Transcript Exploring Mathematics Universe - KSU Web Home

Exploring Mathematics
Universe
The Main Explorer:
Dr. Josip Derado
Kennesaw State University
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The High-School Mathematics
Universe
Open any of the doors in the hall and…
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And You will find yet another
Mathematics Universe
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Leibnitz sequences
1
4
3
1
7
9
5
16
7
9
2
2
2
8
27
64
12
19
6
18
25
37
6
24
36
11
2
61
6
13
2
125
30
49
91
6
64
15
2
216
36
343
127
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Moessner’s Magic
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1
4
9
16
25
36
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Circle
every 2n
number
Form the
cumulative
totals
From 2n to n2 !
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Moessner’s Magic
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 3
7 12 19 27
37 48
1
8
64
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From 3n to n3 !
Circle
every 3n
number
61 75
125
Form the
cumulative
totals
Repeat
the
process
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Moessner’s Magic
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 3 6
11 17 24 33 43 54
1 4
15 32
65 108
175
1
16
81
256
From 4n to n4 !
Circle
every 4n
number
67 81
Form the
cumulative
256 totals
Repeat
the
process
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Moessner’s Magic
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2
6 11 18 26 35
6
24 50
24
46 58 71 85
96 154 225
120 274
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From triangle #s to n! Cool !!!
Circle
every
triangle
number
Form the
cumulative
totals
Repeat
the
process
by circling
the last
member
of every
group
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For more fun and further
reference check
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John Horton Conway
The game of life
Conway's_Game_of_Life
Sprouts
Sprouts
Surreal numbers
And many other things …
Conway’s Lecture series on Web
Convway Lectures
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Can you continue the following
sequence?
1
11
21
1211
111221
?????
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Mathematical Engines:
Imagine and Explore
Euler:
What if there exists a number i such that
i i  1
Impossible, since
1 1 
2
1
2

11 
1 1  i  i  1
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After 250 years of exploration
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Today, complex numbers are applied
everywhere…
The most beautiful formula of all
mathematics:
i
e  1
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Leonhard Euler
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The most productive
mathematician ever to live
Euler's Opus Omnia
Founded the graph theory
Famous Euler formula:
F–E+V=2
300th Anniversary Celebration
of Leonhard Euler, April 27th,
2007 at the German Cultural
Center in Atlanta Georgia.
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Are there any other numbers?
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Hamilton
Quaternions a+b I +c J +d K
Octonions
Hypercomplex numbers
Surreal numbers
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Other wild things in Math
Universe
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There is a positive
number  which is so
small that
  =0
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Impossible!!?? NO, just
imagine such a number.
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Other wild things in Math
Universe
The Pea and Sun
Theorem
(Banach – Tarski
paradox)
You can cut a pea into five
pieces that can be rearranged
into a ball size of the Sun.
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Other wild things in Math
Universe
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Borsuk-Ulam
Theorem
At any instant there
are two antipodal
points on earth which
have the same
temperature.
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Other wild things in Math
Universe
Brower’s
Fixed point
Theorem
Coffee version - Gently stir coffee in a cup. Let it sit until it stops
moving. The fixed point theorem says that there is always one coffee
“particle” which is at the same position where it started.
Crumbled paper - Suppose there are two sheets of paper, one lying
directly on top of the other. Take the top sheet, crumple it up, and put
it back on top of the other sheet. Brouwer's theorem says that there
must be at least one point on the top sheet that is in exactly the same
position relative the bottom sheet as it was originally.
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The Sampling Theorem
If a continuous function is band-limited,
i.e., contains only frequencies within a
bandwidth then it is completely
determined by its values at a series of
points equally spaced less than 1/(2 x
bandwidth) apart.
E.C. Shannon
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Other wild things in Math
Universe
Goedel
Self-referencing
Is this statement true or false?
This sentence is false.
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Other wild things in Math
Universe
The unexpected hanging
A judge tells a condemned prisoner that he will be hanged at noon on one day in the
following week but that the execution will be a surprise to the prisoner. He will not know
the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape
from the hanging. His reasoning is in several parts. He begins by concluding that if the
hanging were on Friday then it would not be a surprise, since he would know by
Thursday night that he was to be hanged the following day, as it would be the only day
left. Since the judge's sentence stipulated that the hanging would be a surprise to him,
he concludes it cannot occur on Friday.
He then reasons that the hanging cannot be on Thursday either, because that day would
also not be a surprise. On Wednesday night he would know that, with two days left (one
of which he already knows cannot be execution day), the hanging should be expected on
the following day.
By similar reasoning he concludes that the hanging can also not occur on Wednesday,
Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur
at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday —
an utter surprise to him. Everything the judge said has come true.
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Why a French clockmaker has
never learned to add fractions?
Achille
Brocost tree
a b ab
 
c d cd
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Further References
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Paul Erdős :
A cocktail Party problem
How many people should be
at the party so we can be
sure that at least 3 guests
will know each other or at
least 3 guests will not know
each other?
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Paul Erdős :
A cocktail Party problem
Answer: 6
For a group of 4 people
the answer is 18.
For a group of 5 people
the answer is not
known.
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Paul Erdős
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Reference
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Other open problems
3 x + 1 puzzle
 3x  1
,
if
x
is
odd
 2
T ( x)  
x
 , if x is even
 2
Conjecture: if you start with any positive integer number x and
iteratively apply T(x), you will reach 1 at some point.
Jeff Lagarias 3x+1 web site
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Million dollar problems
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Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills Theory
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Recent Results
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Andrew Wiles proved
Fermat’s Last Theorem:
There are no non-zero
integer solutions of
x y z
n
n
n
for n > 2.
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Recent Results
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Terence Tao, Brian
Green(2004):
The sequence of prime
numbers contains
arbitrarily long arithmetic
progressions. In other
words, for any natural
number k, there exist kterm arithmetic
progressions of primes.
Terence Tao – Fields Medal 2006
Twin prime conjecture: There are infinitely many integers p such that p
and p+2 are both primes.
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Recent Results
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The Poincare Conjecture
Proven !!?!
The Poincare Conjecture
says that a threedimensional sphere is the
only enclosed threedimensional space with
no holes.
Dr. Grigori Perelman
Fields medalist 2006
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And we are back
OOOps, sorry!!!
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Now we are back!
This presentation will be posted on http://ksuweb.kennesaw.edu/~jderado
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