Lịch sử hình thành công thức Nhị Thức Newton

NEWTON’S BINOMIAL FORMULAKhoa Anh Vo - Hoai Thanh Nguyen February 3, 2012... Khoa Anh Vo - Hoai Thanh NguyenVietnam National University Ho Chi Minh City HCMC University of Science Facult

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 CityVietnam

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 aboutthe journey of John Wallis (1616 - 1703) from the Alhazen’s formulas (965 - 1040), andthe continuation of Issac Newton’s idea (1643 - 1727) Then we give some mathematicalproblems in the educational programs Therefore, we desire to provide more knownledgesfor 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 collaboratorsfor all their helps These include :

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 PedagogyTuan 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 BinomialFormula” strange - looking

The readers can find and download “Newton’s Binomial Formula” at :

http://www.forum.mathscope.org/

or

http://anhkhoavo1210.wordpress.com/

WHAT IS THE NEWTON’S BINOMIAL FORMULA? 5

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

WHAT IS THE NEWTON’S BINOMIAL

FORMULA?

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

using the convention that 0! = 1 to cover the cases where either n, n − k or k is 0

We will also stipulate that x0 = 1 and a0 = 1 These are questionable if x = 0 or a = 0, sothose should be dealt with as separate cases Interpretation of the formula in those casesgives either an = an or xn = xn If all of n = 0, x = 0, and a = 0 then we get the result

00 = 00, which is not particularly meaningful, but as long as we agree on what we mean

by 00we are forced to accept the result

In the generality case, a formula said that : Let r be a real number and z be a complexnumber with magnitude modulus of z less than 1, we have

1 THE INTRODUCTION1.2 THREE PROOFS

The binomial formula can be thought of as a solution for the problem of finding an pression for (x + a)nfrom one for (x + a)n−1or as a way to find the coefficients of (x + a)n

ex-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 matical induction The key calculation is in the following lemma, which forms the basisfor Pascal’s triangle

mathe-According to Pascal’s triangle, we can order the binomial coefficients corresponding to npower(s)

Cmk + Cmk−1= Cm+1kor

mk

Of course, this lemma can be prove clearly And the readers can prove it themself

Lemma For all 1 ≤ k ≤ m Prove that

mk

1 THE INTRODUCTION

mk

We proceed by mathematical induction

Proof For the case n = 0, the formula says

(x + a)0 = 0

0

!

x0a0 = 1Now (x + a)0 = 1 and

0

X

k=0

0k

00

!

= 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!0!a +

1!

0!1!x = a + xwhich is true Thus we have the base cases for our induction For the induction step weassume that

am−kxkand show that the formula is true when n = m + 1

!

am−kxk

#(x + a)

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

nk

n(n − 1) (n − k + 1) = n!

(n − k)!

such ordered k−tuples without repetition Next we observe that the process of plying out (x + a)ninvolves adding up 2nterms each obtained by making a choice for eachfactor too use either the x or the a The choices which result in k x’s and n − k a’s each give

multi-a term of the form multi-an−kxk There are n

k

!distinct ways to choose the k element subset

of factors from which to take the x Thus the coefficient of an−kxk is n

k

! This tells usthat

1 THE INTRODUCTION

1.2.3 DERIVATION USING CALCULUS

Newton’s generalization of the binomial formula gives rise to an infinite series If werestrict to natural number exponents, the convergence considerations are not necessaryand a proof based on the differentiation of polynomials becomes possible

Proof We first note that since (x + a) is a polynomial of degree 1, (x + a)nwill be a nomial of degree n and will thus be determined once we know what the coefficients of each

poly-of the n + 1 possible powers poly-of x are For concreteness let us write

and show how to determine the coefficients bk

Using the power rule and the chain rule for differentiation, we have

p(0) = (0 + a)n= anThen we determine what the coefficients bk must be to satisfy this equation The initialcondition p(0) = an tells us that b0 = an We can relate later coefficients to earlier onesusing the differentiatl equation :

nbn = nbn

Thus for k = 1, n − 1, we get

bk+1= n − k

(k + 1)abkMoreover, using the face that b0 = anthis gives us

it but his study was erratic In 1632, after decision to be a doctor, Wallis was sent in

1632 to Emmanuel College, Cambridge While there, he kept an act on the doctrine of thecirculation of the blood; that was said to have been the first occasion in Europe on whichthis theory was publicly maintained in a disputation He received a Master’s degree in

1640, afterwards entering the priesthood Wallis was elected to a fellowship at Queens’College, Cambridge in 1644, which he however had to resign following his marriage.Wallis made significant contributions to trigonometry, calculus, geometry, and the anal-

ysis of infinite series Especially, Arithfumetica Infinitorum was the most important of

his works In this book, the analytic methods of Descartes and Cavalien was extended In

addition, he also published Algebra, Opera

2 THE SENSITIVITY2.2 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 pher, alchemist, and theologian

philoso-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, placed her firstborn with his grandmother in order to remarry and raise a second family with BarnabasSmith, a wealthy rector from nearby North Witham Much has been made of Newton’sposthumous 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 hersecond husband, Newton was denied his mother’s attention, a possible clue to his complexcharacter Newton’s childhood was anything but happy, and throughout his life he verged

on emotional collapse, occasionally falling into violent and vindictive attacks against friendand 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, andminded mathematics and philosophy more than at any time since Especially in 1666, heobserved the fall of an apple in his garden at Woolsthorpe, later recalling, ’In the same year

I began to think of gravity extending to the orb of the Moon’ In mathematics, Newton laterbecame involved in a dispute with Leibniz over priority in the development of infinitesimalcalculus Most modern historians believe that Newton and Leibniz developed infinitesimalcalculus independently, although with very different notations Moreover, he found thegenerality formula of binomial and give the definition of light theory

He published Philosophiae Naturalis Principia Mathematica in 1687 which was

the important book all over the world In addition, he wrote Opticks

2 THE SENSITIVITY2.3 A JOURNEY OF JOHN WALLIS

Beginning of Alhazen’s Summation Formulas, Ahazen (965 - 1040) - the Iraqi cian who stated some formulas which affected the later results of Wallis Ahazen derivedhis formulas by laying out a sequence of rectangles whose areas represent the terms of thesum

mathemati-• Look at a rectangle whose length is n + 1 and width is n, we divide this rectangle intoseveral rectangles (see Figure 1) Thus its area must equals to

i and width is n + 1, we also divide this

rectangle into several rectangles (see Figure 2) Apply the above formula (i.e

n

X

i=1

i =1

X

i=1

i2+12

• Using this similar method, we can find out the sum of the cubes or the fourth powers

or more if we want Thus we continue to view a rectangle whose length is

n

X

i=1

i2 andwidth is n + 1, we also divide this rectangle into several rectangles (see Figure 3).Apply the above formulas (i.e

X

i=1

i3+ 12

The Geometry, first published in 1638, of René Descartes was the first published treatise

to use positive integer exponents written as superscripts He saw exponents as an index forrepeated multiplication That is to say he wrote x3 in place of xxx Wallis adopted this use

of an index and tried to extend it, and tested its validity across multiple representations.Wallis took from Fermat the idea of using an equation to generate a curve, which was incontrast to Descartes’ work which always began with a geometrical construction Descartesalways constructed a curve geometrically first, and then analyzed it by finding its equa-tion Wallis mixed these ideas, so he defined what is the fractional exponents and provedits existence successful

And the Arithmetica Infinitorum contains a detailed investigation of the behavior

of sequences and ratios of sequences from which a variety of geometric results are thenconcluded We shall look at one of the most important examples Consider the ratio of thesum of a sequence of a fixed power to a series of constant terms all equal to the highestvalue appearing in the sum Wallis researched into ratios of the form :

A = 0

k+ 1k+ 2k+ + nk

nk+ nk+ nk+ + nk

2 THE SENSITIVITYFor each fixed integer value of k, Wallis investigated the behavior of these ratios as nincreases When k = 1, he calculates :

0 + 1 + 2

2 + 2 + 2 =

12

0 + 1 + 2 + 3

3 + 3 + 3 + 3 =

12

0 + 1 + 2 + 3 + 4

4 + 4 + 4 + 4 + 4 =

12 =

This can be seen from the well known Alhazen’s fomulas We have

n (n + 1) =

12

Then Wallis called 1

2 the characteristic ratio of the index k = 1.

When k = 2, Wallis continued to compute the following ratios :

02+ 12

12+ 12 = 1

3 +

16

02+ 12+ 22

22+ 22+ 22 = 1

3 +

112

02+ 12+ 22+ 32

32+ 32+ 32+ 32 = 1

3 +

118

02+ 12+ 22+ 32+ 42

42+ 42+ 42+ 42+ 42 = 1

3 +

124

02+ 12+ 22+ 32+ 42+ 52

52+ 52+ 52+ 52+ 52+ 52 = 1

3 +

130 =

Wallis claimed that the right hand side is always equals 1

3+

16n and this can be checked

As n increases this ratio approaches 1

3 (we can see limn→∞

 1

3 +

16n



= 1

3 now), so Wallisthen defined the characteristic ratio of the index k = 2 as equal to 1

3 In a similar way,Wallos computed the characteristic ratio of k = 3 as 1

4, and k = 4 as

1

5 and so forth.Thus he made the general claim that the characteristic ratio of the index k is 1

k + 1 for allpositive integers

2 THE SENSITIVITYNext, Wallis show that these characteristic ratios yielded most of the familiar ratios

of area and volume known from geometry It means he showed that his arithmetic wasconsistent with the accepted truths of geometry

He assumed that an area is a sum of an infinite number of parallel line segments, andthat a volume is a sum of an infinite number of parallel areas, his basis assumptions were

taken from Cavalieri’s Geometria Indivisibilibus Continuorum which was published

in 1635 Wallis first considered the area under the curve y = xk(see Figure 4) He wanted

to compute the ratio of the shaded area to the area of the rectangle which encloses it

Figure 4Wallis claimed that this geometric problem is an example of the characteristic ratio ofthe sequence with index k In specific, the terms in the numerator are the lengths of theline segments that make up the shaded area while the terms in the denominator are thelengths of the line segments that make up the rectangle (hence constant) He imaginedthe increment or scale as very small while the number of the terms is very large

And then this characteristic ratio of 1

3 holds for all parabolas, not just y = x

2 In detail,

we can use Riemann’s integral for proving that results

Consider a set [0; 1] and a curve y = 5x2, we have an area under this curve (S1) which iscalculated :

= 53

An area of a rectangle which encloses this curve (S2) equals 5, so that we have

S1= 1

3S2This example means that characteristic ratio depends only on the exponent and not onthe coefficient, or we can say that characteristic ratio is not linear

2 THE SENSITIVITY

Figure 5

In addition, that above ratio also shows that the volume of a pyramid is 1

3 of the box thatsurrounds it (see Figure 5) Hence Wallis saw this as another example of his computation

of the characteristic ratio for the index k = 2

These geometric results were not knew Fermat, Roberval, Cavalieri and Pascal had allpreviously made this claim that when k is a positive integer; the area under the curve

2+ 1 The same can be seen for

y =√3

x, whose characteristic ratio must be 3

4 =

11

3 + 1

It was this coordination of two separate representations that gave Wallis the confidence

to claim that the appropriate index of y =√q

xp must be p

q, and that its characteristic ratiomust be p 1

q + 1

Wallis continued to assert that this claim remained true even when the

index is irrational because he gave √3 as an example But in many cases, Wallis had noway to directly verify the characteristic ratio of an index, for example : y = √3 x2

How can we determine the characteristic ratio of the circle? This is the question thatmotivated Wallis to study a particular family of curves from which he could interpolatethe value for circle He wrote the equation of the circle of radius r, as y = √r2− x2, andconsidered it in the first quadrant He wanted to determine the ratio of its area to the r bysquare that contains it

Of course Wallis knew that this ratio is π

4, from various geometric constructions goingback to Archimedes, but he wanted to test his theory of index, characteristic ratio andinterpolation by arriving at this result in a new way Therefore, he considered the family