Pascals Triangle: History and Implications Including Functionality - Essay Example

www.academia-research.com Sumanta Sanyal d: 16/05/2006 Pascal's Triangle: History and Implications including Functionality Introduction This paper investigates the history and implication of Pascal's Triangle which is a number triangle represented as follows in very short form. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 (Mathworld, Pascal's Triangle, 2006) The triangle can be represented by the following equation: (n, r) where (n, r) is a binomial coefficient which signifies coefficient of the number in the row of the triangle. The number rows can be staggered such that they conform to the shape of a triangle in which each subsequent row is obtained by adding the two numbers diagonally above. For example, in the given example, the 3rd number in the 6th row - 10 - can be obtained by the sum of the 2nd and 3rd numbers, those diagonally above, in the 5th row. (Mathworld, Pascal's Triangle, 2006) This may be construed as: (n, r) = = [(n - 1), r] + [(n - 1), (r - 1)] In what may be called the renaissance period in Europe this particular number triangle was first studied by B. Pascal, whose name was subsequently bequeathed to it though Chinese mathematician Yangsui generated it almost 500 years earlier than that. In China it became known as Yangsui's triangle while Persian astronomer-poet Omar Khayyam also studied it in some details.(Mathworld, Pascal's Triangle, 2006) Indian mathematicians like Pingalacharya also knew of it even earlier than the Chinese and all these shall be discussed subsequently together with some of this number triangle's unique properties. Blaise Pascal (1623-1662) Though not much is known about how Pascal exactly came upon this number triangle it is obvious that since he was well-known as a mathematician, philosopher and religious figure in the France of that time he made it famous. This is one of his lesser-known achievements. He is much better known for his discovery of the constant pressure within a static fluid (Pascal's Principle). (Scienceworld, 2006) Nevertheless, Pascal established the principal relationship within this triangle and this can be depicted as per his formula - Pascal's Formula - as below: (n, r) = = [(n - 1), r] + [(n - 1), (r - 1)] The derivation as per Pascal is as follows: (n, r) = = + = + = [(n - 1), r] + [(n - 1), (r - 1)] (Mathworld, Pascal's Formula, 2006) The implication of the formula has already been stated. Now the paper shall proceed to explore the history of the triangle as it goes back beyond the Middle Ages and Pascal. The History of the Triangle The first known description of a binary numeral system, that ultimately generates the triangle, is to be found in the works of Pingalacharya, the famous circa 5th century B.C. Indian scholar on prosody. He is supposed to be the younger brother of the more famous Sanskrit Grammarian Panini, whose grammar is still considered to provide the basic guidelines for that language. Actually, Pingala was exploring the listing of Vedic meters in short and long syllables when he came upon the system of binary numerals. His discussions of the combinations possible for the meters describes the binomial theorem. His works were later taken up by the 10th century Indian mathematician Halayudha whose commentary presents a form of the Pascal's triangle. It is described as the 'Meru-prastaara', as the rudimentary form of the Pascal's triangle was known then in Sanskrit. Pingala was also the first person to make mention of the Fibonacci Numbers, as they are known now. The paper shall touch upon the Fibonacci numbers later on. In Sanskrit, as Pingala would have it, these numbers were known as 'maatraameru'. (Wikipedia, Pingala, 2006) It is noted here that Indian astronomers and mathematicians were quite advanced in those days and quite a few important concepts such as that of zero, attributed to Halayudha who came after Pingala (Wikipedia, Pingala, 2006), was forthcoming from that eminent ancient country. It is also interesting to note that, thus, Pascal's triangle has a much earlier origin than can be envisaged from its apparent history. It becomes evident, as the paper proceeds, that different mathematicians from diverse countries such as India, China and Persia have independently realized it until it ultimately became firmly established as a mathematical entity by B. Pascal of Europe. Yangsui, China Historical records prove that the Chinese as early as in 1261 A.D knew Pascal's Triangle. It appears to a depth of 6 (6 rows) in the works of Yangsui (1261 A.D.) and to a depth of 8 (8 rows) in Zhu Shijiei (1303 A.D.). Yangsui attributes the concept to the 11th century Jia Xian. The Chinese used the triangle to generate binomial coefficients. (Binomial Theorem and Pascal's Triangle) Since there was frequent exchange of academic ideas between the Chinese and the Indians from the 7th century onwards it is not entirely impossible that the works of Pingala influenced the Chinese. There are no exact records to prove this and the general layman view is that this unique triangle is an exclusive Chinese discovery. The Chinese probably made the earliest diagrammatic representation of the triangle reinforcing the popular viewpoint that the triangle was a Chinese invention. The Arab Mathematicians Al-Karaji: It is recorded that the Arabs got their combinatorics from the Indians. (Pascal's Triangle - Mathematics and the Liberal Arts, 2006) Mathematics historian F. Woepcke (1853, Paris) notes in his book of that time that Al-Karaji, a Persian mathematician and engineer, was the first to introduce algebraic calculus to the world. It is already well known that the Arabs invented calculus but what is significant to this paper is that Al-Karaji also investigated binomial coefficients and Pascal's triangle in this context. It may be that some inkling of the triangular pattern may have originated from India but it is certain that independently Al-Karaji was responsible fro introducing it to the then Arab world. Al-Karaji's full name was Abu Bakr Muhammad ibn al-Hasan al-Karaji. It is quite a mouthful but entirely fit for such an excellent mathematician. He lived circa 953 A.D. to circa 1029 A.D. Omar Khayyam: The second Arab of great note who has been known to have delved into the mysteries of the triangle is Omar Khayyam - Persian poet, philosopher, mathematician and astronomer. He lived circa 1044-1123 A.D. (Persian Language and Literature, Omar Khayyam) Khayyam, being a Arab mathematical of his time, did much work on algebra and it is not impossible that he came upon the works of his predecessor Al-Karaji. He makes mention of the triangle in one of his works. The work in which he is supposed to have dealt with the triangle is lost today. (Persian Language and Literature, Omar Khayyam) So nothing definite can be known though it is certain from his own writings that he had become involved with the triangle's intricacies. The Triangle from Algebraic Expansions The paper now explores how the triangle can be derived from algebraic expansions. An algebraic expression is taken: = 1 Note that this is the first number of the triangle. The next expression is taken by increasing the power of the expression by 1. = 1 + x Note here that the coefficients of the expanded expression are 1 and 1. These are the two numbers of the 2nd row of the triangle. Now, the 3rd expression:: = 1 + 2x + The coefficients of the expanded expression give the 3rd row numbers of the triangle - 1, 2 and 1. The subsequent expression derived by increasing the power by 1 of the previous expression: = 1 + 3x + + The coefficients of this expanded expression are 1, 3, 3 and 1. These are the number of the 4th row. (Adapted from: Pascal's Triangle, Krysstal, 2006) Thus, 4 rows have been derived by this method: 1 1 1 1 2 1 1 3 3 1 The generalized version of the expression is: It is noted here that the expression , after expansion, generates coefficients that comprise the numbers of the row of the triangle. It must be noted that 'n' is a positive whole number and the expression is called a binomial expression. This is the simplest manner of generating the triangle and the Indians, Chinese and the Arabs all utilized it to do so. Using Combinatorics Consider the expression and its expansion: = The expression itself signifies the number of ways in which 'r' number of things can be selected from 'n' number of things. That is, from a larger set of 'n' things a smaller set 'r' is to be derived by number of ways. The term n! is pronounced as 'n Factorial' and is utilized to mean that all whole numbers from 1 to n must be multiplied: 1! = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6, and so on. It is to be noted that 0! = 1. Pascal's Formula is derived on this basis. Thus, binomial expressions can be expressed in the form of combinatorics in the following manner: = + x + + Using the factorial translation, = 1 + 3x + + (Pascal's Triangle, Krysstal, 2006) Which gives the desired coefficients required to make up the rows of the triangle. The generalized expression is: = + x + + . + + + The coefficients of the expansion give the numbers of the row of the triangle, as mentioned earlier. (Pascal's Triangle, Krysstal, 2006) In generality, the expression by any other general expression such that: where a and b may have any numerical coefficients as long as they are positive whole numbers. Functionality The primary utility of Pascal's triangle is in probability. It may be used to determine choice in the sense of combination theory. This second may be in the following sense using the given example of the triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Let it be considered that the number of units in each row represents the number of possible events in a given scenario. Thus, if the 5th row represents a set of 5 differently colored marbles and it is sought how many combinations are possible in the selection of 3 marbles it is enough to count up to the 3rd number in the 5th row to find the answer. It is 6. There can be 6 exclusive sets of differently colored marbles that can get selected when 3 marbles are randomly selected from the total set of 5. This is the utility of Pascal's triangle in the theory of combination and probability. (Pascal's Triangle, Krysstal, 2006) Also, the triangle is a very handy tool for teaching binomial expansion to beginners. Other Salient Points Some other salient properties of this unique triangle are bulleted below. The first number after the 1 in each divides all other numbers greater than 1 if it is a prime number. The sum of the numbers along the 'shallow diagonals' of the triangle give the 'Fibonacci numbers' (Mathworld, Pascal's Triangle, 2006). (Mathworld, Pascal's Triangle, 2006). Pascal's Triangle contains the 'figurate numbers' along its diagonals. (Mathworld, Pascal's Triangle, 2006). The sum of the elements of the row is given by (i, j) = . (Adapted from: Mathworld, Pascal's Triangle, 2006) Coloring all the even numbers white and the odd numbers black of the triangle gives the Sierpinski Sieve, a fractal product. (Sierpinski Sieve, Mathworld, 2006) Conclusion There are many other singularities that are associated with the triangle but these are presently considered beyond the scope of this paper. Many of these are available for study at Pascal's Triangle, Mathworld. These singularities do not have much practical use in the world but, nevertheless, to those who are fascinated by numbers Pascal's Triangle represents a sort of magic Pandora's box that contains endless wonders to be carefully and excitingly explored and enjoyed. References . Binomial Theorem and Pascal's Triangle, 2006. Extracted on 12th May, 2006, from: www.roma.unisa.edu.au/07305/pascal.htm Krysstal: Pascal's Triangle, 2006. Extracted on 12th May, 2006, from: http://www.krysstal.com/binomial.html Pascal, Blaise, (1623-1662), Scienceworld, 2006. Extracted on 12th May, 2006, from: http://scienceworld.wolfram.com/biography/Pascal.html Pascal's Formula, Mathworld, 2006. Extracted on 12th May, 2006, from: http://mathworld.wolfram.com/PascalsFormula.html Pascal's Triangle - Mathematics and the Liberal Arts, 2006. Extracted on 12th may, 2006, from: http://math.truman.edu/thammond/history/PascalTriangle.html Pascal's Triangle, Mathworld, 2006. Extracted on 12th May, 2006, from: http://mathworld.wolfram.com/PascalsTriangle.html Persian Language and Literature: Omar Khayyam, 2006. Extracted on 12th May, 2006, from: http://www.iranchamber.com/literature/khayyam/khayyam.php Pingala, Wikipedia Free Encyclopedia, 2006. Extracted on 12th May, 2006, from: http://en.wikipedia.org/wiki/Pingala Sierpinski Sieve, Mathworld, 2006. Extracted on 12th May, 2006, from: http://mathworld.wolfram.com/SierpinskiSieve.html Read More