Transcript m370-notes-newton1

ISAAC NEWTON
1642 -1727
Isaac Newton is possibly the greatest human intellect who has ever lived. Newton's discoveries
in the areas of science and mathematics have been remarkable, prolific and groundbreaking.
Newton's theories are still widely studied today by both mathematics and physics students.
From the mid sixteenth century to the mid to late seventeenth century there were many
significant developments that took place in both the academic and social climate of Europe that
set the stage for Issac Newton. Newton himself once said, "If I have been able to see further, it
was only because I stood on the shoulders of giants.",[1] recognizing the advances he made
could have only come about because of the work of others.
Changes in the status of the church have influenced the papacy to act in a hostile manner toward
academicians. Such examples include the imprisonment of Galileo and Cardano. This causes
the academic development to move away from Italy into northern parts of Europe, in particular
Britain, France and Germany.
Several people during this time had made some key discoveries and innovations in the scientific
and mathematical realm. Before he was imprisoned Galileo began to study the motion of "falling
bodies" and was able to conclude how the rate an object falls is not dependent upon its mass. A
Frenchman by the name of Francois Viete makes huge strides in the development of algebra, in
particular the concept of using words or symbols to stand for numbers or quantities. Rene
Descartes (1596-1650) develops the Cartesian coordinate system among many other ideas. This
associates geometry and algebra so that mathematicians could look at curves from an algebraic
point of view creating an area of study known as analytical geometry. Descartes has a very
intelligent young student by the name of Blaise Pascal (1623-1662) who is primarily interested in
studying theology, but works on mathematics as a "distraction" and
makes some contributions in the area of probability. A French lawyer by the name of Pierre
Fermat formulates a famous "theorem" concerning integer solutions to the equation xn+yn=zn in
which he posed had no positive integer solutions other than the case when n=2. [2]
Issac Newton was born on Christmas day in 1642. He was premature and frail in the early part
of his life. His father had died in October of that year leaving Isaac fatherless. When Isaac was
three his mother remarried a man named Barnabus Smith who wanted nothing to do with Isaac.
Smith forced Isaac to live with his grandmother and have no interaction with his mother, even
though he lived within sight of her. This probably made a significant psychological impact on
Isaac since he has been described as neurotic and a "loner", that he did not have friends. Later
in his life he becomes obsessed with writing down and maintaining long lists of "sins" he
committed such as having indecent thoughts, words or actions. As a boy he enjoys building
working modals of things.[2]
In 1661 Newton goes off to Trinity College (Cambridge) and finds the place in a total mess. This
is the result of several years of political fighting between the Puritans led by Cromwell (who
supported Oxford) and the Royalists (who supported Cambridge). The reformation of the
monarchy has placed Charles II on the throne one year before Newton arrives at Trinity. The
professors who are there have been appointed most for political reasons or religious views. The
climate is not intellectually driven, many professors are rarely on campus spending much of their
time drinking. Few give lectures or work with students.[2]
Newton first begins to study Latin and Aristolelian philosophy. He becomes discouraged and
focuses on problems of a physical nature. The library is one area of the college that has been
better attended to and Newton begins to read the work of Descartes Geometry. His intellectual
studies flourish, he begins to study mathematics, optics, motion, heat and many other topics in
physics. In 1664 Isaac Barrow who is the Lucasian Professor of Mathematics, has recognized
the talent of Isaac and promoted him to the status of scholar, which gave him funds for four
years as he worked toward his master's (PhD) degree.
In the following years came Newton's "period of greatness" in which some of his most famous
results coalesced. In 1665 he develops "Method of fluxions" which is what is known today as
differential calculus and in 1666 he does "Inverse method of fluxions" or known today as
integral calculus.[2]
In 1669 Newton is appointed to Lucasian Professor of Mathematics despite not publishing a
thing because Barrow is so impressed by his actions. In 1689 at the urging of Edmund Halley,
Newton publishes his Principa Mathematica. In 1689 he is appointed a member of parliament.
Between 1692 and 1694 Newton undergoes a period of "derangement" that was probably
associated with ingesting chemicals he was experimenting with. In 1705 Newton is knighted by
Queen Anne. He dies in 1727 and is buried in Westminster Abby.
We now turn our attention to two of Newton's mathematical accomplishments that were a
significant development in the area of solving equations. Methods of solving equations
performed by Cardano and Ferrari all relied on the method of extracting a root. This means the
solution would be expressed as a combination of powers and roots. Newton thought of the
idea of "closing in" on a root to "estimate" its value.
[1] website: http://turnbull.mcs.st-and.ac.uk/~history
[2] Dunham William; Journey Through Genius The Great Theorems of Mathematics; Wiley
(1990).
Newton was able to develop a new way to "solve" equations which today we call
Newton's Method. This was not really a way to solve them in as much as it was a
way to approximate (generate better estimates) solutions to an equation.
When Newton implemented his method he found he needed to multiply out binomial
expressions to very large powers. In order to do this he developed a procedure or
"formula" for this that today we call the Binomial Formula.
The Binomial Formula
The most common version of the Binomial Formula deals with how to expand
expressions of the form (x + y)n. Where x and y are any numbers and n is an integer.
this is sometimes written in the following way.
n
n
n
n
 n  n 1
 xy  y n
    x n  k y k  x n    x n 1 y    x n  2 y 2    x n 3 y 3    
k 0  k 
1 
2
3 
 n  1
n
n
x  y    k!nn! k ! x nk y k  x n  n x n1 y  nn  1 x n2 y 2  nn  1n  2 x n3 y 3    y n
1
2 1
3  2 1
k 0
x  y 
n
n
The symbol
n
n!
  
 k  k!n  k !
is the combinatorial choose. It is sometimes written nCk.
Newton was even able to extend this formula to include fractional and negative
values for n.
Pascal's Triangle
Pascal's Triangle organizes the information in the combination formula into a
triangle where each is a set of a particular size n and each column are the subsets
of a certain size r. Here is the table below.
r
Notice the entry in the 4th row and 2nd column
0
1
2
3
4
5
6
n
is 6. Remember 4C2 = 6.
0
1
1
1
1
2
1
2
1
3
1
3
3
1
4
1
4
6
4
1
5
1
5
10
10
5
1
6
1
6
15
20
15
6
This table is built from the famous "upside
down L" pattern that the entries above and to
the right and directly above give the entry
below.
5 + 10 = 15
1
3+1=4
Each row can be gotten from the previous one.
This table is very useful in figuring out what (x + y)n is multiplied out.
(x + y)2 = x2 + 2xy + y2
(Look in row 2)
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(Look in row 3)
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(Look in row 4)
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy3 + y5
(Look in row 5)
Before we explore what Newton actually did in terms of solving equations lets look at
the modern view of Newton's method which is sometimes called the NewtonRaphson-Simpson method. It is important to remember this is not a method for
generating and exact solution but only an approximation to a solution. The method is
recursive and uses its own results to generate better estimations of a solution.
Problem: Given a differentiable function f(x) approximate a solution to f(x)=0 (i.e. a
root of f(x)).
Newton’s Method
The idea for this method is to use f(x) to build another function h(x) that will generate
a recursive sequence that approximates the root.
The idea here is to keep following the tangent line at a point on the graph down to the
x-axis and use that for the value of x that will approximate the root. In other words h(x)
represents the x-intercept of the tangent line of f(x).
f(x)
f(xn)
root
root
h(x)
x
h(xn)=xn+1
xn
To get what xn is from xn+1 we write the equation of the tangent line at xn, plug in
the point (xn+1,0) and solve for xn+1.
y  f ( x n )  f ' ( xn )( x  xn )
0  f ( xn )  f ' ( xn )( xn 1  xn )
equation of tangent at xn
substitute in (xn+1,0)
 f ( xn )
 xn 1  xn
f ' ( xn )
solve
f ( xn )
xn 1  xn 
f ' ( xn )
this is the h(x)
The equation above gives the recursively defined sequence for xn. This is what is
used for Newton’s Method. This is repeated until the answers that are generated
agree in a certain number of decimal places.
Here are some examples of Newton’s Method applied.
f ( x)  x 2  2, x0  1
x2  2
h( x )  x 
2x
n
xn
h(xn)
f ( x)  x 3  2 x  5, x0  2
x3  2 x  5
h( x )  x 
3x 2  2
n
xn
h(xn)
0
1
1.5
0
2
2.1
1
1.5
1.41667
1
2.1
2.09457
2
1.41667 1.41422
2
2.09457 2.09455
3
1.41422 1.41421
3
2.09455 2.09455
This method has mixed in with it the influence of Raphson and Simpson. Raphson
incorporated the modern idea of derivatives. Simpson saw the generalization as a
recursive method and even explored equations of several variables.
The original Newton's method was used to approximate solutions to polynomial
equations. This seemed a useful idea at the time since a general method of solving
all polynomial equations was not known, only degree 4 or less.
The method that was used I will demonstrate with by applying it to the cubic equation
x3-2x-5=0, but Newton saw how it could be applied to all polynomials.
Newton let f(x)=x3-2x-5 and realized that f(2)=-1 and f(3)=16 so a root must be
between 2 and 3. He then Let x=2+p be the approximation for the root. Plug this back
into the equation drop off any terms whose degree for p is higher than 1 and solve the
linear equation (since linear equations are easiest to solve).
x  2 p
2  p 3  22  p   5  0
8  12 p  6 p 2  p 3  4  2 p  5  0
10 p  1  0
p  101
21
x  2  101  10
 2.1
21
x  10
p
1021  p 3  21021  p   5  0
9261
1000
2
3
63
42
 1323
100 p  10 p  p  10  2 p  5  0
1123
100
61
p  1000
0
 61
p  11230
 61
21
x  10
 11230
 11761
5615  2.09457
Lets compare the modern
Newton's Method and his
original version more closely
by looking at a quadratic
equation. In particular the
equation x2-2=0, letting
f(x)=x2-2 we see the is a root
between 1 and 2.
x  32  p
x  1 p
1  p 2  2  0
 32  p 2  2  0
1 2 p  p2  2  0
9
4
2 p 1  0
p
 3 p  p2  2  0
3 p  14  0
1
p  12
1
2
x  1  12  32  1.5
1
x  32  12
 17
12  1.41667
x  xn  p
Notice the answers that are generated
agree exactly with the modern day
Newton's method. Lets consider a
general quadratic equation:
ax2+bx+c =0
Let f(x) = ax2+bx+c =0 .Lets look at what
is happening from the nth step.
a  xn  p   b  xn  p   c  0
2
axn2  2axn p  ap 2  bxn  bp  c  0
2axn  b  p  axn2  bxn  c  0

 axn2  bxn  c
p
2axn  b



 axn2  bxn  c
f ( xn )
x  xn 
 xn 
2axn  b
f ' ( xn )