Pascal's Triangle

7 Key excerpts on "Pascal's Triangle"

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  A Discrete Transition to Advanced Mathematics

  Second Edition

  • Bettina Richmond, Thomas Richmond(Authors)
  • 2023(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 9 Fibonacci Numbers and Pascal’s Triangle The mathematician’s patterns, like the painter’s or the poet’s must be beauti- ful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. —G. H. Hardy (1877–1947) 371 372 9. Fibonacci Numbers and Pascal’s Triangle The most distinct and beautiful statement of any truth must take at last the mathematical form. —Henry David Thoreau (1817–1862) 9.1. Pascal’s Triangle The array of numbers we call Pascal’s triangle was known long before Blaise Pascal’s 1654 treatise Traité du Triangle Arithmétique (Treatise on the Arithmetic Triangle). The Italian mathematician Niccola Tartaglia (1506–1559) included the triangle in his Gen- eral Trattato di Numeri et Misure (General Treatise on Numbers and Measures) pub- lished in 1556 and 1560. Peter Apian’s 1527 German text Rechnung (Calculating) fea- tured “Pascal’s” triangle on the title page. Even before Johannes Gutenberg’s invention of movable type circa 1450, “Pascal’s” triangle appeared on the cover of the Chinese text Ssu-yüan yü-chien (The Precious Mirror of the Four Elements) by Chu Shih-Chieh in 1303. The entries of Pascal’s triangle are significant as the binomial coefficients—that is, the coefficients in the expansion of ( + )  . We saw in Section 4.1 that ( + )  is the sum of the 2  terms of form  1  2 ⋯   where each   is either  or . The term  1  2 ⋯   will equal  −   if exactly  of the  factors  1 ,  2 , ... ,   are chosen to be . Thus, the coefficient of  −   in the expansion of ( + )  is (   ). The argument above puts the binomial coefficient (   ) in its more general interpre- tation as the number of ways to select a -element subset of an -element set. This combinatorial aspect of the binomial coefficients makes them widely applicable far be- yond the algebraic problem of expanding powers of a binomial.

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  A History of Probability and Statistics and Their Applications before 1750

  • Anders Hald(Author)
  • 2005(Publication Date)
  • Wiley-Interscience

    (Publisher)

  These numbers had been studied before Pascal, partly because they enumerate the number of (equidistant) points contained within certain figures (hence the name) and partly because of their simple additive structure. He arranges these numbers in a two-way table as shown below: Pascal's Table of Figurate Numbers ROOT ORDER I 2 3 4 5 1 1 1 I I 1 2 1 2 3 4 5 3 1 3 6 10 15 4 I 4 10 20 35 Any number, 10, say, at row 4 and column 3 is obtained as the sum of the numbers in the row above ending at the given column 10 = 1 + 3 + 6. This property is, however, exactly that proved as Proposition (2) above. The figurate numbers are therefore in a one-to-one correspondence to the numbers in the arithmetic triangle, which is also obvious by looking at the numbers in the diagonals, and the properties of the figurate numbers therefore follow from the results above. Pascal shows how to solve the equation t r n -l , c < t,,, for given n and c by a simple numerical procedure. He further discusses the properties of the product m(m + l ) . . . ( m + n - 1 ) and shows how the largest integral root of the equation m = c may be found. In Usage du Triangle Aritlimktique pour trouver les Puissances des Bindines et Apotomes, Pascal shows by example the connection between the binomial coefficients and the numbers in the arithmetic triangle, i.e., k (a + b)k = 5.2 PASCAL'S ARITHMETIC 52 THE FOUNDATION OF PROBABILITY THEORY BY PASCAL AND FERMAT In the Combinationes Pascal gives a rather trivial discussion of the properties of C; derived from the properties of tn-k.k. The most important part of the paper is the section on Sommation des Puissances Numdriques in which Pascal solves the problem of summation of powers of the terms of an arithmetic series by recursion. Pascal points out that this result may be used to find the area under a parabolic curve of any order. Pascal's proof is as follows. Let a,=a+id, i = O , 1 , ..., where u and d are positive integers, and set n -1

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  Makers of Mathematics

  • Stuart Hollingdale(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  r at a time, is written as

  and is equal to

  The denominator of this fraction is usually written as r! and called ‘r factorial’. Since

  n

  C

  r

  must be a positive whole number, it follows that the product of any r consecutive positive integers is divisible by r! – an important result in the theory of numbers.

  Pascal made a thorough study of the properties of his triangle, in the course of which he gave the first clear exposition of the principle of mathematical induction. This principle, which asserts the validity of reasoning by recurrence, is now regarded as a fundamental axiom of modern mathematics. It may be stated thus: ‘If a property is true of the number 1, and is also true of the number n + 1 provided it is true of n, then it will be true of all whole numbers.’ Here is a simple example of a proof by induction. If we assume (p. 16) that

  then

  Since S(1) = 12 , the general result is established. Many properties of the numbers in Pascal’s triangle can be conveniently proved by induction. Such proofs rely heavily on a theorem, sometimes known as Pascal’s recurrence law, which asserts that

  n

  C

  r

  +

  n

  C

  r-1

  =

  n+1

  Cr . This law (and other results in what is now called ‘combinatorial mathematics’) was skilfully exploited by the two correspondents. The build-up of their reasoning – to answer objections and to deal with ever more difficult problems, as the exposition passes from one man to the other and back again – makes compelling reading, but we must refer the interested reader to the letters themselves. (There are translations in Reference 8

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  Playing with Infinity

  • Rózsa Péter(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  It follows from this that the sum of the terms in each row is double the sum of the terms in the previous row. For example let us construct the row after the last one we wrote down. It is done in the following way:

  and we can see clearly that in this row every term of the row 1 6 15 20 15 6 1 occurs exactly twice.

  This throws further light on another property of the Pascal Triangle: adding the terms of a row we obtain the successive powers of 2. Since this is the case in the beginning (apart from the uppermost 1) : i.e. 1 + 1 = 2 = 21 , 1 + 2 + 1 = 4 = 22 , we need not trouble to look any farther; if this property is true for one row, then it will be ‘inherited’ by the next row. We know that the sum of the terms of each row is twice as much as the sum of the terms in the previous row, and if we multiply any power of 2 by 2, we shall get a product 2 x 2 x 2 x ... 2 x 2 with one more 2 in it, i.e. we shall get the next power of 2.

  This kind of proof, which is based entirely on the construction of the natural number series, is called mathematical induction. The natural number series begins with 1, and by continuing to count one more, we can reach any member of the series. The idea of mathematical induction is simply that if something is true at the beginning of the number series, and if this is ‘inherited’ as we proceed from one number to the next, then it is also true for all natural numbers. This has given us a method to prove something for all natural numbers, whereas to try out all such numbers is impossible with our finite brains. We need prove only two things, both conceivable by means of our finite brains: that the statement in question is true for 1, and that it is the kind that is ‘inherited’.

  This is a most important lesson, namely that the infinite in mathematics is conceivable by means of finite tools. Those who like to play about with multiplications will be familiar with the first few rows in the Pascal Triangle. If we construct the powers of 11 successively, we find that

  The figures in the results are the very numbers in the Pascal Triangle. Those who had a good look at the multiplications will know straight away why this is so; when we added the partial products, we carried out just the same additions as in the construction of the rows in the Pascal Triangle (in the case of 115

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  Discrete Mathematics For Teachers

  • Ed Wheeler, Jim Brawner(Authors)
  • 2013(Publication Date)
  • Information Age Publishing

    (Publisher)

  T RIANGULAR N UMBERS IN P ASCAL ’ S T RIANGLE We conclude this section by examining Figure 2, which shows some triangular arrangements of dots and some connections with Pascal’s Triangle. Figure 2: The Triangular Numbers 1, 3, 6, 10, … Notice that each triangular arrangement has one more row of dots than the arrangement to its left, and each row of dots in any one triangle has one more dot than the row above it. Besides the obvious similarity in shape between the arrangements and Pascal’s Triangle, let us focus on another connection. If we count the number of dots in each arrangement, we see that there are 1, 3, 6, and 10 dots, respectively, in the triangular arrangements. Consequently, the numbers in the sequence 1, 3, 6, 10, … are called the triangular numbers . From looking at the individual rows in each arrangement, we can see that the n th triangular number is the sum of the first n natural numbers. For example, the third triangular number 6 = 1 + 2 + 3, and the fourth triangular number 10 = 1 + 2 + 3 + 4. Example 9 (a) Find a recurrence relation for the sequence of triangular numbers. (b) Find the fifth, sixth, and seventh triangular numbers. Solution (a) To get the next triangular number, we add the next counting number to the previous triangular number. In other words, if n t denotes the n th triangular number, then 1 n n t t n − = + , for 2 n ≥ . (b) With the notation from part (a), 5 4 5 10 5 15 t t = + = + = , 6 5 6 15 6 21 t t = + = + = , and 7 6 7 21 7 28 t t = + = + = . 2 9 2 C H A P T E R 6 : C O M B I N A T O R I C S If we take another look at Pascal’s Triangle, we find that the triangular numbers appear in a diagonal column heading down and to the left, beginning in Row 2, as highlighted in Figure 3. Of course, as a consequence of the symmetry of Pascal’s Triangle, the triangular numbers also appear in a diagonal column going down and to the right.

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  Principles and Techniques in Combinatorics

  • Chen Chuan-Chong, Koh Khee-Meng(Authors)
  • 1992(Publication Date)
  • WSPC

    (Publisher)

  Pascal triangle, after the renown French mathematician Blaise Pascal (1623-1662) who discovered it and made significant contributions to the understanding of it in 1653. The triangle is also called Yang Hui’s triangle in China as it was discovered much earlier by the Chinese mathematician Yang Hui in 1261. The same triangle was also included in the book “Precious Mirror of the Four Elements” by another Chinese mathematician Chu Shih-Chieh in 1303. For the history of the Pascal’s triangle, the reader may refer to the book [E].

  Figure 2.4.1.

  Figure 2.4.2.

  We now make some simple observations with reference to Figure 2.4.2 .

  (1)  The binomial coefficient , located at the nth level from the top and rth position from the left, is the number of shortest routes from the top vertex representing to the vertex representing . This is identical to what we observed in Example 1.5.1 .

  (2)  As = , the entries of the triangle are symmetric with respect to the vertical line passing through the vertex .

  (3)  Identity (2.3.1) says that the sum of the binomial coefficients at the nth level is equal to 2

  n

  , and identity (2.3.6) says that the sum of the squares of the binomial coefficients at the nth level is equal to .

  (4)  Identity (2.1.4), namely , simply says that each binomial coefficient in the interior of the triangle is equal to the sum of the two binomial coefficients on its immediate left and right “shoulders”. For instance, 21 = 15 + 6 as shown in Figure 2.4.2 .

  2.5.  Chu Shih-Chieh’s Identity

  We proceed with another observation in Figure 2.4.2 . Consider the 5 consecutive binomial coefficients:

  along the NE line when r = 2 from the right side of the triangle as shown. The sum of these 5 number is 1 + 3 + 6 + 10 + 15 = 35, which is the immediate number we reach after turning left from the route 1-3-6-10-15. Why is this so? Replacing by

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  Mathematics and the Divine

  A Historical Study

  • Teun Koetsier, Luc Bergmans(Authors)
  • 2004(Publication Date)
  • Elsevier Science

    (Publisher)

  CHAPTER 21 Pascal’s Views on Mathematics and the Divine

  Donald Adamson Dodmore House, The Street, Meopham, Kent, DA 13 0AJ, United Kingdom

  Contents

  1. Introduction

  2. Characteristics of God

  3. God’s Salvation-Plan

  4. Heart and reason

  5. The Wager argument (418)

  6. Mathematical aspects of presentation

  7. Conclusion

  References

  1. Introduction

  Pascal’s mathematical achievement is fourfold. Lacking Descartes’s algebraic expertise, he chose to work in the traditional field of synthetic rather than analytic geometry, and thus contributed to the study of conic sections. Six years later, in The Treatise on the Arithmetical Triangle and in its first appendix Multiple Numbers , he published his findings on the theory of number (prime numbers and magic squares), propounding the method of combinatorial analysis known as ‘Pascal’s Triangle’, and applying properties of the binomial theorem. In collaboration with Fermat, and at the prompting of his gambling acquaintance the Chevalier de Méré, he laid the foundations of modern probability theory. And towards the end of his life, in his History of the Cycloid , he resolved various problems in the geometry of indivisibles, thereby helping to create the infinitesimal calculus: this, though falling short of the generalized formulation which made Newton’s integral calculus possible, nevertheless established the geometric laws applicable to a curve.

  Such intermittent but highly concentrated scientific activity confirms him as one of the seventeenth century’s greater mathematicians, yet in a famous letter he disparages such intellectual labours and pours scorn upon his arithmetical and geometric achievement. ‘I would not go two steps out of my way for geometry’s sake’, he writes, adding that so different is the work on which he is currently engaged that he has almost forgotten about mathematics; ‘for, to be frank with you about geometry, it is, in my opinion, the highest of mental exercises; but I also see that it is so useless that I draw hardly any distinction between a mere geometrician and a skilled workman’; ‘whilst geometry was a good means of testing one’s mental powers, it was not a good means of employing them’ (522). 1

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