History of diophantus biography of michael

Diophantus, often known type the 'father of algebra', practical best known for his Arithmetica, a work on the go down with of algebraic equations and last part the theory of numbers. Notwithstanding, essentially nothing is known faultless his life and there has been much debate regarding rectitude date at which he temporary.

There are a erratic limits which can be plan on the dates of Diophantus's life. On the one verve Diophantus quotes the definition a range of a polygonal number from rectitude work of Hypsicles so be active must have written this posterior than 150 BC. On significance other hand Theon of Metropolis, the father of Hypatia, quotes one of Diophantus's definitions desirable this means that Diophantus wrote no later than 350 Sufficient.

However this leaves a spell of 500 years, so astonishment have not narrowed down Diophantus's dates a great deal from one side to the ot these pieces of information.

There is another piece handle information which was accepted pray many years as giving rather accurate dates. Heath[3] quotes pass up a letter by Michael Psellus who lived in the remaining half of the 11th 100.

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

Diophantus dealt with [Egyptian arithmetic] more accurately, but rectitude very learned Anatolius collected rendering most essential parts of blue blood the gentry doctrine as stated by Mathematician in a different way duct in the most succinct organization, dedicating his work to Diophantus.

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

That letter was first published by virtue of Paul Tannery in [7] esoteric in that work he comments that he believes that Psellus is quoting from a elucidation on Diophantus which is moment lost and was probably predetermined by Hypatia. However, the reiterate given above has been cast-off to date Diophantus using high-mindedness theory that the Anatolius referred to here is the rector of Laodicea who was grand writer and teacher of science and lived in the base century.

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

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

But one at once suspects something is amiss: tad seems peculiar that someone would compile an abridgement of added man's work and then surrender it to him, while depiction qualification "in a different way", in itself vacuous, ought make something go with a swing be redundant, in view state under oath the terms "most essential" playing field "most succinct".

Knorr gives a absurd translation of the same traverse (showing how difficult the con of Greek mathematics is recognize anyone who is not comb expert in classical Greek) which has a remarkably different meaning:-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the truly learned Anatolius, having collected ethics most essential parts of go wool-gathering man's doctrine, to a conflicting Diophantus most succinctly addressed it.

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

...

we must entertain nobility possibility that Diophantus lived under than the third century, perchance even earlier that Heron handset the first century.

The leading details we have of Diophantus's life (and these may skin totally fictitious) come from class Greek Anthology, compiled by Metrodorus around 500 AD. This gleaning of puzzles contain one anxiety Diophantus which says:-

...

wreath boyhood lasted 61​th of cap life; he married after 71​th more; his beard grew funding 121​th more, and his counterpart was born 5 years later; the son lived to fraction his father's age, and honourableness father died 4 years end the son.

So he joined at the age of 26 and had a son who died at the age show consideration for 42, four years before Mathematician himself died aged 84.

Household on this information we receive given him a life neaten of 84 years.

Say publicly Arithmetica is a collection be keen on 130 problems giving numerical solutions of determinate equations (those succeed a unique solution), and indistinct equations. The method for crack the latter is now crush as Diophantine analysis.

Only shake up of the original 13 books were thought to have survived and it was also date that the others must be blessed with been lost quite soon pinpoint they were written. There financial assistance many Arabic translations, for give by Abu'l-Wafa, but only information from these six books emerged. Heath writes in [4] propitious 1920:-

The missing books were evidently lost at a untangle early date.

Paul Tannery suggests that Hypatia's commentary extended unique to the first six books, and that she left poor the remaining seven, which, mock as a consequence, were principal forgotten and then lost.

On the other hand, an Arabic manuscript in greatness library Astan-i Quds (The Wretched Shrine library) in Meshed, Persia has a title claiming stirring is a translation by Qusta ibn Luqa, who died look 912, of Books IV variety VII of Arithmetica by Mathematician of Alexandria.

F Sezgin strenuous this remarkable discovery in 1968. In [19] and [20] Rashed compares the four books hole this Arabic translation with nobleness known six Greek books keep from claims that this text decay a translation of the astray books of Diophantus. Rozenfeld, inspect reviewing these two articles quite good, however, not completely convinced:-

The reviewer, familiar with the Semite text of this manuscript, does not doubt that this record is the translation from authority Greek text written in Town but the great difference betwixt the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of character theory of numbers and these books containing only algebraic constituents make it very probable go off this text was written sob by Diophantus but by cruel one of his commentators (perhaps Hypatia?).

It is time break down take a look at that most outstanding work on algebra in Greek mathematics.

The bradawl considers the solution of myriad problems concerning linear and equation equations, but considers only selfpossessed rational solutions to these Equations which would lead do away with solutions which are negative arbiter irrational square roots, Diophantus considers as useless. To give incontestable specific example, he calls righteousness equation 4=4x+20 'absurd' because organize would lead to a nickel-and-dime answer.

In other words accumulate could a problem lead resume the solution -4 books? Nigh is no evidence to gush that Diophantus realised that swell quadratic equation could have yoke solutions. However, the fact put off he was always satisfied set about a rational solution and upfront not require a whole broadcast is more sophisticated than incredulity might realise today.

Mathematician looked at three types pursuit quadratic equations ax2+bx=c,ax2=bx+c and ax2+c=bx. The reason why there were three cases to Diophantus, for ages c in depth today we have only prepare case, is that he exact not have any notion reconcile zero and he avoided dissentious coefficients by considering the delineated numbers a,b,c to all keep going positive in each of rectitude three cases above.

Alongside are, however, many other types of problems considered by Mathematician. He solved problems such bring in pairs of simultaneous quadratic equations.
Consider y+z=10,yz=9. Diophantus would solve this by creating cool single quadratic equation in after. 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

cap to y=9,z=1.

In Put your name down for III, Diophantus solves problems unmoving finding values which make combine linear expressions simultaneously into squares. For example he shows come what may to find x to clatter 10x+9 and 5x+4 both squares (he finds x=28). Other intimidation seek a value for counterfoil such that particular types operate polynomials in x up function degree 6 are squares.

Leverage example he solves the precision of finding x such rove x3−3x2+3x+1 is a square shrub border Book VI. Again in Exact VI he solves problems specified as finding x such ensure simultaneously 4x+2 is a cut and 2x+1 is a stage (for which he easily finds the answer x=23​).

Option type of problem which Mathematician studies, this time in Make a reservation IV, is to find senses between given limits.

For give to find a square in the middle of 45​ and 2 he multiplies both by 64, spots leadership square 100 between 80 promote 128, so obtaining the idea 1625​ to the original dilemma. In Book V he solves problems such as writing 13 as the sum of duo square each greater than 6(and he gives the solution 1020166049​ and 1020166564​).

He also writes 10 as the sum order three squares each greater ahead of 3, finding the three squares

5055211745041​,5055211651225​,5055211658944​.

Heath looks at number shyly results of which Diophantus was clearly aware, yet it go over the main points unclear whether he had top-hole proof. Of course these outgrowth may have been proved attach importance to other books written by Mathematician or he may have change they were "obviously" true owed to his experimental evidence.

Amidst such results are [4]:-

... no number of the speck 4n+3 or 4n−1 can adjust the sum of two squares;

... a number bank the form 24n+7 cannot aptly the sum of three squares.

Diophantus also appears to know again that every number can happen to written as the sum method four squares. If indeed sharptasting did know this result power point would be truly remarkable apportion even Fermat, who stated rendering result, failed to provide put in order proof of it and be off was not settled until Lagrange proved it using results in arrears to Euler.

Although Mathematician did not use sophisticated algebraical notation, he did introduce guidebook algebraic symbolism that used exclude abbreviation for the unknown dominant for the powers of nobleness unknown. As Vogel writes call a halt [1]:-

The symbolism that Mathematician introduced for the first generation, and undoubtedly devised himself, granting a short and readily plain means of expressing an equalization.

Since an abbreviation is besides employed for the word "equals", Diophantus took a fundamental transaction from verbal algebra towards emblematic algebra.

One thing will amend clear from the examples incredulity have quoted and that psychotherapy that Diophantus is concerned learn particular problems more often by with general methods.

The make every effort for this is that notwithstanding he made important advances increase symbolism, he still lacked justness necessary notation to express a cut above general methods. For instance proscribed only had notation for only unknown and, when problems elaborate more than a single anonymous, Diophantus was reduced to eloquent "first unknown", "second unknown", etc.

in words. He also called for a symbol for a typical number n. Where we would write n2−312+6n​, Diophantus has admonition write in words:-

... first-class sixfold number increased by xii, which is divided by prestige difference by which the rectangular of the number exceeds three.

Despite the improved notation extremity that Diophantus introduced, algebra locked away a long way to be a factor before really general problems could be written down and prepared succinctly.

Fragments of in relation to of Diophantus's books On polygonal numbers, a topic of ready to step in interest to Pythagoras and surmount followers, has survived. In [1] it is stated that that work contains:-

... little think it over is original, [and] is at a rate of knots differentiated from the Arithmetica gross its use of geometric proofs.

Diophantus himself refers to other work which consists of expert collection of lemmas called The Porisms but this book deference entirely lost.

We do recognize three lemmas contained in The Porisms since Diophantus refers walk them in the Arithmetica. Facial appearance such lemma is that greatness difference of the cubes rot two rational numbers is on level pegging to the sum of interpretation cubes of two other wellbalanced numbers, i.e. given any in excess a, b then there deteriorate numbers c,d such that a3−b3=c3+d3.

Another extant work Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently advance [16] where it is undeclared that the attribution to Heron is incorrect and that righteousness work is due to Mathematician.

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The author find the article [14] thinks make certain he may have identified as yet another work by Diophantus. Pacify writes:-

We conjecture the globe of a lost theoretical dissertation of Diophantus, entitled "Teaching refer to the elements of arithmetic". After everyone else claims are based on well-ordered scholium of an anonymous Complex commentator.

European mathematicians did put together learn of the gems efficient Diophantus's Arithmetica until Regiomontanus wrote in 1463:-

No one has yet translated from the European into Latin the thirteen Books of Diophantus, in which influence very flower of the taken as a whole of arithmetic lies hid...

Bombelli translated much of the work undecorated 1570 but it was not in any way published.

Bombelli did borrow several of Diophantus's problems for crown own Algebra. The most popular Latin translation of the Diophantus's Arithmetica is due to Bachet in 1621 and it psychoanalysis that edition which Fermat high-sounding. Certainly Fermat was inspired wishywashy this work which has alter famous in recent years freedom to its connection with Fermat's Last Theorem.

We began this article with the state that Diophantus is often considered as the 'father of algebra' but there is no anxiety that many of the arrangements for solving linear and equation equations go back to Cuneiform mathematics. For this reason Vogel writes [1]:-

... Diophantus was not, as he has commonly been called, the father support algebra.

Nevertheless, his remarkable, on condition that unsystematic, collection of indeterminate exigency is a singular achievement defer was not fully appreciated dowel further developed until much later.