Wednesday, June 25, 2014

Magic Squares

"In recreational mathematics, a magic square is an arrangement of distinct numbers (i.e. each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number."

Srinivasa Ramanujan (1887-1920) was an Indian mathematician and autodidact who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.  The mathematical community during his time was largely located in Europe so he pretty much developed his own mathematical research in isolation.  On of Ramanujan interesting creation was his super magic square, which looks like this:


What's amazing about Ramanujan's magic square is that not only do all the rows, columns, and diagonals sum to 139 but also the four corners,  the four middle squares, the first rows two middle numbers and the last rows middle numbers as well as the first columns two middle numbers and the last columns middle numbers, and all the four squares that make up each corn.  The best part is that the top row is Ramanujan's date of birth!  Altogether this makes his magic square a super magic square. 

I wanted to make a normal magic square with my birthday as the top row and I came close since the rows and diagonals sum to 134 but not the columns.  The weird part is that the sum of the first column is 123 which is a difference of -11 from 134, the second columns sum is 119 with a difference of -15, the third columns sum is 143 with a difference of 9, and the fourth columns sum is 151 with a difference of 17.  If you add the first two columns together you get -26 and if you add the last two columns together you get 26.  Adding them together gives you 0....strange!
My almost magic square with my birthday as the top row:


To create my magic square I simply put my birthday in the top row and either subtracted or added the difference of my birthday numbers from Ramanujan's.  For example, the first square in my first column is 18 which is 4 less than Ramanujan's first square in his first column; so I subtracted 4 from every number in Ramanujan's first column.  My second number is 7, which is 5 less than Ramanujan's square which is 12 so I subtracted 5 from every number in Ramanujan's second column. I did this until I finished with the last column. Then I tried playing around with the numbers to get the columns to add to 134 and that's when I noticed if you add the differences of the sums of the columns you get zero.




I switched books with Sarah Keilman and skimmed e: The Story of a Number by Eli Maor.  My first impression was it was dry and I had a hard time staying awake to read.  At times it seemed more like a textbook, which I don't typically enjoy reading.  I like narratives better like the first book I read, Love and Math: The Heart of Hidden Reality.  The history of a couple mathematicians was interesting and fun to read; some mathematicians had interesting traits that I didn't know about and I enjoyed reading about them.  I think the number e and its connection to nature is just fascinating so I enjoyed seeing the connection in sunflowers and other aspects of nature.  Also e's connection with interest earned on bank accounts and arches is interesting too.  The number e seems too perfect to me but that makes it intriguing as well.  The book was very informative and had a lot of math so I think it was meant for readers with a mathematics background.  Overall it was a good skim, and I learned a lot! 

Tuesday, June 17, 2014

Women and Mathematics

Mathematics has undoubtedly been a male-dominated field. "Being a woman in mathematics was hard because it was a lose-lose situation.  If women dedicated themselves too much to mathematics and their careers, they were judged harshly as a woman.  But if they embraced the more stereotypical roles and responsibilities of a woman by getting married and having a family, they were not taken seriously as a mathematician" (Henrion, pg. 69). Women were not accepted in the field of mathematics until the twentieth century; however there have been a number of great women mathematicians before then who took the risk of being scrutinized.  Three of the first great woman mathematicians were:

1. The first was Hypatia of Alexandria (370-415).  She was a Greek philosopher, astronomer, and mathematician.   In the year 440 she became head of the Neoplatonic School in Alexandria, Egypt.  Hypatia learned mathematics from her father,a teacher in mathematics, and later by many others. Although none of her written work as survived, she has a commentary on the 13-volume of Arithmetica by Diophantus, commentary on the Conics of Apollonius, edited the existing version of Ptolemy's Almagest, edited her father's commentary on Euclid's Elements, and wrote a text called "The Astronomical Canon".  
Hypatia Raphael Sanzio detail-2.jpg

2. Elena Cornaro Piscopia (1646-1684) was an Italian mathematician, philosopher, and theologian.  Piscopia was a child prodigy who became the first woman to earn a doctoral degree.  She got her degree at the University of Padua in theology and later became a lecturer there in mathematics.  She was the first and last woman to receive her doctoral degree from the University of Padua until the late twentieth century. She also studied many languages including Hebrew, Spanish, French, and Arabic, composed music, sang, and played many instruments. 


3.Maria Agnesi (1718-1799) was an Italian mathematician, philosopher, and philanthropist who was the oldest of 21 children and was considered to be a child prodigy.  She wrote a textbook for her brothers, explaining mathematics which later became the first mathematics book by a woman that still survives today.  The first volume of her book covered arithmetic, algebra, trigonometry, analytic geometry, and calculus.  The second volume covered infinite series and differential equations.  No one before her had published a text on calculus that included the methods of calculus of both Isaac Newton and Gottfried Liebnitz.  She became the first woman appointed as a mathematics professor at a university, the University of Bologna, which was an act on Pope Benedict XIV.  Agnesi also was recognized by the Habsburg Empress Maria Theresa of Austria.  

Maria Gaetana Agnesi.jpg

These women faced discrimination and oppression yet overcame all obstacles to dive into the depths of mathematics.  They were all brilliant and very brave for having been involved in the field of mathematics which was overwhelming dominated by males.  If it hadn't been for these women and others before and after them, women may have never had the courage to stand up to men and expand their knowledge in the mathematics field.  Because of these women I was able to study mathematics without even considering it being a big deal.  But for these women it was and I can't imagine the suppression they endured based on gender alone.  In 2010, women earned 57.2% of bachelor degrees in all fields, 50.3% being in the sciences and engineering fields.  This is a huge accomplishment for women and without the many brave women before us, this dream may never have been a reality. 


Tuesday, June 10, 2014

Love and Math: The Heart of Hidden Reality by Edward Frenkel: Book Review


The book Love and Math: The Heart of Hidden Reality by Edward Frenkel was about a young mans (Edward Frenkel) journey into the depths of mathematics.  Frenkel grew up in Kolomna, Russia, the son of two engineers.  In school, Frenkel hated mathematics and had a passion for quantum physics until a family friend (Evgeny Evgenievich) converted him to mathematics, where he eventually fell in love with it.  Evgenievich was Frenkel's first, and one of many, guides into the world of mathematics.  After high school Frenkel decided to apply to Moscow State University where antisemitism raged rapidly at the time (in the 1980's).  Being Jewish, Frenkel knew he had little hope getting in but applied anyways. Frenkel encountered extreme and unjust antisemitism during his entrance exam and didn't get in because of it.  Frenkel had no choice but to go to one of the only colleges in Moscow that wasn't antisemitic, Kerosinka, in hopes of pursuing his dream of becoming a mathematician.  Frenkel had many teachers and guides giving him research projects including Yakov Isaevich, Borya Feigin, Victor Kac, and Vladimir Drinfeld just to name a few.  Vladimir Drinfeld was perhaps the most influential, since it was him who showed Frenkel the way into the Langlands Program.  Frenkel's dream of becoming a mathematician came true at an early age because he had so many great mathematician realize his potential and educated him further in his research.  After college Frenkel had no hope of going to graduate school in Russia but he was becoming so prominent in the mathematics field that Harvard University invited him to accept a mathematics award.  He decided to stay at Harvard, completing his Ph. D and later became an associate professor.  Because of help from DARPA (Defense Advanced Research Projects Agency) Frenkel and his colleagues were granted millions to further research the Langlands Program.  The langlands programs big question was why the electromagnetic duality leads to the same langlands dual group that mathematicians discovered in a totally different context?  Frenkel has published many works and has won prestigious mathematics awards for his hard work and brilliance.  After Harvard University, he decided to become a professor of mathematics at the University of California, Berkeley where he currently resides.

I thoroughly enjoyed Love and Math: The Heart of Hidden Reality and highly recommend it to everyone I encounter.  He allows you to see mathematics in a light never before seen and in a understandable way for any audience.  I am a pure mathematics major and reading this book just opened my eyes to the true beauty of mathematics that I hadn't realized before.  Frenkel's passion and love really came through in this book.  He shows that mathematical formulas express an eternal truth about the universe and this truth belongs to everyone, everywhere.  Frenkel says it best when he writes, "mathematical formulas are some of the purest most versatile, and most economical expressions of truth known to man.  They convey timeless and precious knowledge, unaffected by fads and fashion, and impart the same meaning to anyone who comes in contact with them.  The truths they express are the necessary truths, steadfast beacons of reality guiding humanity through time and space."  I love the fact that Frenkel sees mathematics as an art form and also inevitable truths about our world.  His words are truly inspirational.  

Some of my favorite thought-provoking quotes from the book are:

Albert Einstein wrote, "How can it be that mathematics begin after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?"

Kurt Godel wrote, "mathematics describes a non-sensual reality, which exists independently both of the acts and of the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind."

Edward Frenkel wrote, "each new result in math pushes back the veil covering the unknown."

Heinrich Hertz wrote, "one cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers."  

Edward Frenkel wrote, "mathematical truths seem to exist objectively and independently of both the physical world and the human brain.  There is no doubt that the link between the world of mathematical ideas, physical reality, and consciousness are profound and need to be further explored." 

David Thoreau wrote, "we have heard about the poetry of mathematics, but very little of it has yet been sung."

This last quote is one of my favorites because it seems so true.  Very few know the true art form of mathematics and it needs to be shown to more of the population in an understandable way.  Frenkel recognized this so wrote a screenplay and created a movie with friend, Reine Graves, called Rites of Love and Math as an allegory showing that a mathematical formula can be beautiful like art.  In this movie the main character finds the formula for love but it also can be used for evil, so he has it tattooed on his girlfriends body to keep it safe and eternal.  

We have only discovered very little of the puzzle of mathematics and are just touching the surface of putting some of the separate pieces together.  I cannot even come close in imaging what the whole picture will look like.  I only wish I could be around for the day the puzzle is complete and the true beauty of mathematics is revealed to the world for everyone to admire.      

Tuesday, June 3, 2014

History of Math: Fibonacci, Ars Magna


"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."

Leonardo Pisano Bigollo or better known as Fibonacci was an Italian mathematician born in the twelfth century in Pisa, Italy.  He traveled around the world to places like North Africa, Algeria, Egypt, Syria, Greece, Sicily, and Provence. When he returned from his travels, he wrote a mathematical book called Liber Abaci (Book of Calculation) relaying what he had learned from his travels.  In Liber Abaci, he is best known for spreading the Hindu-Arabic numeral system in Europe in 1202 and also for a number sequence called Fibonacci numbers.  Fibonacci wrote 5 mathematical works including 4 books and one letter. 


Fibonacci recognized that "arithmetic with Hindu-Arabic numerals was simpler and more efficient than Roman numerals" so he wrote Liber Abaci introducing "numeration with the digits 0-9 and place value."  He showed how "commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other applications" could be done simpler using the Hindu-Arabic numeral system rather than Roman numerals.  Fibonacci therefore, revolutionized the way we do math today. Can you imagine if we still used Roman numerals today? I don't think mathematics would have advanced as fast as it did without Fibonacci's contribution.
Also in Liber Abaci, Fibonacci introduced a problem involving growth among rabbits and solves it using a sequence or Fibonacci numbers.  This sequence was used by Indian mathematicians around the 6th century but it was Fibonacci who wrote about it and introduced it to the West.  Therefore, Fibonacci did not create the sequence he merely passed it on and was later named after it.  The amazing fact about the Fibonacci sequence is that it can be found in nature among bees, shapes of shells, and the spirals of seeds in a seed head (sunflowers).  Out of the Fibonacci sequence comes the golden ratio which comes from taking two successive numbers in Fibonacci's series, dividing each by the number before it, plotting these values and seeing that the numbers are tending to a limit known as Phi.  Dr. R. Knott puts it best when he says "...just as the ratio of successive Fibonacci numbers eventually settles on the golden ratio, evolution gradually settled on the right number too. The legacy of Leonardo Pisano, aka Fibonacci, lies in the heart of every flower, as well as in the heart of our number system."  

Ars Magna:

The Ars Magna or "The Great Art" is a book on Algebra written by Italian Renaissance mathematician, physician, astrologer, philosopher, and gambler, Gerolamo Cardano.  He wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music.  The Ars Magna was first published in 1545 and was titled Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra).  Gerolamo wrote a second edition published in 1570 which is considered one of the three greatest scientific treatises of the early Renaissance. 

The Ars Magna has forty chapters which contains the first published solution to thirteen different types of cubic equations and quartic equations and was the first to mention multiple roots and complex numbers.  Although it contains the solution to cubic equations, who discovered the solution first is still debatable.  In 1535, Niccolo Fontana Tartaglia was famous for solving cubics in the form of x^3+ax=b.  However, he kept his method a secret: that is, until Cardano convinced Tartaglia to reveal his solution by making Cardano promise not to reveal or publish it.  Cardano spent the next two years working on expanding Tartaglia's formula while Cardano's student, Lodovico Ferrari, discovered a way of solving quartic equations that relied on Tartaglia's cubic solution.  However, Scipione del Ferro, an Italian mathematician also solved cubic equations and claimed to have before Tartaglia himself.  When Cardano became aware of this he published Ars Magna and acknowledges that Tartaglia gave him the formula for solving cubic equations and that Scipiano del Ferro also discovered the same formula at the same time.  Because Tartaglia was so secretive of his solution and there are no surviving scripts from del Ferro, who really came up with the solution to cubic equations first is still a mystery. 


Tuesday, May 27, 2014

History of Mathematics: Zero and Brahmagupta

" Zero is both a number and the numerical digit used to represent that number in numerals."

The number zero first appeared in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe) in 628 AD.  Brahmagupta was an Indian mathematician and astronomer who never wrote any proofs for his work so it is unknown how his mathematics were derived.  Most of his works were composed in elliptic verse so it has a poetic ring to them which was common at that time in Indian mathematics. Brahmagupta did a number of important works including algorithms for square roots, the solution of quadratic equations, and the area of a cyclic quadrilateral, known as Brahmagupta's Formula. Brahmagupta established a set of rules for numbers and the modern version includes:
  • The sum of zero and a negative number is negative.
  • The sum of zero and a positive number is positive.
  • The sum of zero and zero is zero.
  • The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
  • A positive or negative number when divided by zero is a fraction with the zero as denominator.
  • Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. 
  • Zero divided by zero is zero.
As you can see, Brahmagupta not only mentions zero but also basic mathematical rules for computing zero as well.  According to Brahmagupta, zero divided by zero is zero which was then accepted for centuries later.  In the 12th Century, almost 500 years after Brahmagupta, Indian mathematician Bhaskara II showed that zero divided by zero should be infinity because 1 can be divided into an infinite number of pieces of size zero.  This answer was considered correct for centuries as well, until the modern view, that any number divided by zero is undefined, was established.  Zero divided by zero is still a controversial topic discussed today by many mathematicians. What do you think the answer should be?

My personal opinion is that zero divided by zero is undefined so I agree with the modern view.  I believe it is undefined because zero divided by zero it is like asking "how many zeros are in zero"; are there 1, 0, infinitely many? Since this question doesn't really make since, I believe zero divided by zero is undefined.