10/22/2015
A Gem from Isaac Newton
“Nature and Natures Laws lay hidden by Night
God said “Let Newton” be and all was light”
- A. Pope
Mathematics of the Heroic Century
Francois Viète (1540-1603)
early use of algebraic notation
John Napier (1550-1617) and Henry Briggs (1561-1631)
logarithms
Rene Descartes (1596-1650)
Discours de la mêthode: analytic geometry
Blaise Pascal (1623-1662)
probability theory; Pascal’s Triangle; cycloid;
Pierre de Fermat (1601-1665)
number theory
Problem of Points
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10/22/2015
Francois Viete
in artem analyticen isagoge
D in R – D in E aequabiture A
quad
or
DR – DE = A2
Calculated π to 9 – 10 digits
Using 393216 = 6×216 sided
polygon
(Archimedes: 96 = 6×24)
Pascal & Fermat: Problem of Points
In a game of chance two players have equal probability of
winning a round (think flipping a fair coin and calling heads or
tails). The two agree to play until one player has won n rounds.
If the game is interrupted before either player has won n
rounds, what is the best (fairest) way to split the pot?
Example: Player A and B each put up 5 dollars and a coin is
flipped. Player A wins if 5 heads comes up before 5 tails;
otherwise Player B wins. The game is interrupted after 3 heads
and 2 tails has occurred. How should the pot be split?
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10/22/2015
Cycloid: path traced by point on a circle as it rolls along a line
•
•
•
•
h = amplitude
a = midline
•
x = at − h sin(t )
y = a − h cos(t )
Proof that x4 + y4 = z4 has no non-trivial solutions
Outline: Show that if a solution exists, the variant x4 + y4 = z2
implies existence of a smaller variant x’4 + y’4 = z’2 (z’ < z) which
implies existence of smaller variant x”4 + y”4 = z”2 (z” < z) etc.
resulting in an infinite descent contradiction.
Primitive Pythagorean triple <a, b, c>: a2 + b2 = c2 and a and b are
relatively prime; Every Pythagorean triple leads to a primitive
Pythagorean triple. (if d = gcd(a,b) then d2|c2 so <a/d ,b/d, c/d>
is primitive).
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10/22/2015
1. For integers u and v, 4uv + (u-v)2 = (u+v)2 is Pythagorean triple
when 4uv is a square
2. If <a, b, c> is a Pythagorean triple then a and b are of opposite
parity.
3. Main Result: The equations
a = 2uv
b = u2 – v2
c = u2 + v2
establish a 1:1 correspondence between primitive Pythagorean
triples and pairs of integers (u,v) of opposite parity
Example: Given <5, 12, 13> u = (13+5)/2 = 9 and v = (13 – 5)/2 = 4
4. Method of descent for x4 + y4 = z2
Assume x4 + y4 = z4 has a non trivial solution and let
<x’2, y’2, z’> be a primitive Pythagorean triple
5. From Main Result assume x’2 is even, y’2 is odd and there are
relatively prime integers p and q such that
x’2 = 2pq
y’2 = p2 – q2
z’ = p2 + q2
Since p and q are relatively prime and 2pq = x’2 is a square either
Case I: p = 2a2 is twice a square and q = b2 is odd
Case II: p = z”2 is an odd square and q = 2c2 is twice an even square
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10/22/2015
Case I: p = 2a2 is twice a square and q = b2 is odd (impossible)
(2a2)2 = 4a4 = p2 = q2 + y’2 = b4 + y’2
Since both b2 (and b4) and y’2 are odd, the sum of two odd integers
b4 + y’2 which equals 4a4 cannot be divisible by 4 #
Recall:
x’2 = 2pq
y’2 = p2 – q2
z’ = p2 + q2
Case II: p = z”2 is an odd square and q = 2c2 is twice an even square
(z”2)2 = p2 = y’2 + q2 = y’2 + (2c2)2
So <y’, 2c2, z”2> is a (w/o log) primitive Pythagorean triple so there
are relative prime integers u and v such that
2c2 = 2uv
y’ = u2 – v2
z”2 = u2 + v2
So 2c2 = 2uv implies u and v are squares so z”2 = u2 + v2 is a sum of
4th powers or z”2 = u’4 + v’4. however z”2 = p (see above) so z”2
< p2 + q2 = z’ ≤ z’2.
Hence for any set of integers x’, y’ and z’ which are a solution to
x’4 + y’4 = z’2 there is another set of integers u’2, v’2 and z” with z”
< z’ which is also a solution. Therefore no smallest solution #
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10/22/2015
Isaac Newton
Born 12/25/1642 Woolsthorpe Lincolnshire
1661 – Trinity College Cambridge (Restoration 1660)
1664 – promoted to scholar (4 yr. support for Master’s degree)
1665- 66 “Plague Years”
1665 – Generalized Binomial Theorem; method of fluxions
1666 – inverse method of fluxions
1669 - De Analysi written (pub 1711)
1668 - elected fellow of Trinity University
1669 – made Lucasian Chair of Math
1671 – display refracting telescoope
Isaac Newton
1670’s – worked on alchemy & theology
1684 – Edmund Halley urges Newton to publish
1687 – Philosophiae Naturalis Principia Mathematica
1688 – “Glorious Revolution”
1689 – elected to Parliament
1696 – resigns Lucasian chair – to London & Warden of Mint
1704 – Opticks; elected President of Royal Society (De
Quadratura)
1705 – Knighted by Queen Anne
Died 1727 age 84 – buried Westminster Abbey
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10/22/2015
Newton on his anni mirabiles
In the beginning of the year 1665 I found the Method of
approximating series & the Rule for reducing any dignity of any
Bionomial into such a series. The same year in May I found the
method of Tangents of Gregory & Slusius, & in November had the
direct method of fluxions & the next year in January had the Theory
of Colours & in May following I had entrance into ye inverse method
of fluxions. And the same year I began to think of gravity extending
to ye orb of the Moon & (having found out how to estimate the force
with wch [a] globe revolving within a sphere presses the surface of
the sphere) from Keplers rule of the periodic times of the Planets
being in sesquialterate proportion of their distances from the center
of their Orbs, I deduced that the forces wch keep the Planets in their
Orbs must [be] reciprocally as the squares of their distances from the
centers about wch they revolve: & thereby compared the force
requisite to keep the Moon in her Orb with the force of gravity at the
surface of the earth, & found them answer pretty nearly.
Sir Isaac Newton in Reflections: Remembering Newton springer.com
Newton’s Generalized Binomial Theorem
m
m−n
m − 2n
BQ +
CQ...
( P + PQ ) n = P n + AQ +
m
m
n
2n
3n
where A, B, C is used for the previous term; that is
=P
m
n
+
m mn
m − n m m n 2 m − 2n m − n m m n 3
P Q+
⋅ P Q +
⋅
⋅ P Q +
n
2n n
3n
2n n
Set Pm/n = 1 to get
m
( )(
(1 + Q ) n = 1 + Q + n n
m
m
n
m
2
− 1)
Q2 +
( mn )( mn − 1)( mn − 2 ) Q 3 + ..
3!
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10/22/2015
Generalized Binomial Theorem
In modern summation notation:
∞
s
s
s
s
s
(1 + x ) = ∑   x k =1 +   x +   x 2 +   x3 + ...
k =0  k 
1  2
 3
s
s
 s  s ( s − 1)( s − 2 ) ... ( s − k + 1)
where   = 1,   = s,   =
k!
0
1
k 
Example (1 – x)1/2
Note alternating signs from –x
1 2 
1 2  1 2 
1 2 
1 2 
k
( − x ) =1 −   x +   x 2 −   x3 +   x 4 ...

k =0  k 
 1   2 
 3 
 4 
∞
12
(1 − x ) = ∑ 
= 1−
(1 2 )
x+
(1 2 )( −1 2 )
1!
2!
(1 2 )( −1 2 )( −3 2 )( −5 2 )
4!
x2 −
(1 2 )( −1 2 )( −3 2 )
3!
x3 −
x 4 − ...
1
1
1
5 4
= 1 − x − x 2 − x3 −
x − ...
2
8
16
128
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10/22/2015
Quick Application: Calculating Square Roots
1
 1
3 = 4 − 1 = 4 1 −  = 2 1 −
4
 4
 1  1  1  1  2 1  1 3 5  1  4

= 2 1 −   −   −   −
− ... 


 2  4  8  4  16  4  128  4 



≈ 1.732116699
3=
49 1
49 
1  7
1
−
=
1−
1 −  =
16 16
16  49  4
49
2
3
4

7 1 1  1 1 
1 1 
5  1 
= 1 −   −   −   −
−
...

 

4  2  49  8  49  16  49  128  49 

≈ 1.732050808
Newton’s Pi (Methodus Fluxionum et Seriersum Infinitarum – written
1671)
1 
1
1
1
5 5 
x 2 1 − x − x 2 − x3 −
x − .. 
8
16
128
 2

expand by binomial theorem
1
2
y = x − x = x ⋅ (1 − x )
radius = 1
( 1 4 ,0 )
1/6 of a circle
of radius 1/2
2
24
2
2
(1,0 )
( 1 2 ,0 )
π
1
1
4
= ∫ x − x 2 dx +
0
area of the 30-60-90 triangle
3
32
Dunham p. 174
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10/22/2015
Newton’s pi
Newton computed pi to 15 digits
writing later
"I am ashamed to tell you to how
many figures I carried these
computations, having no other
business at the time."[
Infinite Series Formulas for π (mid 17th Century)
Arctan Formula
1
= = −1
1
+
Gregory-Leibniz Series
1 =
2 + 1
=1− + − +⋯
converges very slowly
Machin’s Formula
1
1
= 4
−
4
5
239
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10/22/2015
Arctan Series Formula (Details)
x
1.
1
∫ 1+ t
2
dt
t = tan ( u ) dt = sec2 ( u ) du
0
x
t=x
t=x
1
1
=
2
1+ t
sec 2 ( u )
t=x
1
sec 2 u
dt
=
∫0 1 + t 2 t =∫0 sec2 u du = t =∫0 du = u = arctan ( x )
t =0
1
= 1 − t 2 + t 4 − t 6 + ...
1+ t2
2.
x
3.
x
1
2
4
6
8
∫0 1 + t 2 dx = ∫0 1 − t + t − t + t − ...dt
= x−
2 k +1
∞
x3 x 5 x 7 x 9
k x
+ − + − ...∑ ( −1)
3 5 7 9
2k + 1
k =0
In Conclusion
The calculus was the first achievement of modern
mathematics and it is difficult to overestimate its
importance. I think it defines more unequivocally
than anything else the inception of modern
mathematics, and the system of mathematical
analysis, which is its logical development, still
constitutes the greatest technical advance in exact
thinking."
John von Neumann, in: James R. Newman The World of
Mathematics
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