Omar Khayyam’s Binomial Theorem

By Asiah Saiyidah

INTRODUCTION

Omar Khayyam was a scholar who is well-known as an intelligent poet, mathematician, philosopher, astronomer and physician. His full name is Ghiyath Al-Din Abu’l-Fatih ‘Umar Ibn Ibrahim al-Nisaburi al-Khayyami. He was born at Nishapur, Persia (now Iran) in year 1044 A.D. and died in 1123/24 A.D. He was named al-Khayyam, which means tent-maker in Arabic. One of his major work, which he is best known is his collection of poetry in the book of the Ruba’iyyat of Omar Khayyam (A. P. Clifford, 2009).

CONTRIBUTION

Omar Khayyam’s contribution in mathematics is the binomial theorem. It is about how he determined the n-th roots (A. Mehdi and V.B. Glen, 2017). The elaborations will be discussed later in the article.

Omar Khayyam’s other contributions can be seen from his experience in engage to several fields of knowledge which are poetry, philosophy, astronomy, astrology, mathematics and physics. Due to his contributions, he was acknowledged among the Westerners as a great Persian poet and philosopher. Meanwhile in the East, he was well-known as a great astronomer who also have done many great works in various fields, such as mathematics, philosophy and astronomy.

As a scholar in Islamic Mathematics, he was the first scholar who could develop the binomial theorem and obtain binomial coefficients (A. Zahoor, 1997) and his work on Algebra as continuation from Scholar al-Khawarizmi has big influence and noteworthy to be mentioned.

Hence, in the attempt to elaborate and explain his contribution, this paper shall give some examples based on some scholarly articles and books as reference.

BINOMIAL THEOREM

By definition, the Binomial Theorem is a way of expanding a binomial equation which is raised to some (rather large) power (S. Elizabeth, 2009).

Furthermore, binomial is a fundamental block of algebra and many parts in mathematics. Hence, Omar Khayyam put interest on this subject and was the first to understand how general binomial function form works.

In general, the expansion of (x + y)n as its first term will have;

nC0xn = xn                ,and the next term will be

nC1xn–1y = nxn–1y                     ,and lastly it will have

nCn yn = yn                   as its last term.

The coefficient of nCk is the number of chains of x’s and y’s in which y will occur precisely k times and x will occur precisely (n – k) times. These results are found in the following important theorem, which has been first proved by Omar Khayyam. This algebraic foundation is developed by Omar Khayyam. The earlier pioneers did not have the concise algebraic notation to solve the binomial problem with ease in comparison to the current way to solve – which is very easy and compact.

For any nonnegative integer n,

(x + y) n = nC0xny0 + nC1xn–1y1 + nC2xn–2y2 + … + nCkxn–kyk + … + nCnx0yn

= k xn-kyk

For example,

Expand (a+b)n, for n  N   (N = any nonnegative natural number)

Geometrical result for general n, solved by Omar Khayyam can be shown as an example:

For n=2, (a+b)2 = a2 +2ab +b2

Untitled.png

This diagram shows the geometrical result of binomial with n=2.

Omar Khayyam used geometry to express the binomial equation. The actual method of his solution to the problem is unknown because his work on the subject has been lost but we do know from other various sources that he was able to solve the nth problem. In another words, he was able to express or expand (a+b) to the nth.

The expression of (a+b) when n increases from 1 to 5 can be done as:

(a+b)1 = a + b

(a+b)2 =  a2 + 2ab + b2

(a+b)3 = a3 + 3a2b + 3ab2 + b3

(a+b)4 = a4 +4a3b + 6a2b2 + 4ab3 +b4

(a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b– (3)

Formal equation for expansion:  (a+b)n = an + nan−1b + … + bn

These computations uses the Distributive Law. The Distributive Law say, in equation (1), is when multiply a form (a+b) to some other form, that is (c+d), the result will get four different terms.

(a+b) (c+d) = ac + ad + bc + bd   – (1)

(a+b) (c+d+e) = ac+ad+ae+bc+bd+be   – (2)

And so on.

Based on (a+b)n expressions above, observe that all the power terms that are involved, that is the degrees of a and b, will add up to give the n. Such as equation (3), where  n=5, for a3b2 total degrees = 2+3 = 5.

Moreover, is that the coefficients of n=5, is (1, 5, 10, 10, 5, 1). These numerical coefficients are known as binomial coefficients (A. P. Clifford, 2009). These are the values in a row of the Pascal’s Triangle as in Diagram 2 (in Appendix). The question is, how to expand a large value of nCr, such as (a+b)10 to get the coefficients without having do the arithmetic? That is the essence of this binomial problem that Omar Khayyam solved.

The binomial coefficients is expanded from its expression to form a symmetrical triangle, which is named as Pascal’s Triangle or usually called as Khayyam’s triangle in Iran (A. Rooney, 2012) which was studied by India, Persia, China and Italy, including Al-Kharaji. Omar Khayyam had generalized Indian methods for extracting roots of squares and cubes to include the fourth, fifth and higher roots (L. Mastin, 2010). He succeeded in solving cubic equations.

Another way to find nCr, is by using Pascal’s triangle. The starting point is three 1’s as shown in Diagram 1 (in Appendix). When new pair of numbers come, then add another row one by one until it forms a symmetrical triangle as shown in Diagram 2 (in Appendix). In Pascal’s triangle, each number below is the addition of the two numbers above it. This pattern forms the binomial coefficient series.

Omar Khayyam’s work about this contained in his book “Treatise on Demonstration of Problems of Algebra” (1070).

He mentioned about Indian mathematician work (Aryabhata, 500 A.D.) on cubic powers. Hence, Indian mathematicians knew at least how to expand (a+b)3. Omar is actually motivated by problem of finding cube roots and higher powers of a number which can only be understood by someone who have deep knowledge of binomial theory, like himself.

APPLICATION

The Binomial Theorem is widely used in our daily application at home, workplace and so on. There are a lot of areas which requires the application of binomial theorem, such as computing for the distribution of IP addresses. Binomial theorem has also been an important use in the architecture field for the design of infrastructure.

In economics, Binomial theorem is used as one of the vital tools in the nation’s economic predictions. It is used by economists to handle a widely distributed variables in order to count probabilities. This helps to predict the behavior of nation’s economy in the near future. Realistic economic predictions are possible by using this Theorem.

Another application of Binomial Theorem is in the calculation of interest that will be received after several years on a sum of money that has been invested at certain interest rate compounded. Furthermore, the theorem also helps to find the size of population of a country after several years, 10 year as example, if the annual population growth rate is known. This Theorem helps in many ways to calculate the rational powers of any real binomial expression involving two terms. Hence, Binomial Theorem is listed as one of efficient tools, vital for a lot of applications, including economics.

 

REFERENCE

A.P. Clifford, The Math Book: From Phytagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. 2009, Sterling Publishing Company, Inc., page 94.

A. Rooney, The History of Mathematics, 15 July 2012, The Rosen Publishing Group, page 126-129.

A.R. Amir-Moez, A Paper of Omar Khayyám, 1963, Scripta Mathematica 26, pp. 323–337

B. Rachel, Omar Khayyam: Persian Mathematician, Astronomer and Poet., 18 May, 2009, FindingDulcinea. N.p., Retrieved on 22 April 2017, from: http://www.findingdulcinea.com/features/profiles/k/omar-khayyam.html

A. Zahoor, Omar Al-Khayyam (1044-1123 C.E.) 1992, 1997. Retrieved on 26 April 2017, from http://www.unhas.ac.id/rhiza/arsip/saintis/khayyam.html

S. Elizabeth, The Binomial Theorem: Formulas., 2009, Purplemath. Retrieved on 26 April 2017 from http://www.purplemath.com/modules/binomial.htm.

A. Mehdi and V. B. Glen, Umar Khayyam, (Spring 2017 Edition), The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Retrieved on 26 April 2017 from https://plato.stanford.edu/archives/spr2017/entries/umar-khayyam/

L. Mastin, Islamic Mathematics. 2010. Retrieved on 26 April 2017 from http://www.storyofmathematics.com/islamic.html

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