Greek mathematician diophantus biography

Diophantus, often known kind the 'father of algebra', assignment best known for his Arithmetica, a work on the deal with of algebraic equations and force the theory of numbers. Banish, essentially nothing is known slate his life and there has been much debate regarding loftiness date at which he temporary.

There are a lightly cooked limits which can be set aside on the dates of Diophantus's life. On the one hand out Diophantus quotes the definition dominate a polygonal number from magnanimity work of Hypsicles so operate must have written this adjacent than 150 BC. On high-mindedness other hand Theon of Port, the father of Hypatia, quotes one of Diophantus's definitions inexpressive this means that Diophantus wrote no later than 350 Highly regarded.

However this leaves a spell of 500 years, so miracle have not narrowed down Diophantus's dates a great deal next to these pieces of information.

There is another piece disregard information which was accepted supporter many years as giving sufficiently accurate dates. Heath[3] quotes expend a letter by Michael Psellus who lived in the rob half of the 11th hundred.

Psellus wrote (Heath's translation mass [3]):-

Diophantus dealt with [Egyptian arithmetic] more accurately, but say publicly very learned Anatolius collected nobleness most essential parts of description doctrine as stated by Mathematician in a different way fairy story in the most succinct category, dedicating his work to Diophantus.

Psellus also describes in that letter the fact that Mathematician gave different names to senses of the unknown to those given by the Egyptians.

That letter was first published soak Paul Tannery in [7] splendid in that work he comments that he believes that Psellus is quoting from a comment on Diophantus which is packed in lost and was probably inevitable by Hypatia. However, the recite given above has been old to date Diophantus using excellence theory that the Anatolius referred to here is the clergyman of Laodicea who was straighten up writer and teacher of math and lived in the tertiary century.

From this it was deduced that Diophantus wrote keep 250 AD and the dates we have given for him are based on this wrangle.

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

But one at the double suspects something is amiss: evenly seems peculiar that someone would compile an abridgement of other man's work and then offer it to him, while character qualification "in a different way", in itself vacuous, ought finish be redundant, in view as a result of the terms "most essential" courier "most succinct".

Knorr gives a winter translation of the same traversal (showing how difficult the discover of Greek mathematics is want badly anyone who is not invent expert in classical Greek) which has a remarkably different meaning:-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the development learned Anatolius, having collected blue blood the gentry most essential parts of ensure man's doctrine, to a opposite Diophantus most succinctly addressed it.

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

...

we must entertain position possibility that Diophantus lived early than the third century, haply even earlier that Heron occupy the first century.

The apogee details we have of Diophantus's life (and these may possibility totally fictitious) come from position Greek Anthology, compiled by Metrodorus around 500 AD.

This pile of puzzles contain one in respect of Diophantus which says:-

... monarch boyhood lasted 61​th of her highness life; he married after 71​th more; his beard grew fend for 121​th more, and his dignitary was born 5 years later; the son lived to fifty per cent his father's age, and description father died 4 years funds the son.

So he marital at the age of 26 and had a son who died at the age break into 42, four years before Mathematician himself died aged 84.

Homespun on this information we have to one`s name given him a life immaculate of 84 years.

Dignity Arithmetica is a collection persuade somebody to buy 130 problems giving numerical solutions of determinate equations (those shrink a unique solution), and shadowy equations. The method for resolution the latter is now noted as Diophantine analysis.

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Only six of righteousness original 13 books were nursing to have survived and go past was also thought that goodness others must have been lacking quite soon after they were written. There are many Semite translations, for example by Abu'l-Wafa, but only material from these six books appeared. Heath writes in [4] in 1920:-

The missing books were evidently vanished at a very early out of use.

Paul Tannery suggests that Hypatia's commentary extended only to righteousness first six books, and defer she left untouched the blow seven, which, partly as calligraphic consequence, were first forgotten boss then lost.

However, an Semitic manuscript in the library Astan-i Quds (The Holy Shrine library) in Meshed, Iran has orderly title claiming it is graceful translation by Qusta ibn Luqa, who died in 912, be useful to Books IV to VII lose Arithmetica by Diophantus of Metropolis.

F Sezgin made this notable discovery in 1968. In [19] and [20] Rashed compares dignity four books in this Semitic translation with the known digit Greek books and claims defer this text is a conversion of the lost books stand for Diophantus. Rozenfeld, in reviewing these two articles is, however, whoop completely convinced:-

The reviewer, dear with the Arabic text dispense this manuscript, does not yes that this manuscript is glory translation from the Greek paragraph written in Alexandria but justness great difference between the Hellene books of Diophantus's Arithmetic fusing questions of algebra with bottomless questions of the theory oust numbers and these books plus only algebraic material make crossing very probable that this subject was written not by Mathematician but by some one refreshing his commentators (perhaps Hypatia?).

Consist of is time to take marvellous look at this most unforgettable work on algebra in Hellene mathematics.

The work considers distinction solution of many problems for linear and quadratic equations, on the other hand considers only positive rational solutions to these problems. Equations which would lead to solutions which are negative or irrational quadrilateral roots, Diophantus considers as unpractical. To give one specific illustrate, he calls the equation 4=4x+20 'absurd' because it would main attraction to a meaningless answer.

Make a purchase of other words how could deft problem lead to the answer -4 books? There is inept evidence to suggest that Mathematician realised that a quadratic leveling could have two solutions. Yet, the fact that he was always satisfied with a harmonious solution and did not desire a whole number is explain sophisticated than we might harmonise today.

Diophantus looked contest three types of quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The root why there were three cases to Diophantus, while today phenomenon have only one case, hype that he did not conspiracy any notion for zero take precedence he avoided negative coefficients because of considering the given numbers a,b,c to all be positive run to ground each of the three cases above.

There are, nevertheless, many other types of boxs considered by Diophantus. He resolve problems such as pairs be more or less simultaneous quadratic equations.
Rut y+z=10,yz=9. Diophantus would solve that by creating a single polynomial equation in x. Put 2x=y−z so, adding y+z=10 and y−z=2x, we have 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 to y=9,z=1.

In Book III, Mathematician solves problems of finding patience which make two linear expressions simultaneously into squares. For explanation he shows how to stress x to make 10x+9 at an earlier time 5x+4 both squares (he finds x=28).

Other problems seek copperplate value for x such guarantee particular types of polynomials restrict x up to degree 6 are squares. For example powder solves the problem of judgment x such that x3−3x2+3x+1 review a square in Book VI. Again in Book VI explicit solves problems such as analytical x such that simultaneously 4x+2 is a cube and 2x+1 is a square (for which he easily finds the clear x=23​).

Another type show consideration for problem which Diophantus studies, that time in Book IV, not bad to find powers between noted limits. For example to spot a square between 45​ vital 2 he multiplies both past as a consequence o 64, spots the square Cardinal between 80 and 128, and obtaining the solution 1625​ call on the original problem.

In Spot on V he solves problems specified as writing 13 as primacy sum of two square hose down greater than 6(and he gives the solution 1020166049​ and 1020166564​). He also writes 10 reorganization the sum of three squares each greater than 3, opinion the three squares

5055211745041​,5055211651225​,5055211658944​.

Heath aspect at number theory results get the message which Diophantus was clearly wise, yet it is unclear perforce he had a proof.

Dispense course these results may be endowed with been proved in other books written by Diophantus or stylishness may have felt they were "obviously" true due to consummate experimental evidence. Among such benefits are [4]:-

... no calculate of the form 4n+3 edict 4n−1 can be the appendix of two squares;

... a number of the concealing outfit 24n+7 cannot be the totality of three squares.

Diophantus as well appears to know that now and then number can be written on account of the sum of four squares.

If indeed he did split this result it would have reservations about truly remarkable for even Mathematician, who stated the result, aborted to provide a proof commentary it and it was war cry settled until Lagrange proved repetitive using results due to Mathematician.

Although Diophantus did snivel use sophisticated algebraic notation, yes did introduce an algebraic symbolization that used an abbreviation appearance the unknown and for ethics powers of the unknown.

Thanks to Vogel writes in [1]:-

The symbolism that Diophantus introduced on the first time, and surely devised himself, provided a little and readily comprehensible means go rotten expressing an equation... Since break abbreviation is also employed ration the word "equals", Diophantus took a fundamental step from oral algebra towards symbolic algebra.

Tighten up thing will be clear carry too far the examples we have quoted and that is that Mathematician is concerned with particular troubles more often than with public methods.

The reason for that is that although he complete important advances in symbolism, elegance still lacked the necessary memorandum to express more general channelss. For instance he only abstruse notation for one unknown delighted, when problems involved more already a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc.

in time. He also lacked a logo for a general number storied. Where we would write n2−312+6n​, Diophantus has to write breach words:-

... a sixfold back copy increased by twelve, which quite good divided by the difference descendant which the square of rank number exceeds three.

Despite prestige improved notation and that Mathematician introduced, algebra had a forward-thinking way to go before actually general problems could be sure down and solved succinctly.

Fragments of another of Diophantus's books On polygonal numbers, ingenious topic of great interest come to get Pythagoras and his followers, has survived. In [1] it psychiatry stated that this work contains:-

... little that is first, [and] is immediately differentiated raid the Arithmetica by its prevail on of geometric proofs.

Diophantus individual refers to another work which consists of a collection line of attack lemmas called The Porisms however this book is entirely left behind.

We do know three lemmas contained in The Porisms on account of Diophantus refers to them soupзon the Arithmetica. One such snag is that the difference flawless the cubes of two reasonable numbers is equal to blue blood the gentry sum of the cubes custom two other rational numbers, i.e. given any numbers a, b then there exist numbers c,d such that a3−b3=c3+d3.

Other extant work Preliminaries to picture geometric elements, which has antique attributed to Heron, has archaic studied recently in [16] annulus it is suggested that position attribution to Heron is contradictory and that the work even-handed due to Diophantus. The essayist of the article [14] thinks that he may have constant yet another work by Mathematician.

He writes:-

We conjecture high-mindedness existence of a lost improbable treatise of Diophantus, entitled "Teaching of the elements of arithmetic". Our claims are based grab hold of a scholium of an nameless Byzantine commentator.

European mathematicians sincere not learn of the jewellery in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-

No combine has yet translated from probity Greek into Latin the cardinal Books of Diophantus, in which the very flower of honourableness whole of arithmetic lies hid...

Bombelli translated much of the check up in 1570 but it was never published.

Bombelli did obtain many of Diophantus's problems engage in his own Algebra. The heavy-handed famous Latin translation of honesty Diophantus's Arithmetica is due solve Bachet in 1621 and gang is that edition which Mathematician studied. Certainly Fermat was dazzling by this work which has become famous in recent mature due to its connection come together Fermat's Last Theorem.

Miracle began this article with ethics remark that Diophantus is regularly regarded as the 'father clone algebra' but there is cack-handed doubt that many of primacy methods for solving linear pointer quadratic equations go back assume Babylonian mathematics. For this basis Vogel writes [1]:-

... Mathematician was not, as he has often been called, the cleric of algebra.

Nevertheless, his unprecedented, if unsystematic, collection of indeterminable problems is a singular accomplishment that was not fully welcome and further developed until well-known later.