Exploring Mathematics Universe: Exploring Mathematics Universe The Main Explorer:
Dr. Josip Derado
Kennesaw State University
The High-School Mathematics Universe: The High-School Mathematics Universe Open any of the doors in the hall and…
And You will find yet another Mathematics Universe: And You will find yet another Mathematics Universe
Leibnitz sequences: Leibnitz sequences 1 4 9 16 25 36 49 64
3 5 7 9 11 13 15
2 2 2 2 2 2
1 8 27 64 125 216 343
7 19 37 61 91 127
12 18 24 30 36
6 6 6 6
Moessner’s Magic: Moessner’s Magic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Circle every 2n number Form the cumulative totals 1 4 9 16 25 36 49 From 2n to n2 !
Moessner’s Magic: Moessner’s Magic Circle every 3n number Form the cumulative totals Repeat the process 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 3 7 12 19 27 37 48 61 75 1 8 27 64 125 From 3n to n3 !
Moessner’s Magic: Moessner’s Magic Circle every 4n number Form the cumulative totals Repeat the process 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 3 6 11 17 24 33 43 54 67 81 1 4 15 32 65 108 175 256 1 16 81 256 From 4n to n4 !
Moessner’s Magic: Moessner’s Magic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Circle every triangle number Form the cumulative totals Repeat the process by circling the last member of every group 2 6 11 18 26 35 46 58 71 85 6 24 50 96 154 225 24 120 274 120 From triangle #s to n! Cool !!!
For more fun and further reference check: For more fun and further reference check
John Horton Conway: 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
Can you continue the following sequence?: Can you continue the following sequence? 1
11
21
1211
111221
?????
Mathematical Engines:Imagine and Explore: Mathematical Engines: Imagine and Explore Euler:
What if there exists a number i such that
Impossible, since
After 250 years of exploration: After 250 years of exploration Today, complex numbers are applied everywhere…
The most beautiful formula of all mathematics:
Leonhard Euler : Leonhard Euler 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.
Are there any other numbers?: Are there any other numbers? Hamilton
Quaternions a+b I +c J +d K
Octonions
Hypercomplex numbers
Surreal numbers
Other wild things in Math Universe: Other wild things in Math Universe There is a positive number which is so small that
= 0
Impossible!!?? NO, just imagine such a number.
Other wild things in Math Universe: 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.
Other wild things in Math Universe: Other wild things in Math Universe Borsuk-Ulam Theorem
At any instant there are two antipodal points on earth which have the same temperature.
Other wild things in Math Universe: 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.
The Sampling Theorem: 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
Other wild things in Math Universe: Other wild things in Math Universe Goedel
Self-referencing
Is this statement true or false?
This sentence is false.
Other wild things in Math Universe: 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.
Why a French clockmaker has never learned to add fractions?: Why a French clockmaker has never learned to add fractions? Achille
Brocost tree
Further References: Further References
Paul Erdős :A cocktail Party problem: 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?
Paul Erdős :A cocktail Party problem: 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.
Paul Erdős : Paul Erdős
Reference: Reference
Other open problems: Other open problems 3 x + 1 puzzle 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
Million dollar problems: Million dollar problems Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills Theory
Recent Results: Recent Results Andrew Wiles proved Fermat’s Last Theorem:
There are no non-zero integer solutions of for n andgt; 2.
Recent Results: Recent Results 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 k-term 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.
Recent Results: Recent Results Dr. Grigori Perelman Fields medalist 2006 The Poincare Conjecture
Proven !!?!
The Poincare Conjecture says that a three-dimensional sphere is the only enclosed three-dimensional space with no holes.
And we are back: And we are back OOOps, sorry!!!
Now we are back!: Now we are back! This presentation will be posted on http://ksuweb.kennesaw.edu/~jderado