Muslim mathematician and astronomer, who made important contributions to the development of algebra was Omar Khayyam. Born in 1048 in Nishapur, modern day Iran, he obtained his early education from a scholar named Sheikh Mohammad Mansuri, one of the most renowned scholars of Khorasan province. Omar Khayyam started his career with teaching algebra and geometry.
Besides medicine and astronomy, mathematics was his passion. His influential mathematical treatise called “Treatise on Demonstration of Problems of Algebra” which he completed in 1070, highlighted the basic algebraic principles in mathematics. The Pascal’s triangle and his work on triangular array of binomial coefficients, has make a mark in the field of Mathematics. In 1077 another major work was written by Khayyam namely “Sharh ma ashkala min musadarat kitab Uqlidis” meaning “Explanations of the Difficulties in the Postulates of Euclid”, as he was trying to prove the parallels postulate on the properties of figures in the non-Euclidean geometry.
The theory of proportion, geometrical algebra, binomial theorem and cubic equations have been solved with higher order mathematics, most of them used in engineering and technology. Omar Khayyam, the man who remains something of an enigma, as he 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, especially his outstanding contributions in algebra. His poetry is better known in the West than any other non-Western poet.
According to historical encyclopedia, Omar Khayyam joined one of the regular caravans making a three month journey from Nishapur (now a days Iran) to the great city of Samarkand, which is now in Uzbekistan. Samarkand was a center of scholarship, and Khayyam arrived there probably in 1068, at the age of 20. In Samarkand he made contact with his father’s old friend Abu Tahir, who was governor and chief judge of the city. Tahir, observing Khayyam’s extraordinary talent with numbers, gave him a job in his office. Soon Khayyam was given a job in the king’s treasury, were he made a major advance in algebra.
We all are familiar with this fact that during our secondary school, we learn linear equations in one variable as well as linear equations in two variable. The general formula for such equation is in the form of ax2 + bx + c = 0; these are called quadratic equations. As we know that the cubic equations are different from quadratic equation, which is in the form of ax3 + bx2 + cx + d = 0. According to mathematical treatise of Omar Khayyam, cubic equations are harder to solve than quadratic equations and naturally becomes true, as he conjectured correctly that it is not possible to solve cubic equations using the traditional Ancient Greek geometrical tools of straight-edge and compass. But other methods are required to solve such equations.
Khayyam publication “Treatise on Demonstration of Problems of Algebra and Balancing” was one of his greatest works, in it he showed and explained how a cubic equation can have more than one solutions. He also showed how the intersections of conic sections such as parabolas and circles can be utilized to yield geometric solutions of cubic equations. The Archimedes principle had actually started thousand years earlier, when Khayyam considered the specific problem of finding the ratio of the volume of one part of a sphere to another.
Khayyam considered the problem and solved in methodical way. Viz In modern mathematics, and the elements of algebra, Khayyam’s solution to the equation x3 + a2x = b features a parabola of equation x2 = ay, a circle with diameter b/a2, and a vertical line through the intersection point. The solution is given by the distance on the x-axis between the origin and the vertical line. Or
When an equals sign (=) is used, this is called an equation. A very simple equation using a variable is: 2 + 3 = x. In this example, x = 5, or it could also be said, “x is five”. This is called solving for x.
Khayyam’s algebra was not the system of letters and signs we use today. His algebra was expressed in words. So, where today we write: Solve for x: x2 + 6 = 5x. Khayyam wrote: What is the amount of a square so that when 6 dirhams are added to it, it becomes equal to five roots of that square? The structure, relation and quantity of algebra, Khayyam learned algebra in another language. According to Omar Khayyam, “By learning the simple language of algebra, mathematical models of real-world situations can be created and solved. These problems can’t be solved by only using arithmetic. Instead of using words, algebra uses symbols to make statements. In algebra, letters are often used to represent numbers. Algebra also uses the same symbols as arithmetic for adding, subtracting, multiplying and dividing”.
Omar Khayyam’s solutions avoided negative coefficients and negative roots, because negative numbers were not acknowledged in Islamic mathematics. Some cultures, however, had incorporated negative numbers into mathematics – for example Brahmagupta had introduced negative numbers into Indian mathematics 400 years earlier.
Although Omar Khayyam’s achievement was magnificent, he was personally disappointed that he needed to utilize geometry to solve cubic equations – he had hoped to discover an algorithm using only algebra.
(Shah Khalid is afreelance writer, having a Diploma in Mass Communication & Journalism andElectrical Engineering from IUST Awantipora)
peerzadakhalid1545@gmail.com