Download week 6 - NUS Physics

Quest of

Lecture Five
1
Outline
 Review of number representation
 Mathematical classification of
numbers
 Irrationality of 2
 History of

 Methods of computing

2
Number 13 in Various
Representations
Egypt
ΔIII
Babylon
IΓ
Greece
Maya
XIII
Rome
China
Attic
13
HinduArabic
modern
11012 binary
3
Property of Numbers
 The mathematical properties of
numbers are independent of their
representations. E.g., 13 is an odd
number, no matter how it is
represented. 2 + 2 = 4, which
transcends cultures.
4
Natural Numbers N
 The numbers 1, 2, 3, …,
5
Integers Z
 The numbers 1, 2, 3, 4, …
and the number 0,
and negative numbers -1, -2, -3, -4,
…
 It took a long time for people to
understand the necessity of 0 and
negative numbers
 Negative number -n has the property
that (-n) + n = 0
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Rational Numbers Q
 The numbers of the form p/q where p
and q are integers
 The representation of rational
numbers in this form is not unique,
p/q represents the same number as
(np)/(nq) for any n ≠ 0
 We can choose p and q such that
GCD(p,q) = 1
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Adding and Multiplying Rational
Numbers
 p/q  r/s = (pr) / (qs)
 p/q + r/s = (ps + rq)/(qs)
(We use the abbreviated notation p s
= ps)
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Irrationals
 There are numbers that cannot be
written as p/q. These numbers will
be called irrational numbers
 There are countably infinite rational
numbers, but irrational numbers are
not countable
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Count the Rationals
p/q
1
2
3
4
6
7
8
…
5/1
6/1
7/1
8/1
…
…
1
1
1/1
2
2/1
4
3/1
2
3
½
5
2/2
8
3/2
4/2
5/2
6/2
7/2
8/2
3
6
1/3
9
2/3
3/3
4/3
5/3
6/3
7/3
8/3
4
10
¼
2/4
¾
4/4
5/4
6/4
7/4
8/4
5
1/5
2/5
3/5
4/5
5/5
6/5
7/5
8/5
6
1/6
2/6
3/6
4/6
5/6
6/6
7/6
8/6
7
1/7
2/7
3/7
4/7
5/7
6/7
7/7
8/7
8
1/8
2/8
3/8
4/8
5/8
6/8
7/8
8/8
7
4/1
5
11
…
…
10
Proof that
2 is irrational
 Prove by contradiction
p
 Assuming p & q are integers such that  2
q
and GCD(p,q)=1, then
p2
2
2
or
p

2
q
2
2
q
 We use the fact that square of odd number
is odd, square of even is even. So p2 is
even, then p is even.
 Let p = 2s, where s is any integer, so
p2=4s2, and q2=p2/2=2s2. This says q is
also even, which contradicts the assumption
that GCD(p,q) =1
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Square of an even number is even,
and odd is odd
 Represent integer n by its binary
expansion
k 1
n  ak 2  ak 1 2
k

 a1 2  a0 , a j  0,1
 Clearly n is odd if a0=1, and even if
a0=0. Then the last binary digit in n2
is a02=a0 (think of multiplication in
binary)
 Thus the parity of n2 is same as n.
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Real Numbers R
 The collection of rational numbers
and irrational numbers forms the real
numbers. The set of reals are
uncountable.
-2
The real line.
-1
0
1
2
3
x
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Algebraic and Transcendental
Numbers
 The algebraic number x is a real
number that is a root of a polynomial
equation with integer coefficients
c0  c1 x  c2 x 
2
cN x  0
N
 A real number that is not an algebraic
number is called a transcendental
number. E.g., -1, ½, 2 are
algebraic, e and  are transcendental.
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Complex Number C
 A complex unit i has the property
i2 = -1, or i = 1
 A complex number is of the form
a+ib
where a and b are real numbers.
 Solution of a polynomial equation of
real coefficients can be a complex
number.
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Addition and Multiplication of
Complex Numbers
 (a + ib) + (c + id) = (a+c) + i(b+d)
 (a + ib)  (c + id) = (ac +iad + ibc +
i2bd) = (ac-bd) + i(ad + bc)
16
Complex Plane
Im z
z=x+iy =reiθ
r
θ
Re z
x is called real part,
y imaginary part
17
Relation of Types of Numbers
Complex
Real
Rational
Integer
18
Definition of 
S =  D = 2 R
Diameter D
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Area of a Circle
R
A=R  (S/2) =  R2
Approximately S/2
R
R
2 R
r
A   2 rdr  2
2
0
  R2
0
r dr
20
Volume of a Sphere
4
3
V  R
3
Hand with Reflecting Globe,
self-portrait by M. C. Escher,
1935.
21
 in the Bible
The Bible suggested that  = 3.
“Also, he made a
molten sea of ten
cubits from brim
to brim, round in
compass, and five
cubits the height
thereof; and a line
of thirty cubits did
compass it round
about.” — Old
Testament (I
Kings vii.23)
22
Egyptian  (Rhind Papyrus)
Each square is 9 units of
area, each triangle is 9/2
units. Assuming that the
shaded area is
approximately equal to
the area of circle, and
79=63 ≈ 64, then
(9/2)2 ≈ 64 = 82
or
 ≈ 4 (8/9)2 = 3.16049…
3
23
Archimedes Of Syracuse (ca 287212 BC)
Archimedes is
regarded as
one of the
greatest
mathematician
of all time. In
physics, we
have the
Archimedes’
principle for
hydrostatics.
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Archimedes’ Method
n=6
The perimeter of a regular
polygon of n sides
inscribed in a circle is
smaller than the
circumference of the
circle, whereas the similar
circumscribed polygon is
greater than the
circumference. With 96
sides, Archimedes found
3
10/71
<<3
1/7
or
3.14084 <  < 3.142858
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Zu Chongzhi (429 – 500 AD)
Zu Chongzhi (祖冲之) obtained
accurate estimates of
3.1415926 <  < 3.1415927.
This level of accuracy was not
surpassed until early Renaissance
in Europe. Methodology-wise, it
is similar to Archimedes’.
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Modern Methods based on Series
Expansions
 Let f(x) be some smooth function,
then we can write f(x) as a sum of
infinite terms:
f ''(0) 2
f ( x)  f (0)  f '(0) x 
x 
2
where n !  1 2  3   (n  1)  n
and
n
d
f ( x)
(n)
f (0) 
dx n
f ( n ) (0) n

x 
n!
f '(0)  lim
0
f ()  f (0)

x 0
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Example of Series
2
3
n
x
x
x
e  1 x     
2 6
n!
3
5
2 n 1
x
x
x
n
sin( x)  x  
  (1)

6 120
(2n  1)!
x
x3 x5 x7
arctan( x)  x    
3 5 7
2 n 1
x
 (1) n

2n  1
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The Arctangent Function
y /2

θ
(0,0)
3/2
(1,y)
0 x
(1,0)
Given a point (1,y)
on the vertical line
at x=1, the
arctangent of y is
the angle θ, i.e.,
arctan(y) = θ,
–/2 < θ < /2.
The angle θ is
measured in radian,
that is, a full circle
is 2.
29
Arctangent Series at y=1

1 1 1 1 1
 1     
4
3 5 7 9 11
Although this formula for  is simple and easy to
calculate, it requires large numbers of terms for a good
estimate of it.
The rest of the story in the 1700 to 1800 AD was to find
faster convergent series.
30
John Machin (1680-1752)
 Combining two arctangent series with
small arguments gives faster
convergence. Machin obtained 100
decimal places by:

1
1
 4 arctan  arctan
4
5
239
1
1
1
1
 4 



3
5
7
 5 35 55 7 5



1
1
1
 1





3
5
7
 239 3  239 5  239 7  239



31
Other Interesting Formula for 
 William Brouncker (1620-1684 AD)
continued fraction
12
 1
2
3

2
52
2
72
2
92
2
2
4
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S. Ramanujan (1877-1920 AD)
8 1103
8  4!(1103  26390)
8  8!(1103  26390  2)




4
4
8

9801
9801 396
9801 2  396
8 (4n)! (1103  26390n)


4
4n
9801 (n !)
396
1
Srinivasa Ramanujan was one of
India's greatest mathematical
geniuses. He made substantial
contributions to the analytical
theory of numbers and worked on,
elliptic functions, continued
fraction, and infinite series.
33
Leonhard Euler (1707-1783)
Swiss mathematician who in his
numerous works made major
contributions to virtually every
branch of the mathematics of his
day. Euler formula relates the
most common mathematical
constants in a mysterious way:
e
i 
1  0
e = 2.71828182846…
 = 3.14159265358979…
i = 1
34
Computer Age
 Borwein quartic convergence
algorithm (1987)
Let y0  2  1, a0  6  4 2
Iterate (k  0,1, 2,
)
1  (1  yk 4 )1/ 4
yk 1 
,
4 1/ 4
1  (1  yk )
ak 1  ak (1  yk 1 ) 4  22 k 3 yk 1 (1  yk 1  yk 12 )
ak approaches 1/ as k goes to infinity.
35
Working of Borwein Iteration
 Start with y0=0.41421356…,
a0=0.34314575…,
 k=0, compute y1=(1-(1-y04)1/4)/(1+
(1-y04)1/4) = 0.00373489, and a1= a0
(1+y1)4-23y1(1+y1+y12)=0.31831,
1/a1=3.1415926…
 Set k=1, compute y2 and a2 and so
on
  ≈ 1/ak
36
A Brief Summary of History of 
Who?
Babylonian
Egyptians
Archimedes
Zu Chongzhi
Machin
Ferguson
Guilloud
Chudnovsky
Kanada
When?
2000 BC
2000 BC
c.380 BC
c.480 AD
1706
1954
1966
1989
1999
No. of Decimals
1
1
2
6
100
530
250,000
525,229,270
206,158,430,000
37
The

3.1415926535897932384626433832795028841971693993751058209749
44592307816406286208998628034825342117067982148086513282306
64709384460955058223172535940812848111745028410270193852110
55596446229489549303819644288109756659334461284756482337867
83165271201909145648566923460348610454326648213393607260249
14127372458700660631558817488152092096282925409171536436789
25903600113305305488204665213841469519415116094330572703657
59591953092186117381932611793105118548074462379962749567351
88575272489122793818301194912983367336244065664308602139494
63952247371907021798609437027705392171762931767523846748184
67669405132000568127145263560827785771342757789609173637178
72146844090122495343014654958537105079227968925892354201995
61121290219608640344181598136297747713099605187072113499999
98372978049951059731732816096318595024459455346908302642522
30825334468503526193118817101000313783875288658753320838142
06171776691473035982534904287554687311595628638823537875937
51957781857780532171226806613001927876611195909216420198938
09525720106548586327886593615338182796823030195203530185296 …
38
Summary
 Various ways of representing numbers are
equivalent. Mathematical properties are
independent of number representation.
 Numbers are classified according to their
properties.
 The history of computation of the value of 
reflects the development of mathematics
and computing power of the day.
39