France: Day 17- Cedric Villani

Today is the birthday of Alexis Bouvard born 1767 in Contamines, Haute-Savoie, France. Bouvard is famous for the mathematics he used to discover Neptune.

The quote of the day is by Cedric Villani. In his book, Birth of a Theorem, A Mathematical Adventure, he writes, "Far from moving swiftyly between two points, in a straight line, the mathematician moves forward haltingly, along a long and windy road. He meets with obstacles, suffers setbacks, sometimes he loses his way. As we all do from time to time."

Cedric Villani

Cedric Villani was born 1973 in Brive-la-Gaillarde, France. Villani is currently working on partial differential equations and mathematical physics. He was awarded the Fields Medal in 2010 for his work on Landau damping and the Boltzmann equation.

In the April 14, 2015, edition of The New Yorker, an article entitled The Lady Gaga of French Mathematicians Comes Stateside by Thomas Lin, Lin quoted Villani, "We (mathematics) are the most hidden of all fields. We are the one that typically interact the least with the outer world. We are also the field which is the most emblematic of revulsion in school."

Geoffroy Clavel wrote in the May 7, 2014 edition of the Huffington Post, an articled entitled Cedric Villani, 'The Lady Gaga of Mathematics' Wants To Bring The Joy Of His Discipline To Everyone. In the article, Clavel describes Villani as the current ambassador of mathematics. Villani not only loves mathematics, "he also wants to convince the wider public that this dry subject can be fascinating - - as long as you know how to talk about it."

I have another item to add to my bucket list, meeting Cedric Villani, perhaps on my next trip to France. Passez une bonne journee mes amis, jusqu'a la prochaine fois.

France: Day 9 - Chatelet

Today is the birthday of Blaise Pascal born 1623 in Cleremont-Ferrand, France. At the age of 19, Pascal invented the first mechanical calculator that was sold commercially. In physics, he proved that air had weight and that vacuums are possible in nature. In mathematics, he developed an early form of integral calculus and cofounded with Fermat, probability theory. Eventually, Pascal lived as an informal hermit producing two works of religous philosophy: Provential Letters and Thoughts.

Today's quote is written by Voltaire in a correspondance to King Frederick II of Prussia. He wrote that Chatelet "was a great man whose only fault was being a woman."

Emilie du Chatele

Emilie du Chatele was born December 17, 1706 in Paris, France. Chatele was a noblewoman who made contributions in the areas of philosophy, natural science, and mathematics. 

In her writings, she challenged John Locke's philosophy. She was adamant that knowledge could only be verified through experience.

In natural science, she specifically focused on fire. She predicted that there was a special light that emanated heat on objects. This special light is now known as infrared radiation.

In the mathematics, she is most well known for translating Isaac Newton's Principia Mathematica. This translation led French scientists to discard Cartesian physics and adopt Newtonian which was highly controversial at the time. She also corrected Newton. Newton had shown that energy of a moving object was proportional to the mass times the velocity of the object. Chatele demonstrated that energy was proportional to the mass times the square of the velocity of the object. Chatele's work on the relationship between energy and velocity inspired Albert Einstein to formulate the equation, E = mc^2.

Emilie Chatele was born during the Age of Enlightenment. Her father was a courtier for King Louis XIV and her parents used this advantage to educate her in languages, mathematics, and the sciences. Her mother strongly encouraged her to question any stated fact. This encouragement helped foster an independent thinking daughter. Her independence was present in her intellectual works as well as her personal life. She had an intellectual and intimate relationships with Voltaire and the philosophers, Maupertuis, and La Mettrie. She was independent and original woman. She was her own person.

France: Day 8 - Fermat

Today is the birthday of Frieda Nugel born 1884 in Cottbus, Brandenburg, Germany. Nugel was one of the first women to receive a doctorate in mathematics in Germany.

Today's quote is by Pierre de Fermat. He said "I am more exempt and more distant than any man in the world."

Pierre de Fermat

Pierre de Fermat was born August 17, 1601 in Tarn-et-Garnone, France. Fermat, a lawyer, was influential in the early developments of calculus and made significant contributions in the areas of analytic geometry, probability, and optics.

Thirty years after his death, a handed written note by Fermat was found in the margin of a book entitled "Arithmetica" written by Diophantus.  Fermat wrote, "It is impossible for any number which is a power greater than the second to be written as a sum of two like powers.  I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." His quote means that there are no three positive integers, x, y, and z, where x^n + y^n = z^n and n is an integer greater than 2. This "marvelous demonstration" the Fermat refers to, was a mystery for 358 years, spawning a branch of mathematics called algebraic number theory and the modularity theorem.

Two examples I can give are: 3^1 + 4^1 = 7^1 and 3^2 + 4^2 = 5^2 and there are infinite number of examples for powers of 1 and 2 but none greater than 2. In 1994, Andrew Wiles successfully proved this mysterious conjecture that had baffled mathematicians for almost four centuries.

Fermat had made claims of original proofs on many of his theorems but only a few of those proofs are in existance. Many mathematicians doubt that those proofs existed due to the difficult nature of his theorems and the limitations of the mathematics at his time.

What type of man was Fermat? His contributions to mathematics are immense. He and Descartes are considered the emminent mathematicians of their lifetimes. Why did he choose to write that note in the margin of his father's book? Did he feel a need to overexagerrate his abilities? Was he concerned about his legacy and wanted a theorem that would live for centuries? Was he a prankster and some how knew that his brief commentary would drive future mathematicians to the edges of their own sanities? For myself, Fermat is the true mystery.

France: Day 2 - Germain

Today is the birthday of Paul Guldin born 1577 in St. Gall, Switzerland. Guiding made contributions in the areas of volumes and the center of gravity.

Today's quote is by Carl Friedrich Gauss. In a letter to Sophie Germain, Gauss wrote, "The enchanting charms of this sublime science reveal to only those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounter infinitely more obstacles than a man in familiarizing herself with complicated problems, succeeds nonetheless in surmounting these obstacles and penetrating the obscure parts of them, without doubt she has the noblest courage, quite extraordinary talents and superior genius."

Sophie Germain

Sophie Germain was born April 1, 1776 in Paris, France. When she was 13, the Bastille fell and as a result she was required to stay inside. During this isolation, she taught herself Greek, Latin, and mathematics. Her parents disapproved of her passion for mathematics which at this time was considered an inappropriate field of study for women. Her parents attempted to restrict her studies by eliminating the fire in her room and by removing her clothes. However, Sophie's parents relented when they found her asleep with a frozen ink horn in hand and a slate of equations on her desk. 

Sophie was not allowed to attend to attend an university but was able to obtain lecture notes. She started send comments on the lecture notes under the pseudonym Monsieur Antoine-August Le Blanco. Using this pseudonym, she established a relationship with Carl Gaus and Joseph-Louis Lagrange, two prominent mathematicians. 

Germain made contributions in the areas of elasticity theory, differential geometry, and number theory. A Germain prime number and her additional work on Fermat's Last Theorem enhanced the exploration of the subject for hundreds of years.

I became aware of Sophie Germain through the movie, Proof. I wonder if the gender bias that has been prevalent in the STEM occupations has improved. Our mathematics department consists of 9 teachers, 5 of which are female. I wonder about nonteaching occupations and the pay inequity that exists. I know of at least 10 female, former calculus students that are deeply involved in STEM occupations. I wonder how they are coping?

France: Day 1 - Descartes

Today is the birthday of Charles Reyneau born 16546 in Brissac, Maine-et-Loire, France. Reyneau published one of the first influential textbooks on the newly invented calculus.

Today's quote is by Rene Descartes entitled, Cogito ergo sum. "... Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all objects (presentations) that had ever entered into my mind when awake, had in them no more truth than illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth; I think, therefore I am, was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the skeptics capable of shaking it, I counted that I might, without scruple, accept it as the first principle of the philosophy of which I was in search."

As witnessed when I watch a recent news segment, I am on "assignment" which means I am on vacation, and will be writing about my five favorite French mathematicians; Rene Descartes, Sophie Germain,  Pierre de Fermat, Emilie du Chatelet, and Cedric Villani.

Rene Descartes

Descartes was born on March 31, 1596 in La Haye, France. As a student, he studied rhetoric, logic, and the "mathematical arts". "Mathematical arts" included mathematics, music, astronomy, metaphysics, natural philosophy, and ethics. He attended the University of Poitiers and obtained a law degree.

After graduation, he studied medicine and theology. He traveled and joined the army. During his stint in the army, he met scientist and philosopher, Isaac Beeckman. Beeckman convinced Descartes that he should apply mathematics and logic to understand the natural world.

Descartes is considered the father of modern philosophy. He believed that all truths are ultimately linked and that the rational structure of mathematics and science could be used to uncover the meaning of the natural world.

Descartes introduced Cartesian geometry which is a fusion of geometry and algebra. I call Cartesian geometry, coordinate geometry. I love coordinate geometry and I believe its application allowed me to have an easy transition in to trigonometry and polar coordinates. Descartes also developed laws of refraction and formed a practical understanding of rainbows. Pope Alexander VII placed Descartes' writings on the Index of Prohibited Works.

Deriving While Driving

Today is the birthday of Max Zorn born in 1906 in Krefeld, Germany. Zorn is responsible for Zorn's lemma which states that if a partial ordered set exists and any subset of that set has an upper bound then the original set has a maximum element.

The quote of the day is by Srinivasa Ramanujan. Replying to G.H Hardy's suggestion that the number of a taxi (1729) was "dull", "No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being (1 + 1728) and (729 and 1000)."

A week ago, our school held its commencement for the class of 2015. During the week prior to graduation, I spent a great deal of time with the seniors that were in my class and in the organizations that I advised. Our conversations were reflective on their growth and their excited and nervous anticipation of their upcoming educational journeys. Each class attempts to leave a legacy their senior year and some classes do have a more resilient legacy to the corrosiveness of time.

Each numerical year seems to have its own identity. My high school graduation year, 1976, was the bicentennial year. The year 1984, was wrapped in the Orwellian concept of Big Brother. 1999 was made famous by the singer, Prince. The year 2000 was the millennial year. 2008 had the 8 rotated so that it became the infinity symbol. Members of the class of 2012 asked the question, "Don't you want to be 1  2?" 2013 ends in 3 and is also divisible by 3! 2015 was the year of π because on March 14 at 9:26 am, the first eight digits of that irrational number occurred. 

I was contemplating these numerical wonderments at a stoplight on my way to school the day after the commencement ceremony and was drawn to the class of 2016. In my meandering thoughts, I jumped to the factorization of the 2016 and was shocked in my determination that 2016 has 36 factors. Factors are natural numbers that 2016 can be divided by without any remainder. The factors of 2016 are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. I started to wonder about the number of factors in each of the years that I have taught. The chart below depicts those graduation years and the number of their factors.

To determine the number of factors, I found the prime factorization of each year. The prime factorization of 2016 would be 2 x 2 x 2 x  2x 2 x 3 x 3 x 7 or in exponential form, 2^5 x 3^2 x 7^1. I used the exponents in the exponential form to find the total factors that exist in 2016. The total number factors can be found by using the following arithmetic: (5 + 1) x (2 + 1) x (1 + 1). Using my high school graduation year, 1976, I found that its prime factorization is 2^3 x 13^1 x 19^1. The exponents are 3, 1, and 1. The number of factors in 1976 is 16; [(3 +1) x (1 + 1) x (1 + 1)].

When I examined the list, I noticed there are five prime years, 1987, 1993, 1997, 2003, and 2011. I believe the next graduating class that will have exactly 36 factors will be the class of 3168. The prime factorization for 3168 is 2^5 x 3^2 x 11^1. The last graduating class to have 36 factors was the class of 1800, 2^3 x 3^2 x 5^2. I was not teaching in 1800 and am not planning on teaching in 3168!

2015 Calculus Presentations

Today is the birthday of James Hutton, born 1726, in Edinburgh, Scotland. Hutton is famous for his theory of the age of the Earth.

The math quote of the day is by Rene Descartes who said, "Perfect numbers like perfect men are very rare."

Every year the final for my calculus classes consists of finding a partner, agreeing on a topic of interest, finding the mathematics that exists in it, and creating a presentation on that topic. This year's classes formed twenty presentations.

I have linked the title of each presentation to its Youtube site or the reader may visit my Youtube site: Charles Kruger.

The topics are: Aquariums, Aviation, Car Audio, Computer Animation, Dance 1, Dance 2, Discus, Farm, Golf, Guitars, Hydroelectric Power, Luminance in Art, Natural Disasters, Photosynthesis, Rollerblading, Soccer, Swimming, Tangled, Theory of Relativity, and Track & Field.

I enjoy these presentations for many reasons. Students take ownership of their presentations. The students and I usually learn something new. The students enjoy finding the mathematics in their topics. The presentations allow the students to be creative. The presentations are a showcase or a capstone of what the students have learned. The students use skills they have learned in other classes.

Enjoy!