Exemplars

History of Math: Janos Bolyai

There are so many mathematicians that have made huge contributions to mathematics. One of the mathematicians that I find the most interesting is Janos Bolyai. Bolyai was a Hungarian mathematician with an interesting story. His father, Farkas Bolyai, wanted Janos to grow up to be a mathematician. In 1816, Farkas wrote to his friend and great mathematician, Carl Gauss, and asked if Janos could study under him. By 1817, Janos went to Calvinist College in Marosvásárhely to study under Gauss. 

Around 1820, Bolyai started to take after his father and work on Euclid's fifth postulate. The goal for Bolyai was to change to postulate so that it could be derived from the other postulates. In 1823, Janos wrote to his father saying

"...created a new, another world out of nothing..."

When Janos was trying to change Euclid's fifth postulate to be easier to use, he instead found the "opposite" of Euclidean geometry which is now known as non-Euclidean geometry. 

Non-Euclidean geometry is essentially the opposite of Euclidean geometry. One of the main forms of non-Euclidean geometry that is used is hyperbolic geometry. This is the type of non-Euclidean geometry that Janos Bolyai discovered. What makes these two geometries so different is only caused from one difference. The meaning of “parallel” in these two geometries negate one another. In Euclidean geometry, there exists exactly one parallel line through a point that is parallel to a line. In non-Euclidean geometry, there can be no parallel lines or more than one parallel line through a point that is parallel to the original line.

Whichever axioms or postulates are true in strictly Euclidean geometry, the negation of that axiom or postulate is true in hyperbolic geometry. A lot of hyperbolic geometry is done on a Poincare Disk Model to show the curvature of the geometry. 

Although Bolyai had studied under Gauss and considered him a friend, when Bolyai told Gauss about his discovery Gauss was unimpressed. According to Gauss, he had discovered everything that Bolyai had been working on and said that he never wrote it down. Bolyai was so distraught by this that he gave up all of his work in math and left the country. He then became a very bitter man who was hard to be around. 

Today, people around the world respect non-Euclidean geometry as a field of study. Bolyai is often associated with the discovery of non-Euclidean geometry along with Nikolai Lobachevsky and also Carl Gauss.

Sources

Janos Bolyai Biography-http://www-history.mcs.st-andrews.ac.uk/Biographies/Bolyai.html

Communicating Mathematics: The Math Book Book Review

For the past several weeks, I have read The Math Book by Clifford A. Pickover. Overall, I like the idea of this book. It is a book of the 250 most important mathematical innovations. Each page gives you a new and exciting discovery in math. This all happens in the order they were discovered.

I feel as though the discoveries at the beginning of the book were way more interesting than the later ones. This is probably because the discoveries at the beginning lead towards the discoveries that happened later in history.

The invention that I felt was the most interesting was the simple game of Tic Tac Toe. This game is traced back to 1300 BC.  It is considered the "atom" of board games because many other games are based off of Tic Tac Toe. Interestingly, there are 362,880 ways to place X's and O's. Of these possibilities, there are 255,168 possible games that can be played that end in 5,6,7,8, and 9 moves.

Along with Tic Tac Toe, I found the pages that were on items such as the Mobius Strip and the Klein Bottle were interesting. They are objects that are in at least three dimensions but they are only one sided objects.

This book shows us many interesting discoveries in math and tells us everything they can in a one page summary.

One of the major flaws with this book is that the author deviates from the math quite often. He tells us a lot about the mathematicians as well. The flaw comes in the way he does this. He touches on the families of the mathematicians but only seems to do it for the female mathematicians. He also points out the religions of the mathematicians if they are not a white Christians. This happened mostly when they were Jewish. I feel as though it is demoralizing being a female mathematician that he would say this stuff. It is more important to talk about their contributions than to tell us how many children the females have or of what religion they are.

Overall, I do believe that this book has a good idea about it. I just feel that it would be better if it stuck more to the math than the other stories that aren't of any importance.

Doing Math: The Pythagorean Theorem for Winning Percentage

Baseball season is officially upon us. At the start of every season, baseball fans everywhere talk about how it's their team's year to win it all. But how do we really know whose year it will be with 161 games left to play in the regular season? There are many ways to try and determine who the next world champ would be. Here is just one of those options.

The Pythagorean Theorem for Winning Percentage

Everybody knows the famous Pythagorean Theorem: a^2+b^2=c^2. So how does this help us to determine a winner? Well with slight modifications, Bill James created the Pythagorean Theorem for Winning Percentage based off of the well-known formula so that it states,

W%=[(Runs Scored)^2]/(Runs Scored)^2 + (Runs Allowed)^2].

This formula does not base the winning percentage of a team based on just their Win-Loss record. It gives us a better understanding of how well a team is actually doing based on the runs that they score and the runs that they allow the other team to score.

Many expert analyst are currently saying that the L.A. Dodgers are going to be the 2014 World Series champions. Let's looking into this a little further. According to MLB.com, the Dodgers had a spring training record of 98 runs scored and 118 runs allowed. Using the formula, we can say

W %=[( 98) ^2]/ (98) ^2 + (118) ^2]
       =.4082.
This would mean that we can expect the dodger to win 40.82% of their games. According to MLB.com, the actual win percent of the dodgers was .368 or 36.8%. Now let's take a look into the Red Sox spring training results. They had a record of 117 runs scored and 149 runs allowed. When plugging these values into our equation we obtain

W %=[( 117) ^2]/ (117) ^2 + (149) ^2]
       =.3814.
This would mean that we can expect a winning percentage of 38.14% from the Red Sox. According to MLB.com, the Red Sox had a winning percentage of .393 or 39.3%.

As we can see, this formula is pretty accurate at calculating expected winning percentages for baseball. The math that is done here is working with spring training statistics which don't necessarily show us an accurate representation of how the season will actually unfold. In spring training, there are so many factors that change in the regular season that make it not an accurate representation of the regular season. Many of the elite teams such as the Red Sox do not use many of their starters or their normal starting lineup. This is so that the manager can see who works well together on the field. They also want to “test out” the AAA minor leaguers with the major league players. This is in case of an injury in the regular starting lineup. The managers want to know that if a key player is injured, there is somebody in the farm system that could be put into that spot and perform just as well. Having these low stats in spring training doesn’t give us a representation of what will happen during the regular season. It does show us that the managers are trying to find the perfect combinations of batting order and defensive plays. It also shows us that there is a good chance that many of the AAA minor leaguers are ready for the majors and can be called up at any moment. Once the regular season really gets underway, we will more accurately be able to pick who will win the 2014 World Series. In my opinion, the Red Sox will make it back to back champions.

Source

www.MLB.com

History of Math: Emmy Noether

Emmy Noether was an influential female mathematician. She was so intelligent that Einstein said she was a genius. Not everything is Noether’s career was so great though. Her father disapproved of her wanting to learn and study math. College prep schools did not want her to study math either. Girls were not allowed to attend a college prep school. This did not stop Noether from doing what she wanted to do. She took a test to get into college anyways and received her PhD in math in 1907. With her degree, Noether worked at the Mathematical Institute at Erlangen without pay or a title. She wanted to teach so badly that she did it anyways. During this time, Noether was able to work with other mathematicians on different projects and topics. Some of the mathematicians she worked with were Klein and Hilbert. These three worked together on Einstein’s Theory of Relativity. Emmy Noether even has a theorem named after her, Noether’s Theorem. Even through all of this, we was not able to teach at a university because of her gender. In 1919, she gained permission to lecture on her own but again, it was without salary. She worked on group theory and abstract algebra that we use in today’s physics. She wanted to teach in Germany but wasn’t allowed to by the Nazi’s and ended up have to leave Germany so that she could eventually teach.

Being a female in mathematician, I find it absolutely ridiculous that so many men would go to the extremes to not let a women in. Even if they are as brilliant as Emmy Noether was. It is sad that this still happens today. Luckily not nearly to the same extent as it has been in the past. Great mathematicians such as Noether and Sophie Germain carved the way for mathematicians such as myself to be able to study this wonderful field. Emmy Noether is inspirational to me because she continued to fight for what she wanted to do and make herself happy despite all of the negativity that she faced trying to do so.

Source

www.sdsc.edu

How Infinity Blows My Mind

Infinity is a number, oh wait, no its not. Or is it? This is one of the many debates still causing mathematicians problems. What really is infinity? To me, infinity is an idea. It is a something that we can never actually reach but we pretend that it is there.

I recently watched a lecture by Manil Suri that blew my mind. He talked about different paradoxes that involve infinity. The first one was the Hotel Infinity. If every room in the Hotel Infinity is booked and somebody comes in and asks for a room. The receptionist says "Here's your room key" and gives the man a key. How is this possible if the hotel is booked? Well the answer is surprisingly simple. Instead of making the man walk down the hallway to his room, they make everybody in every room move down one. This way the first room in the hotel is vacant for the man and the man won't die making it up to his room.

Another concept the Suri talked about that we covered in class is that when we look at the set of the natural numbers versus the set of the even natural numbers, which one is bigger? They each go to infinity right? The more obvious choice would be to choose the set of natural numbers since there are twice as many as just the even natural numbers. In fact, we are able to match the set of even natural numbers in one to one correspondence with the natural numbers such that 2 matches up with 1, 4 matches up with 2, 6 matches up with 3 and so one. In this case, they both go to the same infinity and neither set is larger! So that leads to the question, is there any set of numbers that go larger than infinity? The answer is yes. The set of real numbers from 0 to 1 is larger than infinity. This is because there are so many real numbers between 0 and 1 that it is impossible to set them up in a one to one correspondence with any other set. This then causes the cardinality of this set to be larger than any other cardinality that we can find.

The idea of infinity was credited to Georg Cantor. Cantor's work was mostly in set theory. This is how the thoughts of infinity started to come about. He started to ask questions about continuity and infinity. Working with one-to-one corresponding sets made him start to think that they could go on forever. This idea did not sit very well with most people around his time.

Infinity is such a crazy number or idea. It is hard to wrap my head around something so large that once you do, infinity is actually bigger than that.

http://www.britannica.com/EBchecked/topic/93251/Georg-Cantor

Book Review; e: the Story of a Number

e: the Story of a Number is a book about the number e. Each chapter is a new time period in which e is found or used.

My favorite chapter talked about the "forefather of Calculus. They talked about Leibniz and Newton and how they contributed to calculus and logarithms.What I found was the most surprising is that they said it wasn't Leibniz or Newton who invented calculus. It goes back to the Greeks and Archimedes.

Another chapter that I found interesting was a chapter dedicated to Newton and everything that he discovered. I knew that he discovered a lot but this put all of that into perspective. He worked a lot with Pascal's Triangle and logarithms that we use a lot. Along with all of the calculus too.

Overall, I thought this was a good book. Very interesting. There was a lot of history that was cool to read about. I also thought it was cool that e wasn't "discovered" til later in history but it was used a long time ago. It is everywhere. I would recommend this book to math and science people. Definitely an interesting read.

Doing Math: Pythagorean Theorem for Winning Percentage

Baseball season is officially upon us. At the start of every season, baseball fans everywhere talk about how it's their team's year to win it all. But how do we really know who's year it will be with 161 games left to play in the regular season? There are many ways to try and determine who the next world champ would be. Here is just one of those options.

The Pythagorean Theorem for Winning Percentage

Everybody knows the famous Pythagorean Theorem: a^2+b^2=c^2. So how does this help us to determine a winner? Well with slight modifications, Bill James created the Pythagorean Theorem for Winning Percentage based off of the well known formula so that it states,

W%=[(Runs Scored)^2]/(Runs Scored)^2 + (Runs Allowed)^2].

This formula does not base the winning percentage of a team based on just their Win-Loss record. It gives us a better understanding of how well a team is actually doing based on the runs that they score and the runs that they allow the other team to score.

Many expert analyst are currently saying that the L.A. Dodgers are going to be the 2014 World Series champions. Let's looking into this a little further. According to MLB.com, the Dodgers had a spring training record of 98 runs scored and 118 runs allowed. Using the formula, we can say

W%=[(98)^2]/(98)^2 + (118)^2]
       =.4082.
This would mean that we can expect the dodger to win 40.82% of there games. According to MLB.com, the actual win percent of the dodgers was .368 or 36.8%. Now let's take a look into the Red Sox spring training results. They had a record of 117 runs scored and 149 runs allowed. When plugging these values into our equation we obtain

W%=[(117)^2]/(117)^2 + (149)^2]
       =.3814.
This would mean that we can expect a winning percentage of 38.14% from the Red Sox. According to MLB.com, the Red Sox had a winning percentage of .393 or 39.3%.

As we can see, this formula is pretty accurate at calculating expected winning percentages for baseball. The math that is done here is working with spring training statistics which don't necessarily show us an accurate representation of how the season will actually unfold. Once the regular season really gets underway, we will more accurately be able to pick who will win the 2014 World Series. In my opinion, the Red Sox will make it back to back champions.

Book Review-The Math Book

For the past several weeks, I have read The Math Book by Clifford A. Pickover. Overall, I like the idea of this book. It is a book of the 250 most important mathematical innovations. Each page gives you a new and exciting discovery in math. This all happens in the order they were discovered.

I feel as though the discoveries at the beginning of the book were way more interesting than the later ones. This is probably because the discoveries at the beginning lead towards the discoveries that happened later in history.

The invention that I felt was the most interesting was the simple game of Tic Tac Toe. This game is traced back to 1300 BC.  It is considered the "atom" of board games because many other games are based off of Tic Tac Toe. Interestingly, there are 362,880 ways to place X's and O's. Of these possibilities, there are 255,168 possible games that can be played that end in 5,6,7,8, and 9 moves.

Along with Tic Tac Toe, I found the pages that were on items such as the Mobius Strip and the Klein Bottle were interesting. They are objects that are in at least three dimensions but they are only one sided objects.

This book shows us many interesting discoveries in math and tells us everything they can in a one page summary.

One of the major flaws with this book is that the author deviates from the math quite often. He tells us a lot about the mathematicians as well. The flaw comes in the way he does this. He touches on the families of the mathematicians but only seems to do it for the female mathematicians. He also points out the religions of the mathematicians if they are not a white Christians. This happened mostly when they were Jewish. I feel as though it is demoralizing being a female mathematician that he would say this stuff. It is more important to talk about their contributions than to tell us how many children the females have or of what religion they are.

Overall, I do believe that this book has a good idea about it. I just feel that it would be better if it stuck more to the math than the other stories that aren't of any importance.

History of Math-Fibonacci

Growing up, we always here about this thing called a Fibonacci sequence. I never really understood it until I reached high school. Now I look at this sequence and these numbers all the time. But where did these magic numbers come from?

Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician from the 13th century. He is considered one of the greatest mathematicians from the medieval times. In the year 1202, Fibonacci traveled around Europe and Northern Africa promoting his book, Liber Abaci (The Book of Abacus). This book gained widespread recognition. First of all, in this book Fibonacci introduced the Hindu-Arabic number system into Europe. Secondly, n this famous book was the Fibonacci sequence; one of the world's most famous sequences.

This sequence is famous for a couple of different reasons. I would say that the most important reason is because we see this sequence of numbers everywhere in nature. Even in Fibonacci's original question asking "How many rabbits are created in one year with one pair of rabbits?"

It is unknown how Fibonacci created this sequence. Some believe that he was not the one to create it. He is just the one to make it famous.

Nature of Mathematics-Algebra Vs Geometry

Today I was thinking about the differences between algebra and geometry. It is clear that they are both part of mathematics but why? They are considered two different subjects. Or at least that is what we think. When really looking into the basics and the foundation, they are more similar than people or even myself think. I have come to realize that we can't have geometry without algebra and we can't have algebra without geometry.

The obvious one to discuss is that we can't have geometry without algebra. This is seen within every calculation for geometry. For instance, we try to find the angle measurement of a circle by setting an equation equal to alpha and solving for alpha. Or if we know that volume of a cylinder and we can find the height. We do all of this using algebra.

Looking at the other direction is a bit more difficult. How can we relate algebra to geometry. We learn this in the most simple form. We use it when we are looking for a variable, x, and are given a square or a rectangle. We also see it in slopes. Finding the "rise over run" of a simple geometric figure, a line.

There are many difference when comparing algebra to geometry but there are even more similarities. These are just a few examples. Algebra and geometry go hand 'n hand.