Omar Khayyam - mostly known as a poet but also a great mathematician.

One can find several brands of wine named after the great poet and mathematician Omar Khayyam who lived in present Iran during the 9^th century. The word "khayyam" means "the tent-maker", and although generally considered as Persian, it has also been suggested that he could have belonged to the Khayyami tribe of Arab origin who might have settled in Persia. Little is known about his early life, except for the fact that he was educated at Nishapur and lived there and at Samarqand for most of his life. Many stories are although told about him. Muslims are according to their belief, not allowed to drink wine, but for Omar Khayyam there were special rules. Wine was a divine gift, he argued and should as such naturally be offered to and used by mankind.

Every time of the day is the right time for wine, especially during daytime, in he evening and the nights. One should take every opportunity to live a good life because life is short and no one knows how long it will last. And then we will sit there and feel sad about all the wine we haven’t drunk and all the women we haven’t loved.

He expressed his thoughts in short poems of four rows "rubaiyat".

Some for the Glories of This World; and some
Sigh for the Prophet's Paradise to come;
Ah, take the Cash, and let the Promise go,
Nor heed the rumble of a distant Drum!

His poems were discovered and translated by Edward Fitzgerald, during the 17^th century. The poems very quickly became enormously popular. It is however not entirely clear how much of the text that is Omar Khayyams and how much that has been added by Fitzgerald or others.

The Arabs made a great contribution to the development of mathematics, is said to be their role as intermediary agent between Indian and Greek mathematics and the modern world. They had access to the great Indian and Greek mathematicians works in the original language and translated them to Arabic.

The Arabs did not make full use of the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns.

The Arabs solved some cubic equations algebraically, and gave a geometrical explanation. This was done, for example, by Tabit ibn Qorra (836-901) and by al-Hasan ibn al-Haitham (965-1039).

`Umar al-Khayyami (Omar Khayyam) however, who lived c. 1048-c. 1125, had an immense influence on the development of mathematics in general and analytical geometry, in particular. His work remained ahead of others for centuries till the times of Descartes, who applied the same geometrical approach in solving cubics.

He introduced a geometrical solution, basically using a graphical method of completing the square, to equations of second degree. His method can easily be followed by this example below where it´s important to note that he only deals with numbers as representing lengths or areas.

To solve x²+10x=39 :
The expression x² is represented by the darkest area in the figure below, and 10x is divided into two pieces of the area 5x each, in order to create the shape below. The figure is then made into a complete square by adding the  square of 5*5, (shaded in the figure).

Draw the figure:

x^2+ 5x + 5x = 39 x^2+ 5x + 5x +25 = 39+25 x^2+ 5x + 5x +25 = 64 The side in the largest square is then 8, which gives us x =3 as x + 5 = 8. He could anyhow only get positive answers as he was not familiar with negative numbers..

Khayyam went further and used conical sections to to solve general equations of third degree. He used pure algebra to solve quadratics, but he verified the results using constructions. He also solved particular cubics using constructions. He admitted that he could not find an algebraic formula for the general cubic, but he invited other mathematicians to look for one. He also classified cubic equations, and he described those which he could solve and those which he could not.

Khayyam believed cubic equations could be solved only geometrically, by using conic sections.

His method, used in Algebra (c 1079) was based on a geometrical construction where the solution was found at the intersection between a parabola and a semicircle.

He solved the equation: x³ + Bx = C

by rewriting it as x³ +p²x = p²*q

where B = p² and C = p² *q

He then constructed a parabola with equation x² = py and

a circle with equation x² + y² = qx

He then found a positive solution of the equation in the intersection between the curves.

The following example shows the solution of x³ + 4x = 16

The equation x^3 + 4*x = 16 is first rewritten to x^3+2^2 *x=2^2 *4. The equation for the parabola will then be x^2 =2y and the equation for the circle x^2 +y^2 = 4x which gives us x^2 - 4x +y^2 = 0  leading to  x^2 -4x+4 +y^2 = 4 and hence the equation of the circle becomes; (x-2)^2 +y^2 =2^2 : a circle with centre in (2,0) and radius =2. The circle cuts the parabola in the points (0,0) and ( 2,2). Only positive answers were of course regarded. The solution therefore is x = 2.

This graphical solution is easily demonstated on modern, advanced 
calculator such as Texas T81, T82 or similar.

Omar Khayyam also reformed the calendar. To accomplish this task, he began his work at the new observatory at Ray in 1074 C.E. His calendar ‘Al-Tarikh-al-Jalali’ is superior to the Gregorian calendar and is accurate to within one day in 3770 years. Specifically, he measured the length of the year as 365.24219858156 days. It shows that he recognized the importance of accuracy by giving his result to eleven decimal places. As a comparison, the length of the year in our time is 365.242190 days. This number changes slightly in the sixth decimal place, e.g., in the nineteenth century it was 365.242196 days.

Back