Communicating Mathematics

Amicable Numbers

For my seventh and final blog post, I decided to address an idea which falls under Communicating Mathematics to complete my exemplars.  Amicable numbers were a very interesting topic to me because I could not imagine how much time it took to find such a special trait between these numbers.  Amicable numbers are numbers whose sum of the proper divisors of each minus the number itself is equal to the other number. 

Amicable numbers were known to the Pythagoreans, who were quite interested in special in rare numbers.  They found an interesting trait for numbers 220 and 284.

Let me know break down the process for showing that (220, 284) are amicable. 

Let us first find the factors of both numbers > 

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220 

284: 1, 2, 4, 71, 142, 284 

We will now add up the factors.  Both factors added up give us 504.  Now subtract both 220 and 284 from 504 t0 see that we get the other number.

504-220=284

504-284=220

Thus, the two numbers are amicable.    

Around 850 AD a general formula by which we can find some of these numbers was invented by mathematician Thabit ibn Qurra.  

Thabit ibn Qurra's theorem states that if 

p = 3 × 2^n − 1 − 1,

q = 3 × 2ⁿ − 1,

r = 9 × 2^2n − 1 − 1, 

where n > 1 is an integer and p, q, and r are all prime numbers, then  2ⁿ×p×q and 2ⁿ×r are amicable numbers. 

This theorem work for pairs (220,284) when n=2, (17296, 18416) when n=4, and (9363584, 9437056) when n=7. 

Later on this theorem was rediscovered by Fermat and Descartes whom it is sometimes ascribed and even later on extended by Euler.

Euler extended Thabit's theorem saying that if

p = (2^(n - m)+1) × 2^m − 1,

q = (2^(n - m)+1) × 2ⁿ − 1,

r = (2^(n - m)+1)² × 2^m + n − 1,

where n>m>0 and p, q, and r are prime numbers, then 2ⁿ×p×q and 2ⁿ×r are amicable numbers.  Euler's rule creates additional pairs for (1,8) and (29, 40) with no other pairs being known.  It is claimed in a video by William Dunham that Euler found 58 pairs bringing the total number of existing pairs to 61.  The video can be found by clicking the following link:

 http://www.youtube.com/watch?v=TEh_4LQkkHU

It is still an open question as to how many pairs of amicable numbers there are but many believe there are infinitely many.  With the help of computers, it has been discovered that there are over 7500 amicable numbers.  Another interesting result is that with every known pair of amicable numbers, both numbers are either even or odd.  It is unknown whether an even-odd pair exists.  Amicable numbers play an important role in mysticism and numerology.  Amicable numbers were thought by mystics to possess magical powers.  Astronomers actually used amicable numbers for preparing talismans and horoscopes.  It was believed these numbers had the power to create special ties between individuals.  The mystical notoriety of amicable numbers caused them to be studied more carefully by number theorists.  Numerology is the study of the purported, divine, mystical, or other special relationship between a count of measurement and observed or perceived events.  I guess these numbers matter to me because they are such an interesting find.  With all my research I really didn't find any other significance other than what I noted above but it just goes to show you there is still more in math to be discovered.  I think the relationship these numbers have with other fields, such as astronomy, is quite interesting as well.         

Sources:

http://mathworld.wolfram.com/AmicablePair.html

http://en.wikipedia.org/wiki/Amicable_numbers
http://www.osdata.com/programming/numbertheory/amicablenumbers.html
http://britton.disted.camosun.bc.ca/amicable.html
http://www.eg.bucknell.edu/~ncr006/talks/numerology.pdf

Book Skimpression

The Math Book

The Math Book is written by Clifford A. Pickover and it is an excellent book which puts the different major discoveries in mathematics in chronological order.  The book was an awesome and easy skim through book because half the pages contain short summaries of mathematical discoveries and half the pages contain pictures related to those discoveries.  There were 250 million different discoveries mentioned in The Math Book and they range from 150 Million BC to 2007.  

I did not read each and every discovery that this book discussed but while skimming through I did stop for a few that caught my attention. Enjoy!

Ant Odometer (150 Million BC)

> A type of ant called the cataglyphis fortis would travel immense distances and were able to return to the nest using a direct route.  It was believed there was a built in "computer" that functioned as a pedometer.  Researchers then studied the effects of either giving the ant stilts or amputating their legs to study the results.  Results showed that ants with stilts traveled passed the nest, while ants with amputated legs didn't reach the nest.  Another interesting result was that if the ant started with modified legs, the journey back to the nest could be completed.  

Wheat on a Chessboard (1256)

> This was used for centuries to demonstrate the nature of geometric growth/progression

> It is the earliest mention of chess in puzzles

> Basically each square multiplies by the number of grain in the prior square and therefore each square has 2^n number of grains

> If the the number of grains on a chessboard were put onto trains, that train would be able to reach 1000 times around the Earth

Prisoner's Dilemma (1950)

> The following scenario is given:

You have two suspects that are to be tried.  If both suspects confess, the punishment is five years of wandering in a desert.  If only one suspect confesses, the confessor is free to go and the other is doomed to crawling/eating the dust for 30 years.  If neither suspects confess the punishment is just six months wandering in the desert.  The most forward solution to this problem is that neither suspect should confess and you would get the least lengthy term.  However, research done to analyze the non-zero-sum-squares shows that the suspects are more likely to both confess in the hopes the other doesn't so they can achieve freedom.  This study in mathematics helped in a number of other fields including psychology, sociology, biology, political science, and economics.

Again, those are just a few major milestones which struck my interest in my skim through.  I think this book would be great for any audience.  It is a very easy read and it is quite interesting to see how some of our everyday things came from these mathematical concepts.  I think the most interesting thing about this book is that there are concepts addressed in here that seem so miniscule but when you think about if that concept wasn't discovered, it could change the way you think about life!

History of Math

Women in Mathematics

Sofia Kovalevskaya 

 
For this weekly post I decided to research "Historical Women in Mathematics."  I began reading tiny summaries of the work these women did and came across one, Sofia Kovalevskaya.  The reason I gained interest in this woman was because like the gentleman in the first book I read, Love & Math, she was unable to pursue an education in mathematics because she lived in Russia.  While his hardship was due to his nationality, Kovalevskaya's was because of her gender.  

Sofia's exposure to mathematics began at a very young age.  She credits her Uncle Peter for her interest in mathematics.  Uncle Peter would discuss numerous abstractions and mathematical concepts with her.  After teaching herself trigonometry at only fourteen years old, she went off to study in St. Petersburg.  After finishing her secondary schooling, she was reading to study at a university level but such things were not permitted for women in Russia.  

Kovalevskaya contracted a marriage to Vladimir Kovalevsky and moved to Germany.  While she was still unable to attend university lectures, she was tutored privately and eventually received her doctorate after writing treatises on partial differential equations, Abelian integrals, and Saturn's rings.  After her husband's death, she was appointed to a position as a lecturer in mathematics at the University of Stockholm.  She became the first woman in that region of Europe to receive a full professorship. 

Her written paper titled, "On the Theory of Partial Differential Equations," was published in 1875.  In this paper, Kovalevskaya generalized a problem that was posed by Cauchy.  Kovalevskaya had significantly simplified the proof of this problem and gave the theorem its definitive form.  The theorem on the existence and uniqueness of solutions of partial differential equations can sometimes be referred to as the Cauchy-Kovalevskaya Theorem.

I researched what this theorem was and Wolfram-Alpha gave me this...
   
"This theorem states that, for a partial differential equation involving a time derivative of order , the solution is uniquely determined if time derivatives up to order of the dependent variable are specified at a single surface, provided the surface is a free surface i.e., not a characteristic surface. (In wave problems, a characteristic surface is the same as a wavefront. In problems of dimension greater than three, replace "surface" with "hypersurface.")" 

A visual representation of this can be seen on the following link:
http://www.encyclopediaofmath.org/index.php/Cauchy-Kovalevskaya_theorem

This theory is important because it shows that within a class of analytic solutions of analytic equations the number of arbitrary functions needed for a general solution is the same as the order of the equation and these arbitrary functions involve one less independent variable than the number occurring in the equation.

Later in 1880, a Russian mathematician, Chebyshev, invited Kovalevskaya to present a paper at his conference.  She presented her dissertation paper on Abelian integrals.  The paper was well received and a fellow mathematician offered to help find Kovalevskaya a job in his country.  It was in 1884 that she was appointed her position as a professor at the University of Stockholm.  A few years later, a competition was announced for  the Prix Bordin.  It was a prize competition for papers on the theory of the rotation of a solid body.  Sofia was engaged in her research for this paper.  Sofia submitted fifteen papers anonymously and one of her papers was considered so outstanding she received an even higher prize.  Prior to the submission of this work, there were only two other solutions to the theory of the rotation of a solid body and both cases were when the body was symmetric.  These cases were developed by Euler and Lagrange.  Sofia described the first solvable case for an unsymmetrical top.  Her work on Saturn Ring's is only briefly reviewed but she found the stability of motion of liquid ring-shaped bodies. Laplace found the form of the ring to be a skewed cross section of an ellipse. Kovalevskaya,using a series expansion, proved that the rings were egg-shaped ovals symmetric about a single axis. However, the subsequent proof that Saturn’s rings consist of discrete particles and not a continuous liquid made this work inapplicable.

In 1891, Sofia passed away.  Her contributions to the mathematical field are remarkable.  I was so interested to see yet again so many connections between different fields.  She worked on a theorem with Cauchy which is a mathematician I learned about in 408, she also attended one of Chebyshev's lectures and we use one of his theorems in STA 412.  Sofia Kovalevskaya was a great mathematician who opened the doors at universities to women.  All of my findings talk about how passionate she was to her work and I can absolutely see that.  Her story shows the struggles she was faced with being a female and how she overcame them.  I think this is one of many female mathematician's whose work is inspiring and relate-able.



Sources:
http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?no-ist
http://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma072.pdf
http://www.agnesscott.edu/lriddle/women/kova.htm
http://www.encyclopediaofmath.org/index.php/Cauchy-Kovalevskaya_theorem

Love & Math

 Review of "Love & Math"

The story “Love & Math” is about a young high school boy from the Soviet Union who has a love for quantum physics.  After being persuaded by Evgeny Evgenievich to pursue a love for mathematics, this story embarks on Edward Frenkel’s journey of finding his love for math. 

            

 After only speaking with Evgeny Evgenievich for one meeting, Frenkel was already inspired by the reading Evgenievich asked him to do.  He found his interest in math almost immediately.  It was then that he decided to pursue his education by applying to Moscow University.  During this time, the Soviet Union did not treat Jews fairly.  Frenkel had come from a Russian-Jewish background.  After submitting his application to Moscow University, he was told that he would have no way of getting in because of his nationality.  He decided to give it a try anyway and take the administered exams that needed to be passed in order to get into Mekh-Mat (the school of mathematics).  After succeeding with his written mathematics exam and getting all of his answers correct he was later harassed during his oral mathematics exam and decided to withdraw his application.  He ended up studying applied mathematics at Kerosinka.  Kerosinka could not offer all that was available at Moscow University and so Frenkel snuck into the university to attend lectures.  He sat in on a famous mathematician Kirillov who asked him for help working on a problem.  He connected with a guy Fuch who gave him an article to read dealing with braid groups.  He was able to solve the problem correctly and later presented his findings at a well known lecture ran by Gelfand.  He published his first research in Gelfand’s journal, “Functional Analysis and Applications.”  He then worked on a second problem given to him by Feigin and Fuch.  Feigin went on to be Frenkel’s mentor.  With  Feigin, he researched Kac-Moody algebra where he wrote a journal of his results that was published within a year into Russian Mathematical Survey’s.  While doing all of this side mathematical research and maintaining the position at the top of his class, he also worked with his advisor from Kerosinka, Yakov Isaevich, on three projects in urology.  He referred to applied math as his spouse, while pure math was his secret lover.  After help from Yakov, Frenkel was able to get a job as his assistant at Kerosinka. 

           

 Frenkel then received a letter from Harvard University asking him to accept the Harvard Prize fellowship.  He spoke at many seminars about his work with Kac-Moody algebra.  He was then invited to attend Harvard to obtain his Ph.D, which was not something he was able to get back in Russia.  Frenkel obtained his Ph.D in one year and wrote his theses on Langland’s Dual Groups.  After receiving a grant, where he contributed most of his work, to study the link between Langland’s Problem and quantum physics, he then decided to portray his love of math in the form of a movie.  The screenplay was first published as a book, “The Two-Body Problem.”  His movie creates a sort of formula for love that is kept secret by being tattooed on the mathematician’s wife. 

           

 I enjoyed reading this book overall.  I found some of the mathematical concepts portrayed in it to be sort of difficult to follow but at the very beginning of the book, Frenkel says that this book will make you want to become a mathematician and I could definitely feel that throughout the reading.  I think the journey Frenkel went through to do what he loves is very inspiring to read.  I also found in very interesting how he was able to connect his first love of physics with his new love, math.  It just goes to show you how many connections there are between subjects and even within the subject of math itself.  Overall, it was a pretty fun and easy read. 

History of Math

Who is Fibonacci?
Leonardo Pisano Bigollo, known as Fibonacci, was born in 1170 and died in 1250.  He was an Italian mathematician known by some to be the "most talented western mathematician of the Middle Ages."  He is best known to the modern world for the spreading of the Hindu-Arabic numeral system by the use of his book, Book of Calculation, and for a number sequence named after him called the Fibonacci sequence. 

After Fibonacci recognized that using the Hindu-Arabic numeral system was easier than roman numerals, he traveled throughout the Mediterranean world under one of the leading Arab mathematicians of that time.  When he returned back, he wrote down what he had learned in what is called the Book of Calculation.  The Book of Calculation advocated the use of digits 0-9 and place value.  This book also showed the practical importance of the numeral system by applying to things such as bookkeeping, conversion of weights, and the calculation of interest.  Fibonacci solved a problem involving growth of population of rabbits based on idealized assumptions.  The solution, known as the Fibonacci sequence, was known to Indian mathematicians as early as the 6th century.  I found it surprising that the sequence is known after Fibonacci but he was not technically the first to discover it.  The reason the sequence is named after him, however, is because his book introduced this idea to the West. 

I came across a source that introduced how Fibonacci sequence can be used with gambling and lotteries --Who doesn't want an easier way to win!?  While many speculate that the use of the Fibonacci sequence will provide an edge in picking lottery numbers or bets in gambling, the truth is that the outcome is all chance.  However, there is a betting system based off the Fibonacci system called the Martingale progression that is used in online roulette and casinos where the pattern of bets placed follows a Fibonacci progression, that is each new bet is the sum of the two previous bets.  If a number wins, the bet goes back two numbers in the sequence because their sum was equal to the winning bet.  
The Fibonacci system is like the Martingale progression but the bets stay lower which doubles up each time.  The downside is that the Fibonacci roulette system does not cover all the losses in a bad streak.  Again, odds are always in favor of the casino or lottery, these are just ideas to make the playing of bets more methodical.    

Sources:
http://en.wikipedia.org/wiki/Fibonacci
http://www.goldennumber.net/fibonacci-series/

Round	Scenario 1	Scenario 2	Scenario 3
Bet 1	Bet 1 and lose	Bet 1 and lose	Bet 1 and win
Bet 2	Bet 1 and lose	Bet 1 and lose	Bet 1 and win
Bet 3	Bet 2 and win	Bet 2 and lose	Bet 1 and lose
Bet 4	-	Bet 3 and win	Bet 1 and lose
Bet 5	-	-	Bet 2 and win
Net Result	Even at 0	Down by 1	Ahead by 2

Nature of Mathematics

Understanding Our Number System

In order to be confident in our use of numbers, we must understand how our number system works and how numbers are related to each other.  We use a single number structure.  All numbers in our single number structure can be represented using combinations of a finite set of digits--0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
An important understanding of how our number system works comes from understanding "place value." This means that a digit can carry a different value depending on where in a number it is located.  We use a base 10 system.  Depending on where a digit is placed, that digit represents a power of 10.  

 

When learning about number systems, it is important to remember 3 things.

1.)  Use of numbers in the world around us can take on many different forms.

2.)  The same digit takes on different values depending on where it is placed .

3.)  The number system is finite.  

Below is an interactive chart that helps explain the different ideas behind the number system.

Ideally, what our numeral system does is it

• Represents a useful set of numbers
• Give every number represented a unique representation 
• Reflect the algebraic and arithmetic structure of the numbers. 

Our numeral system can represent a useful set of numbers such as integers, rational numbers, or irrational numbers, for example. Our numeral system gives every number represented a unique representation.  For example, the usual decimal representation of whole numbers gives every non zero whole number a unique representation as a finite sequence of digit, beginning by a non zero digit.  However, when decimal representation is used with rational or real numbers, such numbers have an infinite number of representations (2.31, 2.310, 2.309999999).  The following link gives great insight into how our numeral system reflects the algebraic and arithmetic structure of numbers.  

http://academic.sun.ac.za/mathed/malati/Files/Structure98.pdf

Sources:

http://www.nationalnumeracy.org.uk/userfiles/Images/Numbers_-_learning_pathway_1_web.jpg
 http://www.nationalnumeracy.org.uk/understanding-numbers-and-our-number-system/index.html
http://en.wikipedia.org/wiki/Numeral_system

Doing Math

Tessellations

The Islamic Empire has made significant contributions to the study of mathematics.  Islamic mathematics allowed math to be expressed in an art form.  There was an extensive use of geometric patterns that was used to help decorate buildings.  

Around 810 AD, the House of Wisdom was set up in Baghdad and the work done there to translate the major Greek and Indian mathematics and astronomy into Arabic began immediately.  An early director of the House of Wisdom, Muhammad Al-Khwarizmi, made an extremely important contribution to mathematics with his adaption of the Hindu numerical system (numbers 1-9 and 0).  Some 400 years later this was adopted by the entire Islamic world and Europe.

To make sense of Islamic mathematics, I obviously looked online for information.  I came across a website which I found to be extremely easy to follow and shows an example of how algebra and geometry are entwined in a tessellation.  Let a, b, and c each be the color represented in the second triangle.  When you add the color a+b, you are given this new color.  Each different triangle in the second BIG triangle is represented by some algebraic expression. 

Below is the tessellation that I worked on in class.  I had a lot of fun with this activity which is why I decided to research it further. 

Within this blog, I also touched a little bit on the history of the House of Wisdom, as well as the connection between geometry and algebra.  Tessellations are such an interesting subject.  A tessellation is known as pictures or tiles, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.  Symmetry is a big mathematical idea in tessellations.  There are 17 possible ways for a pattern to tile a surface without overlapping or leaving gaps.  There are also four different ways of moving a motif to another position in the pattern: translation, reflection, rotation, and glide reflection.  I absolutely encourage anyone to visit this website (first website listed under source) which was used as a source for the information in this blog post.  It has some awesome examples of tessellations and is filled with even more information.  Below are some tessellations that really appealed to me.

                                                  

Source:

http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html

http://www.storyofmathematics.com/islamic.html