Lịch sử hình thành công thức Nhị Thức Newton

KHOA A. VO – HOAI T. NGUYEN NEWTON’S BINOMIAL FORMULA Khoa Anh Vo - Hoai Thanh Nguyen February 3, 2012 Khoa Anh Vo - Hoai Thanh Nguyen Vietnam National University Ho Chi Minh City (HCMC) University of Science Faculty of Mathematics and Computer Science 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City Vietnam 2 PREFACE This book is intended as our first English thematic for students who study in high school or people who want to research into the history of mathematics. In detail, this talks about the journey of John Wallis (1616 - 1703) from the Alhazen’s formulas (965 - 1040), and the continuation of Issac Newton’s idea (1643 - 1727). Then we give some mathematical problems in the educational programs. Therefore, we desire to provide more knownledges for the positive vision that pure mathemtics bring it. This book is also a gift which we award to our forum MathScope.Org on New Year 2012 - the Year of Dragon. So we and collaborators send all nice greetings to the readers. Acknowledgement. We (i.e. Khoa Anh Vo - Hoai Thanh Nguyen) thank the collaborators for all their helps. These include : Name High School/ University Thien Huu Vo Truong HCMC University of Science Truong Nhat Thanh Mai HCMC University of Science Quang Dang Nguyen HCMC University of Science Minh Nhat Vu To HCMC International University Phong Tran HCMC University of Pedagogy Tuan Thanh Nguyen HCMC University of Economics and Law Trang Hien Nguyen Phan Boi Chau High School for The Gifted Huyen Thanh Thi Nguyen Luong The Vinh High School for The Gifted Especially, that is the approval of Dr. David Dennis (4249 Cedar Drive, San Bernardino, USA) for our translation of his documents. Furthermore, this makes “Newton’s Binomial Formula” strange - looking. The readers can find and download “Newton’s Binomial Formula” at : http://www.forum.mathscope.org/ or http://anhkhoavo1210.wordpress.com/ 3 Contents PREFACE 3 WHAT IS THE NEWTON’S BINOMIAL FORMULA? 5 1 THE INTRODUCTION 6 1.1 A FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 THREE PROOFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 INDUCTION PROOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 COMBINATORIAL PROOF . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 DERIVATION USING CALCULUS . . . . . . . . . . . . . . . . . . . . 10 2 THE SENSITIVITY 12 2.1 JOHN WALLIS (1616 - 1703) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 ISAAC NEWTON (1643 - 1727) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 A JOURNEY OF JOHN WALLIS . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 THE CONTINUATION OF ISSAC NEWTON’S IDEA . . . . . . . . . . . . . . 25 4 WHAT IS THE NEWTON’S BINOMIAL FORMULA? 5 1 THE INTRODUCTION 1.1 A FORMULA As a result, Newton’s Binomial Formula was proved by two scientists : Isaac Newton (1643 - 1727) and James Gregory (1638-1675). This is really a formula which uses for expansion of a binomial n power(s) that is become a polynomial n + 1 terms. (x + a) n = n  k=0 C k n a n−k x k In order to make sense of the theorem we need to agree on some conventions. First, we define the binomial coefficients C k n =  n k  = n! (n −k)!k! using the convention that 0! = 1 to cover the cases where either n, n −k or k is 0. We will also stipulate that x 0 = 1 and a 0 = 1. These are questionable if x = 0 or a = 0, so those should be dealt with as separate cases. Interpretation of the formula in those cases gives either a n = a n or x n = x n . If all of n = 0, x = 0, and a = 0 then we get the result 0 0 = 0 0 , which is not particularly meaningful, but as long as we agree on what we mean by 0 0 we are forced to accept the result. In the generality case, a formula said that : Let r be a real number and z be a complex number with magnitude modulus of z less than 1, we have (1 + z) r = ∞  k=0  r k  z k Remark. A general formula for m (a i )’s term(s)  m  i=1 a i  n =  n! n 1 !n 2 ! n m ! a n 1 1 a n 2 2 a n m m where n 1 + n 2 + + n m = n. 6 1 THE INTRODUCTION 1.2 THREE PROOFS The binomial formula can be thought of as a solution for the problem of finding an ex- pression for (x + a) n from one for (x + a) n−1 or as a way to find the coefficients of (x + a) n directly. In this section, we have three mathematical proofs which are taken from a small topic Aesthetic Analysis of Proofs of the Binomial Theorem of Lawrence Neff Stout, Department of Mathematics and Computer Science, Illinois Wesleyan University. 1.2.1 INDUCTION PROOF Many textbooks in algebra give the binomial formula as an exercise in the use of mathe- matical induction. The key calculation is in the following lemma, which forms the basis for Pascal’s triangle. According to Pascal’s triangle, we can order the binomial coefficients corresponding to n power(s). n = 0 1 n = 1 1 1 n = 2 1 2 1 n = 3 1 3 3 1 n = 4 1 4 6 4 1 n = 5 1 5 10 10 5 1 It’s easy to observe that the pattern (4 + 6 = 10) is exactly a case of Pascal’s lemma. C k m + C k−1 m = C k m+1 or  m k  +  m k − 1  =  m + 1 k  Of course, this lemma can be prove clearly. And the readers can prove it themself. Lemma. For all 1 ≤ k ≤ m. Prove that  m k  +  m k − 1  =  m + 1 k  Proof. This is a direct calculation in which we add fractions and simplify. 7 1 THE INTRODUCTION  m k  +  m k − 1  = m! (m −k)!k! + m! (m −k + 1)!(k − 1)! = m!(m −k + 1)!(k − 1)! + m!(m −k)!k! (m −k)!k!(m − k + 1)!(k −1)! = m!(k − 1)!(m − k)! [k + (m −k + 1)] (m −k)!k!(m − k + 1)!(k −1)! = m! [k + (m − k + 1)] k!(m − k + 1)! = m!(m + 1) k!(m − k + 1)! = (m + 1)! k!(m − k + 1)! =  m + 1 k  We proceed by mathematical induction. Proof. For the case n = 0, the formula says (x + a) 0 =  0 0  x 0 a 0 = 1 Now (x + a) 0 = 1 and 0  k=0  0 k  a 0−k x k =  0 0  a 0 x 0 = 1 Here we are using the conventions that  0 0  = 1 and that any number to the 0 power is 1. Given the artificiality of these assumptions, we may be happier if the base case for n = 1 is also given. For the case n = 1 the formula says (x + a) 1 = 1  k=0  1 k  a 1−k x k =  1 0  a 1 x 0 +  1 1  a 0 x 1 This is equivalent to x + a = 1! 1!0! a + 1! 0!1! x = a + x which is true. Thus we have the base cases for our induction. For the induction step we assume that 8 1 THE INTRODUCTION (x + a) m = m  k=0  m k  a m−k x k and show that the formula is true when n = m + 1. (x + a) m+1 = (x + a) m (x + a) =  m  k=0  m k  a m−k x k  (x + a) = m  k=0  m k  a m−k x k+1 + m  k=0  m k  a m−k+1 x k =  m 0  a m+1 x 0 + m  k=1  m k  +  m k − 1  a m−k+1 x k +  m + 1 m + 1  a 0 x m+1 Completing the proof by induction. 1.2.2 COMBINATORIAL PROOF The combinatorial proof of the binomial formula originates in Jacob Bernoulli’s Ars Con- jectandi published posthumously in 1713. It appears in many discrete mathematics texts. Proof. We start by giving meaning to the binomial coefficient  n k  = n! (n −k)!k! as counting the number of unordered k−subsets of an n element set. This is done by first counting the ordered k−element strings with no repetitions : for the first element we have n choices; for the second, n − 1; until we get to the k th which has n − k + 1 choices. Since these choices are made in succession, we multiply to get n(n −1) (n −k + 1) = n! (n −k)! such ordered k−tuples without repetition. Next we observe that the process of multi- plying out (x + a) n involves adding up 2 n terms each obtained by making a choice for each factor too use either the x or the a. The choices which result in k x’s and n −k a’s each give a term of the form a n−k x k . There are  n k  distinct ways to choose the k element subset of factors from which to take the x. Thus the coefficient of a n−k x k is  n k  . This tells us that (x + a) n = n  k=0 C k n x n−k a k 9 [...]... Wallis led the young Isaac Newton to his first profound mathematical creation; the expansion of functions in binomial series Thus Wallis’ method of interpolation became for Newton the basis of his notion of continuity So Newton wanted to generalize the methods of Wallis 24 2 THE SENSITIVITY 2.4 THE CONTINUATION OF ISSAC NEWTON S IDEA In 1661, the nineteen-year-old Isaac Newton read the Arithmetica... wealthy rector from nearby North Witham Much has been made of Newton s posthumous birth, his prolonged separation from his mother, and his unrivaled hatred of his stepfather Until Hanna returned to Woolsthorpe in 1653 after the death of her second husband, Newton was denied his mother’s attention, a possible clue to his complex character Newton s childhood was anything but happy, and throughout his... SENSITIVITY 2.2 ISAAC NEWTON (1643 - 1727) “If you ask a good skating how to be successful, he will say to you that fall, get up is a success.” Isaac Newton was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian He was born in 1643, Lincolnshire, England The fatherless infant was small enough at birth When he was barely three years old Newton s mother, Hanna,... interpolation and extrapolation It was here that Newton first developed his binomial expansions for negative and fractional exponents Newton made a series of extensions of the ideas in Wallis He extended the tables of areas to the left to include negative powers and found new patterns upon which to base interpolations Perhaps his significant deviation from Wallis was that Newton abandoned the use of ratios of areas... of Newton s table The binomial pattern of formation is now such that each entry is the sum of the entry to the left of it and the one above that one So that we can find that the ? must be equal to −1 Hence Newton filled in the table of coefficients for the area expressions under the curves (1 + x)p (see Table 5) 27 2 THE SENSITIVITY Table 5 1 what we 1+x now call the natural logarithm of 1 + x, and Newton. .. letters to Oldenburg in which Newton explained his binomial series at the request of Liebniz in 1676 Hence, using the methods of Newton, we can represent the binomial expansion for all real numbers; this is very important in our life In detail, this helps us investigate the graph or the approximately values of more functions, so this is also a basis of all after series Newton is such a great mathematician... appear in the graph, Newton went on to write down more area expressions for curves in this family He obtainedd the following area expressions by first expanding and then finding the area term by term • Third power : x + 3x2 3x3 x4 + + 2 3 4 • Fourth power : x + • Fifth power : x + 4x2 6x3 4x4 x5 + + + 2 3 4 5 5x2 10x3 10x4 5x5 x6 + + + + 2 3 4 5 6 26 2 THE SENSITIVITY At this point, Newton wanted to find... he verged on emotional collapse, occasionally falling into violent and vindictive attacks against friend and foe alike In 1665 Newton took his bachelor’s degree at Cambridge without honors or distinction Since the university was closed for the next two years because of plague, Newton returned to Woolsthorpe in midyear For in those days I was in my prime of age for invention, and minded mathematics and... 5, we can evaluate the area under the hyperbola y = SABED = x − x2 x3 x4 x5 x6 x7 + − + − + = 2 3 4 5 6 7 ∞ (−1)n n=0 xn+1 , −1 < x < 1 n+1 Next, Newton returned to the table of characteristic ratios made by Wallis (see Table 3) As discussed previously, Newton abandoned Wallis’ use of area ratios and set out to make a table of coefficients for a sequence of explicit expressions for calculating areas... the one to the left and the one above that Newton saw that old method is not appropriate for the integers In specific, the entries in the top row must all be 1 in all the interpolated tables (i.e a = 1) and the increment fo the second row must be 1 So this is unreasonable because the distance of coefficients of p = 0 and p = −1 are 1 Table 7 This forced that Newton must found other methods which filled . 14 2.4 THE CONTINUATION OF ISSAC NEWTON S IDEA . . . . . . . . . . . . . . 25 4 WHAT IS THE NEWTON S BINOMIAL FORMULA? 5 1 THE INTRODUCTION 1.1 A FORMULA As a result, Newton s Binomial Formula was. translation of his documents. Furthermore, this makes Newton s Binomial Formula” strange - looking. The readers can find and download Newton s Binomial Formula” at : http://www.forum.mathscope.org/ or http://anhkhoavo1210.wordpress.com/ 3 Contents PREFACE. 12 2 THE SENSITIVITY 2.2 ISAAC NEWTON (1643 - 1727) “If you ask a good skating how to be successful, he will say to you that fall, get up is a success.” Isaac Newton was an English physicist,