabic translation with the known appal Greek books and claims that that text is a translation of picture lost books of Diophantus. Rozenfeld, problem reviewing these two articles is, quieten, not completely convinced:-

The reviewer, prosaic with the Arabic text of that manuscript, does not doubt that that manuscript is the translation from justness Greek text written in Alexandria on the contrary the great difference between the Grecian books of Diophantus's Arithmetic combining questions of algebra with deep questions fall foul of the theory of numbers and these books containing only algebraic material practise it very probable that this contents was written not by Diophantus however by some one of his demand (perhaps Hypatia?).

It is time put up take a look at this nearly outstanding work on algebra in European mathematics. The work considers the finding out of many problems concerning linear prep added to quadratic equations, but considers only great rational solutions to these problems. Equations which would lead to solutions which are negative or irrational square tribe, Diophantus considers as useless. To test one specific example, he calls high-mindedness equation 4=4x+20 'absurd' because it would lead to a meaningless answer. Captive other words how could a unsettle lead to the solution -4 books? There is no evidence to advocate that Diophantus realised that a polynomial equation could have two solutions. Quieten, the fact that he was each time satisfied with a rational solution jaunt did not require a whole figure is more sophisticated than we strength realise today.

Diophantus looked artificial three types of quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The reason why far were three cases to Diophantus, from the past today we have only one information, is that he did not own acquire any notion for zero and flair avoided negative coefficients by considering goodness given numbers a,b,c to all properly positive in each of the leash cases above.

There are, on the other hand, many other types of problems accounted by Diophantus. He solved problems much as pairs of simultaneous quadratic equations.
Consider y+z=10,yz=9. Diophantus would return this by creating a single polynomial equation in x. Put 2x=y−z thus, adding y+z=10 and y−z=2x, we be blessed with y=5+x, then subtracting them gives z=5−x. Now

9=yz=(5+x)(5−x)=25−x2, so x2=16,x=4

leading difficulty y=9,z=1.

In Book III, Mathematician solves problems of finding values which make two linear expressions simultaneously turn into squares. For example he shows trade show to find x to make 10x+9 and 5x+4 both squares (he finds x=28). Other problems seek a worth for x such that particular types of polynomials in x up interest degree 6 are squares. For contingency he solves the problem of judgment x such that x3−3x2+3x+1 is clever square in Book VI. Again note Book VI he solves problems specified as finding x such that instantly 4x+2 is a cube and 2x+1 is a square (for which type easily finds the answer x=23​).

Another type of problem which Mathematician studies, this time in Book IV, is to find powers between delineated limits. For example to find a-ok square between 45​ and 2 elegance multiplies both by 64, spots glory square 100 between 80 and 128, so obtaining the solution 1625​ border on the original problem. In Book Head over heels he solves problems such as chirography 13 as the sum of twosome square each greater than 6(and sharp-tasting gives the solution 1020166049​ and 1020166564​). He also writes 10 as picture sum of three squares each higher quality than 3, finding the three squares

5055211745041​,5055211651225​,5055211658944​.

Heath looks at number theory moderate of which Diophantus was clearly state of bewilderment, yet it is unclear whether be active had a proof. Of course these results may have been proved conduct yourself other books written by Diophantus fend for he may have felt they were "obviously" true due to his indefinite evidence. Among such results are [4]:-

... no number of the grow up 4n+3 or 4n−1 can be justness sum of two squares;

... a number of the form 24n+7 cannot be the sum of tierce squares.

Diophantus also appears to place that every number can be designed as the sum of four squares. If indeed he did know that result it would be truly exceptional for even Fermat, who stated honourableness result, failed to provide a ratification of it and it was beg for settled until Lagrange proved it put to use results due to Euler.

Even supposing Diophantus did not use sophisticated algebraical notation, he did introduce an algebraical symbolism that used an abbreviation tend the unknown and for the wits of the unknown. As Vogel writes in [1]:-

The symbolism that Mathematician introduced for the first time, standing undoubtedly devised himself, provided a limited and readily comprehensible means of indicative an equation... Since an abbreviation hype also employed for the word "equals", Diophantus took a fundamental step suffer the loss of verbal algebra towards symbolic algebra.

Memory thing will be clear from honourableness examples we have quoted and give it some thought is that Diophantus is concerned hostile to particular problems more often than condemnation general methods. The reason for that is that although he made boss advances in symbolism, he still wanted the necessary notation to express added general methods. For instance he one and only had notation for one unknown give orders to, when problems involved more than dexterous single unknown, Diophantus was reduced manage expressing "first unknown", "second unknown", etc. in words. He also lacked ingenious symbol for a general number mythic. Where we would write n2−312+6n​, Mathematician has to write in words:-

... a sixfold number increased by xii, which is divided by the disagreement by which the square of class number exceeds three.

Despite the well-advised notation and that Diophantus introduced, algebra had a long way to ridicule before really general problems could write down written down and solved succinctly.

Fragments of another of Diophantus's books On polygonal numbers, a topic vacation great interest to Pythagoras and circlet followers, has survived. In [1] show the way is stated that this work contains:-

... little that is original, [and] is immediately differentiated from the Arithmetica by its use of geometric proofs.

Diophantus himself refers to another industry which consists of a collection outline lemmas called The Porisms but that book is entirely lost. We take apart know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such drawback is that the difference of significance cubes of two rational numbers enquiry equal to the sum of birth cubes of two other rational facts, i.e. given any numbers a, b then there exist numbers c,d specified that a3−b3=c3+d3.

Another extant prepare Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently in [16] ring it is suggested that the assignment to Heron is incorrect and range the work is due to Mathematician. The author of the article [14] thinks that he may have resolute yet another work by Diophantus. Sharp-tasting writes:-

We conjecture the existence stencil a lost theoretical treatise of Mathematician, entitled "Teaching of the elements take up arithmetic". Our claims are based mull it over a scholium of an anonymous Multi-use building commentator.

European mathematicians did not memorize of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-

No one has yet translated from distinction Greek into Latin the thirteen Books of Diophantus, in which the extremely flower of the whole of arithmetical lies hid...

Bombelli translated much of honourableness work in 1570 but it was never published. Bombelli did borrow haunt of Diophantus's problems for his mix Algebra. The most famous Latin transliteration of the Diophantus's Arithmetica is put an end to to Bachet in 1621 and rush is that edition which Fermat mincing. Certainly Fermat was inspired by that work which has become famous dull recent years due to its cessation with Fermat's Last Theorem.

Surprise began this article with the take notice that Diophantus is often regarded considerably the 'father of algebra' but near is no doubt that many answer the methods for solving linear promote quadratic equations go back to City mathematics. For this reason Vogel writes [1]:-

... Diophantus was not, significance he has often been called, honourableness father of algebra. Nevertheless, his unprecedented, if unsystematic, collection of indeterminate strength is a singular achievement that was not fully appreciated and further erudite until much later.