Professor Blanton

Professor Blanton came in to talk to us about how we have a huge problem with Algebra in this country. She showed us a new way in looking at algebra through materials such as the string problem. She showed us clips of her research of children in the lower grades doing algebraic problems.

up until the 1900’s algebra was for the wealthy, common folk did not learn it because we were based off a more agricultural society and didn’t pursue education as strongly.  Only about 3-5% of the population was learning Algebra.

Now we see Algebra in a different light. We are taught with the “Algebra for All,” concept.  Because of the way we go about teaching Algebra all of a sudden starting in maybe middle school but mostly high school it causes a higher drop out rate because students arent grasping these concepts so late. We are taught and drilled in arithmetic so when we learn the concept of Algebra so late, the two have nothing to do with us in our minds and we become more and more confused.

Professor Blanton has been working and researching the “new” Algebra and has gotten high success rates with this more hands on approach. We saw very young kids doing Algebra with no problem, so why dont we implament put Algebra concepts in the classroom at a younger age like we do with arithmetic?

Professor Saeja Kim

Professor Kim was born in Korea. She was very interested in mathematics and got her high school degree in the mathematics field. She gother B.S. at the Seoul National University in Mathematics in 1975. When she moved to the  U.S. she obtained her Masters in Pure Mathematics at Brown University in 1985. She then got her Ph.D. in Communitive Algebra from the University of Illinois at Urbana-Champagne  in 1988.

She has three girls.  She  moved back to  New Englandand worked for Brown as an engineer. In 1990, she began teaching at UMass Dartmouth as an associate professor in mathematics. She talked a lot about her  interests in math. She is specifically interested in Applied Mathematics. She now teaches it here.  She then showed us how to crack codes in cryptology through ciphering and deciphering.

Primes and Perfect numbers

Prime number: a number that can only be divided by 1 and by itself but the number must be a whole, real or natural number

Perfect number: number whose product or sum of its factors are divisible and equal to itself

An example of this is the number 6

6= 2 x 3 or 1 x 6    this sums up to 1+2+3= 6

The first four perfect numbers you can find are  6, 28, 496, 8 and 128

A Conjecture is an idea that there aren’t any ‘odd’ prime numbers

We learned about Goldbach’s conjecture which is a theorem that every even number n greater than or equal to 4 can be written on the sum of two primes

An example of this is 4=2+2, 6=3+3, 8=5+3, 10=7+3 an so on

Mersenne Primes- we  learned how to do the table for finding Mersenne primes

French monk/mathematician found the gaps between primes so  Mersenne Primes were named after him.

The theorem states that if n is an even perfect number, it can be written as:

2p-1=Mp

P          Mp         2(p-1)

2          3           6

3          7           28

5          31         496

7          127       8,128

Cryptography and Mod clocks

To be able to understand cryptography you need to know background of prime numbers.

-there  are  an  infinite number of prime numbers

-  we can assume that there are a finite number of primes and that the last  of all prime number  is P^m.

If your have your variable as Q then it would be

Q = P1 • P2 • P3 • P4 • P5• P6 • P7 etc. etc. until you get Pm + 1

so the Q > Pm.

So if we have  Q/P1 = remainder of 1; Q/P2 = remainder of 1.

It is just plain old Q though with no prime number factors which makes Q itself prime. And if the case is Q is prime than Pm cant be the biggest prime number.

We then learned about Modular arithmetic. We use Modular arithmetic everyday. If you look at a clock the clock clock is really on a 24 hour basis. It cycles back again.

13 = 1 mod 2
13 = 1 mod 3
13 = 1 mod 4
13 = 3 mod 5
13 = 1 mod 6
13 = 6 mod 7

3 = 1 mod 2
3 = 0 mod 3
3 = 3 mod 4

3 = 1 mod 2
4 = 0 mod 2
5 = 1 mod 2
6 = 0 mod 2
7 = 1 mod 2

3 = 0 mod 2
4 = 1 mod 3
5 = 2 mod 3
6 = 0 mod 3
7 = 1 mod 3
8 = 2 mod 3
9 = 0 mod 3

modular arithmetic is used in cryptology. If x mod 5 = 2; x = 7; x = 22. This is all about finding the remainders.  x³ mod p = 2 is what Sophie Germain used to try and find with her primes.

You have to find the remainder of 2.

Cryptology is a means of finding a way to transfer a code between 2 people to deter a third party.

You would first you the letter A Alice and B Bob and E Eve to represent the three parties involved in the code transfer.

The name of this procedure is the Diffy Helman Exchange. Both parties have their own lock and put them on.

Alice takes off her lock and sends the box back to Bob

Bob then takes off his lock.

They then both choose 2 numbers

A and B chose 2 different numbers t 7^x mod 11.

Eve can find out these numbers but she doesn’t know the  function  that is pre-determined a = 7^A mod 11; a = 7³ mode 11 b^A mod 11 = 4³ mod 11 = 64 mod 11 = 9.

When this is complete Alice and Bob can send each other messages without Eve being able to infiltrate the sending of the mesaage

- if a = 7^A mod 11; b = 7^B mod 11 then a^B mod 11 = b^A mod 11.

At the end of this the key is secure for both Alice and Bob without Eve knowing it.

Last Class

Grace Chisholm Young , Sonia Kovalevskaya and Emmy Noether are three women we learned about on the last day. These three women went to Göttingen University in Germany. The university was known for its focus on Math during WWII.

Sonia Kovalevskaya was born in 1850 to a  noble Russian family. She had a lot of struggles trying to pursue her interest in math. She was banned from matriculating at Heidelberg. Sonia audited lectures and later studied privately with Weierstrass, receiving a doctorate (in absentia) from Goettingen University for an important paper on partial differential equations. In 1884 she obtained a position in Stockholm. She then won honors, including the Prix Bordin in France.

Grace Chisholm Young was born in Haslemere, England. She was educated by a governess.  In these times, this was the custom. Grace became involved in working with the poor in London. She dreamed of studying medicine.However her family would not condone it.  She went to Girton College in Cambridge University and studied math. Girton was the first school at the university level that was dedicated to educating women. In 1893, Grace passed her final examinations and got a degree but it was  not considered a formal degree for women.

Grace went to Gottingen in Germany to continue her graduate program because women couldn’t get a graduate degree in England at the time.
Grace married  William Young who was a tutor at Girton a year after she got her Ph.D. They were together for 44 years. Together the couple wrote  more than 200 mathematical papers and books.   She wrote a paper in 1915 on the foundations of calculus. Her paper  won the Gamble Prize at Cambridge. William and her also wrote books on geometry and set theory.
Grace and William had six children. Most of their children also became mathematicians.

Grace also did everything to get a  medical degree,but she didnt do the internship. Grace knew six languages and taught each of her children a musical instrument. Before  World War II, Grace left Switzerland in 1940 to take two of her grandchildren to England. Because the fall of France she could not return and William died alone two years later. Two years after William died Grace died of a heart attack.

Emmy Noether was born in Erlangen, Germany in 1882. Emmy was the oldest of four children, Her father was Max Noether who was a well known mathematician. When Emmy was younger she didnt have very much interest in  mathematics. She studied languages instead. She concentred on French and English. Her mother taught her to cook, clean, and play the clavier. When she finally graduated high school she passed an exam for her to teach Eglish and French to women.

When Emmy was 18 she took classes  in mathematics at the University of Erlangen.  Because she was a woman, the university wouldnt allow her to actually take classes, in stead they let her audit classes. Emmy sat in on classes for two years. She then passed the exam for her to be a doctoral student in mathematics. Emmy passed the test, studied for five years, and was the second woman to be granted a doctoral degree in mathematics.

Emmy then pursued looking for a teaching job now that she had her doctorate. She decided to help her father at the Mathematics Institute in Erlangen. She did research and taught her fathers classes if he was sick. Then she began to publish her work. She worked on further defining one of Einstein’s theories at the University of Gottingen.
While at  the University of Gottingen, she had  a group of students known as Noether’s boys that followed her work. They traveled from as far as Russia just to have the chance to study with her. She was very involved with her students. Her teaching style was very difficult to understand, but those who did became followers. Many of her students when on to become great mathematicians. Noether’s died in 1935. Most were saddened and shocked because only her closest friends knew about her illness

Ada Byron Lovelace’s work

Ada worked in Finite Difference Algebra

-difference between consecutive terms until it becomes one consecutive difference pattern

-it is a recurrence relation

-Her work was related to the puzzle of the Tower of Hanoi.

This puzzle was invented by a French mathematician Édouard Lucas in 1883. The legend is about an Indian temple which has a large room with three  posts in it surrounded by 64 golden disks. The priests of Brahma, who had a command from an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle, since that time. According to the legend, when the last move of the puzzle is completed, the world will end. It is not clear whether Lucas invented this legend or was inspired by it. The reason why puzzles like these is because Ada Byron Lovelace was intrigued with such puzzles.

borrowed from- Ealmeida’s blog

We also learned about the Fibonacci Sequence

- A single pair of rabbits (male/female) are born at the beginning of a year. Assume the following conditions: 1) Rabbit pairs are not fertile during their first month of life, but there after give birth to one new pair of (male/female) at the end of every month, with a two month maturity. 2) No rabbits die.

How many rabbits (pairs)are there at the end of the year.

Fn= # of rabbit pairs alive at the end of n-month

The pattern goes F0=1 F1=1 F2=2 F3=3 F4=5

to get F5 you would add F3+F4 which would be 8.

so F6= F4=F5

F6= 13

F7=21 the last to Fns you have you add together to get the new Fn for the pairs of rabbits alive

Ada Byron Lovelace

Ada Byron Lovelace was born December 10, 1815. Her mother was Annabella Milbanke. Her father was Lord Byron. Her parents were together for about a year before they had her. Lord Byron was a famous poet. Her parent relationship was tumultuous because Lord Byron had a temper. Her mother was very collected and analytical and had a love for math.  Lord Byron left her mother shortly after she was born. Ada was raised by her mother alone raised. Lord Byron  died in Greece  in 1824.
Ada  was mathematician and a violinist and linguist.  Ada was encouraged by her mother to pursue her studies in math.

Ada married  William Lord King at the age of 19. He later became  Earl of Lovelace. William  was 10 years older than her. Despite the age difference they got along well and had a good relationship. William was very proud of Ada’s  achievements. Together they  had 2 children.

Ada began to get to know Charles Babbage. He was her teacher and eventually coworker. Babbage designed and constructed one of the first Difference Engines. It later was called an Analytical Engine. Ada was very interested in  it. She and Babbage had a light hearted and flirty relationship.  She did all she could to help Babbage and achieved outstanding work. Ada  translated work about the Analytical Engine to help explain it in  English so others were able to comprehend it.

The last years of Ada were full of financial issues and health problems, she died at the age of 36. She chose to be buried next to her father because of many similarities despite the fact they couldn’t be together in life. Lord Byron also died at 3

Mary Sommerville’s work

Professor Kim taught us about Chladni Diagrams. The cover of our book is from Mary Sommerville’s work with Chaldni’s work on vibration.

Chladni was a German physicist. He conducted an experiment that shows if you spread fine powder over glass plates, and run a violin bow over the glass plates edges, then a pattern from the vibrations will emerge.

From Mary’s studies she noticed a reflection along the line, which is one type of plane of symmetry.

“To Chladni is due the whole discovery of the symmetry of nodal lines in vibrating plates”

we learned that isometry is when you move two points the distance stays the same

symmetry is is when you move a point on a plane and the other points match back to that shape

a reflection is a mirror image along the line

- if its horizontal it through the center of the square

- if its vertical is through the center of the square

-if its a diagonal line the sides do a flip over diagonally through the square

there are only four reflections of a square

with an equilateral triangle you get 3 reflections

Mary wrote 4 popular books for the people:  Mechanism of the Heavens talks about the motion of a pendulum and how it comes to the conclusion of the shape of the earth. She proposed to investigate the curve of how a particle may move to oxilate the amplitude of the cycloid.

Professor Kim gave us an example with a sprite can. Take make a mark on the edge of the top. Then roll the can across a straight surface, the motion creates continuous overlapping circles.

cycloid- is the curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line

oxilating means repeating

this is all known as the Helen of Geometry, as in Helen of Troy.

Mary Fairfax Grieg Sommerville

Mary Fairfax Greg Somerville was born on December 26, 1780. She lived in Jed burgh Scotland. She was the daughter of Margaret Charters and Lieutenant William George Fairfax. Her father was apart of  the British Navy. he was gone for long periods of time and she did not really know him very well. Mary went to an all girls boarding school in Musselburgh. Mary was not very happy and was not able to obtain a sufficient education because she was a woman. Mary was not what we would call a genius but she worked her way through and strived hard. She first studied simple arithmetic and algebra at the age of thirteen.

At the age of 24 Mary went on to marry her cousin, Samuel Greig. Greig was a member of the Russian Navy when they married. He didn’t have any interest in math or the work Mary was so keen on. Mary and Samuel had two sons named Woronzow and William George. During their third year of being married, Samuel Greig passed away. In light of this, Mary was now able to have the  freedom to do as she pleased. This meant she was able to continue her studies.

Mary was able to  master J. Ferguson’s Astronomy and then became a student of Isaac Newton’s Principia. Over the years she corresponded  with Scotsman William Wallace. Wallace was a mathematics master at a military college. Then in 1812, she re-married. William Somerville, who was also her cousin, became her second husband. Luckily for Mary, Sommerville was supportive of her studies, so she was able to continue. The couple had  four children together during their marriage.

During  the summer of 1825, Mary began to research magnetism. In 1826, Mary produced a paper on magnetism. It was called “The Magnetic Properties of the Violet Rays of the Solar Spectrum”.  In 1834, she published her second book, “The Connection of the Physical Sciences”. In 1835, Mary and Caroline Herschel were the first women to be elected to the Royal Astronomical Society. This was a huge feat for Mary who struggled to do her studies for a good part of her life. Mary continued her studies well into her later years. She became deaf and was able to live until the age of 92, all the while still doing studies.

Sophie Germain

Sophie Germain was born in 1776. She lived in Paris, France. Her father worked at the Bank of France. Sophie started showing signs of brilliance at a young age. She was precocious and wanted to learn as much as she could.  This however did not go over so well with her parents. During this time period, women were not really “supposed” to learn. They were to be charming and witty, but learning wasn’t an acceptable idea for the time. It was believed that if a women was educated it would put a strain on her brain and wouldn’t be able to handle it. Thus, the women would go insane. Her parents loved her very much but highly discouraged her learning.

In 1789, France was in the midst of the French Revolution.  Sophie was thirteen at the time. It was an unstable  time for Paris and  her father feared for his family and would keep them in the house to protect them.  She loved to read and study math and was happy that she could stay indoors with all of her fathers books.  Her parents feared for their daughter’s health because this stress of learning could potentially harm there daughter. They would take her candle away at night  and her clothes as a means to keep her inside her bed so she wouldn’t harm her brain. But they’re efforts failed. They would  find her asleep  at her desk with the ink frozen in her pen because she would refuse to stop studying. So they finally gave up, and let her study, but did not necessarily condone it.

At the age of 18, Sophie was itching to learn more and have actual classes. At the time, women were not allowed in Universities. She had heard of a class taught by Professor LeGrange. She then sneakily found out that a student of his class had dropped out of his class. She then got the idea to take the boy’s name and submit work for the class herself. Sophie then began to write work under the name of Monsieur LeBlanc. Prof. LeGrange was so impressed with this work of this Monsieur LeBlanc he just had to meet him. Well of course he then found out the the one submitting these papers was a women. He thought she was brilliant and introduced her to his colleagues. Though he didn’t believe women should be at the poly-tech University he still tried all he could to help her and give her problems to work on. She was very shy with her work, but she did well all the time.

In 1801 at 25 years old she started a correspondence  with  German mathematician, Carl Friedrich Gauss.  This correspondence lasted for nearly 5 years. Gauss had no idea that Sophie was a women. Most people said he would have nothing to do with her if he knew she was a women. However, during the revolution Hanover was invaded, which is where Gauss is from. A general that invaded there was a friend of the Germain family and Sophie asked him to protect Gauss. When the general said that Sophie had made him protect Gauss, Gauss replied with the fact that he had no idea who this Sophie Germain was. That’s when  Gauss put two and two together, and despite that she was a women supported her because of her brilliant mind.

Sophie Germain ended up getting breast cancer in her early 50’s in the midst of her work.  Sophie had committed her life to her work in math and science and never did marry or have  children.  She was in extreme pain for two years, and then finally died at the age of 55.  Prior to her death, she was awarded a prize from the French academy of sciences.  Gauss was also working on getting her an honorary degree for all her hard work. The major aspects of her study was her work in the patterns of soap solution across a frame, and her work in number theory that included whole numbers and integers which now helps modern day cryptography.