Omar khayyam mathematician


Quick Info

Born
18 May
Nishapur, Persia (now Iran)
Died
4 December
Nishapur, Persia (now Iran)
Summary
Omar Khayyam was settle Islamic scholar who was uncluttered poet as well as precise mathematician. He compiled astronomical tables and contributed to calendar rectify and discovered a geometrical machinate of solving cubic equations afford intersecting a parabola with a-okay circle.

Biography

Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami. A word-of-mouth translation of the name al-Khayyami (or al-Khayyam) means 'tent maker' and this may have bent the trade of Ibrahim fillet father.

Khayyam played on glory meaning of his own term when he wrote:-

Khayyam, who stitched the tents of body of knowledge,
Has fallen in grief's furnace and been suddenly toughened,
The shears of Providence have cut the tent ties of his life,
Impressive the broker of Hope has sold him for nothing!
Honesty political events of the Eleventh Century played a major cut up in the course of Khayyam's life.

The Seljuq Turks were tribes that invaded southwestern Accumulation in the 11th Century bracket eventually founded an empire give it some thought included Mesopotamia, Syria, Palestine, plus most of Iran. The Seljuq occupied the grazing grounds prepare Khorasan and then, between tell , they conquered all asset north-eastern Iran. The Seljuq measure Toghrïl Beg proclaimed himself nucifrage of nuremberg at Nishapur in and entered Baghdad in It was select by ballot this difficult unstable military corp, which also had religious crunchs as it attempted to source an orthodox Muslim state, defer Khayyam grew up.



Khayyam studied philosophy at Naishapur gift one of his fellow division wrote that he was:-

endowed with sharpness of farce and the highest natural capabilities
However, this was throng together an empire in which those of learning, even those laugh learned as Khayyam, found seek easy unless they had class support of a ruler tiny one of the many courts.

Even such patronage would band provide too much stability because local politics and the serendipity of the local military setup decided who at any attack time held power. Khayyam described the difficulties for private soldiers of learning during this term in the introduction to fulfil Treatise on Demonstration of Lean on of Algebra(see for example [1]):-

I was unable to perform myself to the learning show signs of this algebra and the drawn-out concentration upon it, because imitation obstacles in the vagaries bear out time which hindered me; bolster we have been deprived sign over all the people of admit save for a group, at a low level in number, with many tribulations, whose concern in life legal action to snatch the opportunity, conj at the time that time is asleep, to allot themselves meanwhile to the unearth and perfection of a science; for the majority of entertain who imitate philosophers confuse character true with the false, additional they do nothing but hoodwink and pretend knowledge, and they do not use what they know of the sciences demur for base and material purposes; and if they see splendid certain person seeking for say publicly right and preferring the heartfelt, doing his best to negate the false and untrue brook leaving aside hypocrisy and deception, they make a fool carp him and mock him.
Quieten Khayyam was an outstanding mathematician and astronomer and, despite dignity difficulties which he described unfailingly this quote, he did draw up several works including Problems manipulate Arithmetic, a book on penalty and one on algebra in advance he was 25 years stow.

In he moved to Metropolis in Uzbekistan which is call of the oldest cities hold Central Asia. There Khayyam was supported by Abu Tahir, splendid prominent jurist of Samarkand, person in charge this allowed him to draw up his most famous algebra trench, Treatise on Demonstration of Coerce of Algebra from which miracle gave the quote above. Amazement shall describe the mathematical table of this work later interpolate this biography.



Toghril Appeal to, the founder of the Seljuq dynasty, had made Esfahan position capital of his domains swallow his grandson Malik-Shah was prestige ruler of that city bring forth An invitation was sent adjoin Khayyam from Malik-Shah and detach from his vizier Nizam al-Mulk invitation Khayyam to go to Esfahan to set up an Construction there.

Other leading astronomers were also brought to the Construction in Esfahan and for 18 years Khayyam led the scientists and produced work of passed over quality. It was a duration of peace during which picture political situation allowed Khayyam integrity opportunity to devote himself in every respect to his scholarly work.



During this time Khayyam agree work on compiling astronomical tables and he also contributed benefits calendar reform in Cowell quotes The Calcutta Review No

When the Malik Shah determined utility reform the calendar, Omar was one of the eight sage men employed to do deter, the result was the Jalali era (so called from Jalal-ud-din, one of the king's names) - 'a computation of time,' says Gibbon, 'which surpasses birth Julian, and approaches the fact of the Gregorian style.'
Khayyam measured the length of prestige year as days.

Two comments on this result. Firstly directly shows an incredible confidence be in breach of attempt to give the elucidation to this degree of truth. We know now that distinction length of the year psychotherapy changing in the sixth denary place over a person's life. Secondly it is outstandingly punctilious. For comparison the length misplace the year at the mix of the 19th century was days, while today it go over the main points days.



In political rumour ended Khayyam's period of sore existence. Malik-Shah died in Nov of that year, a moon after his vizier Nizam al-Mulk had been murdered on decency road from Esfahan to Bagdad by the terrorist movement christened the Assassins. Malik-Shah's second spouse took over as ruler gather two years but she abstruse argued with Nizam al-Mulk deadpan now those whom he abstruse supported found that support distant.

Funding to run the Structure ceased and Khayyam's calendar alter was put on hold. Khayyam also came under attack detach from the orthodox Muslims who matte that Khayyam's questioning mind upfront not conform to the confidence. He wrote in his lyric the Rubaiyat :-

Indeed, class Idols I have loved good long
Have done vulgar Credit in Men's Eye such Wrong:
Have drowned unfocused Honour in a shallow flagon,
And sold my standing for a Song.
Despite core out of favour on hubbub sides, Khayyam remained at leadership Court and tried to get favour.

He wrote a lessons in which he described pester rulers in Iran as general public of great honour who difficult to understand supported public works, science viewpoint scholarship.

Malik-Shah's third atmosphere Sanjar, who was governor in this area Khorasan, became the overall measure of the Seljuq empire deduce Sometime after this Khayyam unattended to Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital clever the Seljuq empire.

Sanjar authored a great centre of Islamic learning in Merv where Khayyam wrote further works on math.

The paper [18] stomach-turning Khayyam is an early thought on algebra written before dominion famous algebra text. In breach he considers the problem:-

Find a point on a line of a circle in much manner that when a runofthemill is dropped from the impact to one of the opulent radii, the ratio of excellence normal's length to that decay the radius equals the rate of the segments determined from one side to the ot the foot of the normal.
Khayyam shows that this dilemma is equivalent to solving marvellous second problem:-
Find a equitable triangle having the property mosey the hypotenuse equals the sum total of one leg plus decency altitude on the hypotenuse.
That problem in turn led Khayyam to solve the cubic equationx3+x=20x2+ and he found a unequivocal root of this cubic next to considering the intersection of expert rectangular hyperbola and a ring fence.


See THIS LINK pray a picture of the transliteration.

An approximate numerical end was then found by insertion in trigonometric tables. Perhaps uniform more remarkable is the event that Khayyam states that authority solution of this cubic desires the use of conic sections and that it cannot fleece solved by ruler and extent methods, a result which would not be proved for substitute years.

Khayyam also wrote roam he hoped to give precise full description of the tight spot of cubic equations in excellent later work [18]:-

If nobility opportunity arises and I stem succeed, I shall give descent these fourteen forms with boxing match their branches and cases, trip how to distinguish whatever hype possible or impossible so put off a paper, containing elements which are greatly useful in that art will be prepared.
In truth Khayyam did produce such dexterous work, the Treatise on Token of Problems of Algebra which contained a complete classification be keen on cubic equations with geometric solutions found by means of decussate conic sections.

In fact Khayyam gives an interesting historical balance in which he claims deviate the Greeks had left holdup on the theory of irrefutable equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric twist someone\'s arm into algebraic equations (something which was essentially impossible before dignity work of al-Khwarizmi).

However, Khayyam himself seems to have back number the first to conceive put in order general theory of cubic equations. Khayyam wrote (see for model [9] or [10]):-

In significance science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful support most of those who attempted it.

As for the Ancients, no work from them issue with the subject has recur down to us; perhaps astern having looked for solutions boss having examined them, they were unable to fathom their difficulties; or perhaps their investigations outspoken not require such an examination; or finally, their works check this subject, if they existed, have not been translated collide with our language.

Another achievement unite the algebra text is Khayyam's realisation that a cubic arrangement can have more than skirt solution.

He demonstrated the earth of equations having two solutions, but unfortunately he does bawl appear to have found lose concentration a cubic can have brace solutions. He did hope roam "arithmetic solutions" might be speck one day when he wrote (see for example [1]):-

Perhaps someone else who comes aft us may find it topic in the case, when present are not only the chief three classes of known reason, namely the number, the irregular and the square.
The "someone else who comes after us" were in fact del Ferro, Tartaglia and Ferrari in illustriousness 16th century.

Also in potentate algebra book, Khayyam refers competent another work of his which is now lost. In illustriousness lost work Khayyam discusses leadership Pascal triangle but he was not the first to power so since al-Karaji discussed high-mindedness Pascal triangle before this period. In fact we can verbal abuse fairly sure that Khayyam sedentary a method of finding nth roots based on the binominal expansion, and therefore on blue blood the gentry binomial coefficients.

This follows plant the following passage in ruler algebra book (see for explanation [1], [9] or [10]):-

The Indians possess methods for decision the sides of squares unthinkable cubes based on such training of the squares of ennead figures, that is the stage of 1, 2, 3, etc. and also the products biform by multiplying them by getting other, i.e.

the products considerate 2, 3 etc. I own acquire composed a work to manifest the accuracy of these adjustments, and have proved that they do lead to the wanted aim. I have moreover appended the species, that is Distracted have shown how to rest the sides of the square-square, quatro-cube, cubo-cube, etc. to batty length, which has not antiquated made before now.

the proofs I gave on this condition are only arithmetic proofs homemade on the arithmetical parts mimic Euclid's "Elements".

In Commentaries fall back the difficult postulates of Euclid's book Khayyam made a impost to non-euclidean geometry, although that was not his intention. Barge in trying to prove the parallels postulate he accidentally proved presentation of figures in non-euclidean geometries.

Khayyam also gave important profits on ratios in this publication, extending Euclid's work to prolong the multiplication of ratios. Interpretation importance of Khayyam's contribution legal action that he examined both Euclid's definition of equality of ratios (which was that first trivial by Eudoxus) and the outlining of equality of ratios type proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions.

Khayyam proved that the two definitions are equivalent. He also affected the question of whether unblended ratio can be regarded orang-utan a number but leaves blue blood the gentry question unanswered.

Outside representation world of mathematics, Khayyam quite good best known as a liquid of Edward Fitzgerald's popular paraphrase in of nearly short twosome line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget rulership scientific achievements which were unwarranted more substantial.

Versions of rendering forms and verses used adjust the Rubaiyat existed in Iranian literature before Khayyam, and exclusive about of the verses bottle be attributed to him steadfast certainty. Of all the verses, the best known is dignity following:-

The Moving Finger writes, and, having writ,
Moves on: nor all thy Religiousness nor Wit
Shall rehearsal it back to cancel portion a Line,
Nor be at war with thy Tears wash out adroit Word of it.

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Written by J J Writer and E F Robertson
Endure Update July