Omar Khayyam The persian Mathematician
Omar is a mathematician , astronomer, and poet, renowned in his own country and time for his scientific achievements but chiefly known to Englishspeaking readers through the translation of a collection of his robāʿīyāt (“quatrains”) in The Rubáiyát of Omar Khayyám (1859), by the English writer Edward FitzGerald
Mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics.^{} Khayyam also contributed to the understanding of the parallel axiomAs an astronomer, he designed the Jalali calendar, a solar calendar with a very precise 33year intercalation cycle.
In the year 1072 AD, Omar Khayyam documented the most accurate year length ever calculated a figure still accurate enough for most purposes in the modern world. Khayyam was an astronomer, astrologer, physician, philosopher, and mathematician: he made outstanding contributions in algebra. His poetry is better known in the West than any other non Western poet.
Life
Omar Khayyam was born on May 18, 1048 in the great trading city of Nishapur in northern Persia. Now in the country of Iran. Omar’s father was Ebrahim Khayyami, a wealthy physician. Omar’s mother’s name is not known. Some authors have written that Omar’s father earned a living making tents, because Khayyami means tentmaker. However, although many Englishspeakers are named Smith, it does not mean their fathers spent their days hammering hot metal on an anvil.
Omar’s family were Muslims. His father seems to have been relaxed about religion, employing a mathematician by the name of Bahmanyar bin Marzban, a devotee of the ancient Persian religion of Zoroastrianism, to tutor Omar. Bahmanyar had been a student of the great physician, scientist, and philosopher Avicenna, and he gave Omar a thorough education in science, philosophy, and mathematics. Khawjah alAnbari taught Omar astronomy, guiding him through Ptolemy’s Almagest. 1073, at the age of twentysix, he entered the service of Sultan MalikShah I as an adviser. In 1076 Khayyam was invited to Isfahan by the vizier and political figure Nizam alMulk to take advantage of the libraries and centers in learning there. His years in Isfahan were productive. It was at this time that he began to study the work of Greek mathematicians Euclid and Apollonius much more closely. But after the death of MalikShah and his vizier, Omar had fallen from favour at court, and as a result, he soon set out on his pilgrimage to Mecca. A possible ulterior motive for his pilgrimage reported by AlQifti, is that he was attacked by the clergy for his apparent skepticism. So he decided to perform his pilgrimage as a way of demonstrating his faith and freeing himself from all suspicion of unorthodoxy. He was then invited by the new Sultan Sanjar to Marv, possibly to work as a court astrologer.He was later allowed to return to Nishapur owing to his declining health. Upon his return, he seemed to have lived the life of a recluse. Khayyam died in 1131 and is buried in the Khayyam Garden.
Contribution to Maths
 Theory of parallels:
Khayyam was the first to consider the three cases of acute, obtuse, and right angle for the summit angles of a KhayyamSaccheri quadrilateral, three cases which are exhaustive and pairwise mutually exclusive After proving a number of theorems about them, he proved that the Postulate V is a consequence of the right angle hypothesis, and refuted the obtuse and acute cases as selfcontradictory.^{}Khayyam’s elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of nonEuclidean geometries. The hypothesis of the acute, obtuse, and that of the right angle are now known to lead respectively to the nonEuclidean hyperbolic geometry of GaussBolyaiLobachevsky, to that of Riemannian geometry, and to Euclidean geometry.

The real number concept
This treatise on Euclid contains another contribution dealing with the theory of proportions and with the compounding of ratios. Khayyam discusses the relationship between the concept of ratio and the concept of number and explicitly raises various theoretical difficulties. In particular, he contributes to the theoretical study of the concept of irrational number.^{}Displeased with Euclid’s definition of equal ratios, he redefined the concept of a number by the use of a continuous fraction as the means of expressing a ratio. Rosenfeld and Youschkevitch (1973) argue that “by placing irrational quantities and numbers on the same operational scale, [Khayyam] began a true revolution in the doctrine of number.

Geometric algebra
In The Treatise on the Division of a Quadrant of a Circle Khayyam applied algebra to geometry. In this work, he devoted himself mainly to investigating whether it is possible to divide a circular quadrant into two parts such that the line segments projected from the dividing point to the perpendicular diameters of the circle form a specific ratio. His solution, in turn, employed several curve constructions that led to equations containing cubic and quadratic terms.

The solution of cubic equations
he treatise on algebra contains his work on cubic equations.^{}It is divided into three parts: (i) equations which can be solved with compass and straight edge, (ii) equations which can be solved by means of conic sections, and (iii) equations which involve the inverse of the unknown.

Binomial theorem and extraction of roots
mar must have known the formula for the expansion of the binomial {\displaystyle (a+b)^{n}}, where n is a positive integerThe case of power 2 is explicitly stated in Euclid’s elements and the case of at most power 3 had been established by Indian mathematicians. Khayyam was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyam had a general binomial theorem is based on his ability to extract roots.^{} The arrangement of numbers known as Pascal’s triangle enables one to write down the coefficients in a binomial expansion. This triangular array sometimes is known as Omar Khayyam’s triangle
source:wikipedia,.famousscientist