MTH 495 Recap

Growing up in grade school I was never a fan of history. Yeah, I think some of the events in history were interesting to learn and hear about, but as a whole it did not grab my attention. A typical history class in grade school I thought of was just grading you on how well you knew the sequence of events and the dates in which they happened, who was in involved, and the what the outcome was. It was a memorization type course. There is a reason why I am a Math major. That is because my mind and brain thinks in a systematic process. 
I would have to admit, I was a bit apprehensive about taking this course. Not because this was my senior capstone course to graduate, but by the title of the course: Nature in Mathematics. This title means history. A word that I am not a huge fan of. 
After the first couple of weeks of class, I realized that although we are talking about the history of math, I really enjoyed learning about who and how math was discovered or invented (depending on what you believe). 

One topic that we discussed that I will never forget is infinity. We as a class discussed and debated if infinity was a number. If I was a elementary student first seeing the infinity symbol:

∞

I would say that yes, it is a side-ways eight. Although, we all know that infinity is not a side ways eight...it is bigger than the biggest number we know of. Similarly, negative infinity is smaller than the smallest number we know. I believe that infinity is not a number, it is just a symbol representing a limit beyond what we would expect. I believe this because we cannot do operations on infinity and obtain another number. For example, What is infinity*infinity? maybe it is infinity squared, but then what is that? 

Another topic within infinity that still blows my mind is the Aristotle's Wheel. 

Just like this one day in class talking about infinity, there were multiple other days that I really enjoyed attending. All the topics covered helped me completely understand the topic at hand. Although, I have learned about most the topics before hand in other courses, I only knew the concept, process, why, when, and where you apply it. I did not know where it came from, who originally proved that the topics are true. 

Learning the back story behind the topics was eye opening. Some of the mathematicians that we discussed this semester spent most of their lives learning, teaching themselves, and researching math. Mathematicians are truly dedicated to their work. I still think it is crazy that it took Andrew Wiles seven years to complete Fermat's Last Theorem and when written completely out it is over 150 pages long. 

Looking at on this semester in class, I really have enjoyed it and feel like I know the concepts and topics more even though we hardly did any math or problems throughout the course. Simply learning about how the concepts became was the icing on the cake for me. 

And yes I may like history a little bit more now. :) 

The History of Women in Mathematics

As a female in a Science, Technology, Engineering, and Mathematics (STEM) field I am very interested and aware of the way women are viewed in these fields. Upon graduation in April I hope to inspire other females to pursue whatever field of study they want and don't be afraid of stepping outside the gender norms. Being a resident and peer mentor of the Women in Science and Engineering (WISE) resident hall, has helped me achieve my goals. I have had multiple discussions with other women on our studies and how at times it can be hard, but in the end we achieve what we set out to do.  I think a majority of this is from the support of other females in STEM fields and peers. I know for myself and I am sure of other females, we like to reach out and ask for help when we don't understand; males not so much.
During my math capstone course we took a day to look at Sophia Germain. This was the first time I learnt about female in the math field. Before, I knew that mathematics had some female mathematicians in their history, but I never knew who and what they contributed. This class got me interested in learning about other female mathematicians. Below are some that I found a read up on their life, what they contributed, and when.

Hypatia ~370 AD - 415 AD

Born in Alexandria, Egypt to a father whom was also a mathematician, she had a great supported. Theon, her father, had a different philosophy to raising his daughter than most others at the time. He wanted her to know and think that she could do anything that she put her mind to. Thus, expressed education and physical activity were good to have to help you succeed. Following in her fathers footsteps, Hypatia, eventually surpassed his knowledge and at this point he sent her to Athens to study mathematics. After completely her studies, she returned to Egypt and taught. While teaching, she continued to do research. In her research, she is most known for coming up with the idea of hyperbolas, parabolas, and ellipses. Because of her teaching abilities she was able to explain the tough, dense math topics and ideas to other easily. Although she contributed a lot to the math community her life ended tragically in a murder, because others thought she was a witch and doing black magic, when she was just developing math concepts!

Emmy Noether 1882 - 1935

Noether was born in Germany to her parents. From a young age, she was always interested in math. However, she received her teaching certification for foreign languages. Even after this period in her life, she had that passion and love for math she wanted to pursue. Although she was not able to registrar for her classes because of her gender, she was allowed to sit in on the course; so she did. Finally she was able to enroll in classes and after about three years of studies she received her math degree. After graduating, she continued to study and research mathematics. She is best known for her research in abstract algebra. The Noetherian rings are named after her. She also helped develop the axiomatic approach to math. 

Sofia Kovalevskaya 1850 - 1891 

Kovalevskaya's bedroom wall paper were her fathers calculus notes. This is where she started to learn math and teach it to herself. Although, this is not what her parents wanted her to do. They forbidden her from teaching herself mathematics and studying the subject, because they believed that women should not have a higher education. That's where Kovalevskaya hid in her room and studied mathematics. She taught herself trigonometry in her teen years and after completing her secondary education continued on to an university. She complete her degree and went on to do research.

Sophie Germain 1776 - 1831 

Germain is a French, self-taught mathematician. All of her material for which she taught herself was from her male friends whom were able to attend the school. Her parents were opposed of her studying mathematics so they took her light and candles away at night. But at night she would use the candles she smuggled and continued her studies. She is best known for a limited proof Fermat's last theorem, for prime numbers under 100. She has won prizes for her work. Germain was praised by male mathematicians of her work. Gauss especially liked her number theory proofs. She died at the age of 55 due to breast cancer.  

It was interesting to look and learn about some women mathematicians and the struggles that they had to go through to pursue their passion. From females not even allowed to enroll in a university, to parents saying no, they all overcame the obstacles and pursued their dreams. I couldn't imagine going through what all three of the above women did. But it also made me realize that if you truly believe in what you love, you can achieve it. Learning about these three women have added to my experience as a female math student to work hard and never give up no matter what others think or feel.

Source: http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1154&context=cmc_theses
and MTH 495 notes

Communicating Euler's Circuits and Paths

One may look at the image below and ask what does this represent, why is this useful, or how is this math?

This is what mathematicians would call a graph in graph theory. It contains vertices (the blue points) and edges (line connecting vertices). This type of graph can be defined as a visual representation of the relations between quantities (vocabulary.com). This particular graph above is the graph of the Seven Bridges of Konigsberg, a mathematics problem by Leonardo Euler. Here is a website that describes this particular graph.  A graph can represent a real world problem. For example, a graph can represent airplane traffic between cities. Airlines use graphs to come up with the most efficient travel path to get from point A to point B. To get to B from A, it may not be a straight and simple path. There maybe stops on the way, for fuel or picking up more passengers. A path of travel for an airplane may not be completing straight either, because they want the safest path. Meaning that in case of an emergency they want to fly over other airports for a safer landing. Not only is the most efficient path effective, airlines have to communicate with each other and make sure the traffic high above the clouds isn't crowded at one particular time. Euler, a great mathematician, was the first to look at graph theory. He created the Euler circuit and path. Let's first start with some terminology.

Path: is connected edges with no repeated vertices. Paths can be closed and open
Euler Path: is a path that uses every edge of a graph exactly once. 
Example- BBADCDEBC
Circuit: is a closed path
Euler Circuit: is a circuit that uses every edge of a graph exactly once.
Example- CDEBBADC

How can one just look at a graph and tell that it contains an Euler circuit or path without tracing over the edges and trying to find it? Euler found some patterns and came up with Theorems on these topics.

Theorem 1

If a graph has any vertices of odd degree, then it cannot have an Euler Circuit.
If a graph is connected and every vertex has even degree, then it has at least one Euler Circuit. 

Theorem 2
If a graph is connected and has exactly 2 vertices of odd degree, then it has at least one Euler Path.

I found this table that sums up the Theorems nicely in a more visual format: 

# of ODD Vertices	Implication (for a connected graph)
0	There is at least one Euler Circuit.
1	THIS IS IMPOSSIBLE!
2	There is no Euler Circuit but at least 1 Euler Path.
more than 2	There are no Euler Circuits or Euler Paths.

Although all of this may seem useless and not "mathy," this is development in math has grown over the years. Graph Theory has evolved and more and more people are seeing the value in it. And yet this is another invention that Euler discovered in his life. To learn about everything that Euler discovered click here.

So the next time you look at an image that is simply just lines and points, there is more than just the simple lines and points behind it. It may represent an real world problem or used to show routes. It's math. 

References: http://www.ctl.ua.edu/math103/euler/euler.htm#3 Important Questions
https://www.math.ku.edu/~jmartin/courses/math105-F11/Lectures/chapter5-part2.pdf

The Joy of X: A Guided Tour of Math, from One to Infinity

I recently just finished reading The Joy of X: A Guided Tour of Math, from One to Infinity, by Steven Strogatz. Strogatz is an applied mathematician, but he knows that some people may not understand math right away. While reading his book he takes you on a journey through the progression of math as you would learn it as a student. He breaks the book down into six parts.

1. Numbers - what are they? how do we use them? and why they are important.
2. Relationships - algebra, functions, story problems
3. Shapes - triangles, pi, trigonometry
4. Change - calculus, e, vectors
5. Data - normal distribution
6. Frontiers - primes, group theory, sequences.

Throughout these sections he gives multiple examples to help explain the concept and what is going on. He does this in a manner that everyone can understand. The his creative imagines he uses throughout the book and his sense of humor it is an easy read for even a non-mathematician to enjoy.

Even has a person whom has studied math for now 17 years, I learned some new ways to explain some math concepts that others cannot wrap their mind around. For example, why is a negative times a negative positive? This is how he showed it:

-1 x 3 = -3

-1 x 2 = -2

-1 x 1 = -1

-1 x 0 = 0

-1 x -1 = ?

If you follow the pattern on the right hand side, it is counting up by one so the ? would be a positive 1. I have never thought about a way to explain this to others that didn't believe this. However, I now have some what of a reasoning to show them why this is true. 

Overall, this book was an easy read and I would recommend this book to anyone who would like to know a little bit more behind how math works. He gives great examples that will stick with you. And you will sure get a kick out of his humor throughout the book. 

Some of my favorite quotes from the book:

"...math always involves invention and discovery..." -pg 5

"...even wrong answers can be educational...as long as you realize they're wrong." -pg 63

"Proofs can cause dizziness or excessive drowsiness. Side effects of prolonged exposure may include night sweats, panic attacks, and, in rare cases, euphoria. Ask you doctor if proofs are right for you." -pg 93

Is Math a Science? Is Math an Art?

This is a question that has been debated for while. Everyone has a different perspective, opinion, and thought about the way they look at math as a subject matter. In order to answer these questions I feel like we need to define math, science, and art. According to dictionary.com the following are the definitions:

Mathematics: the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.

Science: a branch of knowledge dealing with a body of facts or truths systematically arranged and showing the operation of general laws.

Art: the quality, production, expression, or realm, according to aesthetic principles, of what is beautiful, appealing, or of more than ordinary significance.

Besides from looking up the definitions of the three words of interest, I researched others opinions and thoughts on the two questions, and also conducted a short poll (from my friends, not random!).  

Looking straight from the definitions I would say math is science and not art. This is because of the word systematically stands out in both definitions for me. Both science and math have a very systematic approach you take to solve a problem. For example, the scientific method is a systematic way carry-out an experiment and the order of operations is a systematic way to solve an equation. 

Art and math do not seem to line up, according to the definitions. In the definition of art, it seems to be expressing how art looks once it is done. Although I do not agree with this definition 100%, I had to be on fair playing field and grab them from the same resource. Therefore, comparing definitions I do not see a link between math and art. 

The results are in.....

	Yes	No
Is Math Science?	3	2
Is Math Art?	1	4

I asked 5 of my friends both questions. Only one person said that math is art, which I found interesting. And it was almost a 50/50 split for if math is science. 

In my research online, I found that many other people who answer these questions are drawn down the middle as well. And almost all responses said it determines on how you think about math and how you define science and art. It's a tough question to tackle and come to a hard-set conclusion. 

Here is my opinion: 

Math is science because we follow laws, theorems, and other properties of mathematical concepts. 

Math is art because as a mathematician you have to use your imagination, creativity, and recognizance to see how to solve a problem. To help support this claim, I will provide you with the definition of art from the Merriam-Webster: "Something that is created with imagination and skill and that is beautiful or that expressed important ideas or feelings."

An example is proving a theorem. You have to first have the will, imagination, and creative thinking skills to work your way through the problem and then you then must also follow all the rules of math and other proven properties to finish the proof properly. And finally, once the proof is written formally with all justifiable reasoning, on clean-crisp paper, you can say that you have now created a beautiful piece of work that took skills and systematic methods. 

Now lets take a poll again....Is Math Science and/or Art? 

2-D to 3-D; 3-D to 2-D

To a visual learner, mathematical nets are easier to visualize and recognize how the 2-D net will fold up into a 3-D shape. Although I would consider myself a visual learner, this activity we did in class was a challenge. In class, we were tasked with creating 3 nets so when they are put together they would form a cube.
In order to complete this task we constructed a triangular prism (half the area of the cube), a triangular pyramid (a third of the area of the cube), and a square pyramid (a sixth of the area of the overall cube). We had to think about how all three of these shapes would fit together to form a cube. 
In our group we first started by making the triangular prism (which will account for 1/2 the area of the cube). This was pretty easy to think about and draw out the net because it is our base point. Below is an image of how we drew out our net (left) and a net that we were given at the end of class (right). Comparing the two nets you can see they will make the same 3-D figure. 

                         

After looking at both of these nets, I feel like the way our groups did it was easier to draw out with the least amount of measuring. You can see that to get the rectangle at the top right, we just needed to draw a 2in line that was 90 degrees from the base already created. Where as in the one we were provided after class, you would need to know the exact length of the rectangle, although this wouldn't be difficult to find, it is one extra measurement. This length for a 2in cube would be the square root of three. 

Next, we created the square pyramid. I had a hard time thinking about this one, but once one of my group members drew it out and folded it into the pyramid I had the light bulb moment of where this figure fit in with the triangular prism to form the cube. Also, as a side note, I think this is the coolest net, because it looks like a pinwheel. 

Finally, we made the net for the triangular pyramid. This net was hard to visualize and create. I think what was the hardest about this net was the lengths of the sides and no one side had a full face of the cube. This is due to the fact that it is only 1/6th the area of the cube. Once we created the net and folded it up, it took some time for me to figure out how the shape fit into the other two pieces to form the cube. Once the cube was complete it was a sigh of relief and accomplishment (just like finishing a proof). 

As a class, we shared each of our nets and noticed differences in some, but they all ended up working out. Throughout our discussion someone said "Mathematicians find a solution, prove it, and then accept it, where engineers find the best possible solution," this statement really rang true to me. 
When I first heard this I thought about all the possibilities for a net of a cube. 

In my eyes I feel like the second column, second row net is the most efficient. If you were to draw multiple nets on page you could easily interlock them to get more nets on one page. However, I think the easiest one to visualize that it will get folded up into a cube is the first one. Because you roll up the middle row and then the two extra  side squares make up the base and top surface of the cube. Which one do you like the most?

In the end, everyone thinks and visualizes of things differently. Although, it was hard for me visualize the three pieces of the cube at first in the end I wrapped my brain around it and our cube was complete. Another example that math will stretch your brain and will make you think. 

Doing Math