Communicating Mathematics

Fermat's Little Theorem:

As I was looking through different proofs to use for my communicating mathematics post, I came across a proof that I had never even heard of before. This proof was called Fermat’s Little Theorem, which is not to be confused with Fermat’s Last Theorem. This proof states the following

Let p be a prime number which does not divide the integer a, then a^p-1 = 1(mod p).

True to Fermat’s form, a proof of this was not provided. Euler was the first to finish and publish the proof. Although a proof almost identical to Euler’s was found in a manuscript of Leibniz fifty years prior to that of Euler. 

To begin this proof, we will list the first p-1 multiples of a, which looks like this:

a, 2a, 3a, … (p-1)a

After this, assume that we have some ra and sa that are the same modulo p. This means that we can conclude

r = s(mod p)

and also the p-1 multiples of a above are distinct and nonzero. This means that they must be congruent to 1, 2, 3, … ,(p-1) in some way. The next step of this proof is to multiply each of these congruencies together in order to obtain the following

a*2a*3a*…*(p-1)a = 1*2*3*…*(p-1)(mod p).

We can further simplify this as the following

a^p-1(p-1)! = (p-1)!(mod p)

Now, the last step of this proof is to divide each by (p-1)!. By doing this, we can see that we will get

a^p-1 = 1(mod p)

This is exactly what we were trying to show. Therefore, we have just proved Fermat’s Little Theorem.

Fermat’s Little Theorem is important to math because a^p-1 is so easy to calculate, that most primality tests are built using a version of this theorem.

History of Math

Pascal's Triangle:

Throughout this semester I began to research Pascal's triangle. At first after leaving office hours, I felt that doing 16 hours of research of Pascal's triangle would be nearly impossible. I quickly realized that I had been very, very wrong about the lack of information on the triangle. I already knew that Pascal's triangle contained a great deal of math patterns and sequences, and that it could be used for a few different mathematical purposes. I had no idea to what extent how awesome this triangle actually was. I began by googling "Pascal's triangle patterns" and thought that maybe a few things would pop up. The third site seemed the most interesting, so I chose that. I had expected a few patterns, most of which I had already known, to be displayed before me. I scrolled through the first three and thought "yap! I was right." Then a scrolled through the fourth and looked over it for about ten minutes trying to understand the pattern that was trying to be conveyed. Finally, I got it. I checked the pattern for a few numbers on the top row of the triangle and then moved my way down around the 10th row. The pattern still worked. Now I was pretty certain that the pattern was true, because I would be utterly shocked if I just happened to stumble upon and disprove a well known pattern in Pascal's triangle, but I still had to continue to look. I found Pascal's triangle filled in to the 27th row and checked for some of the lowest numbers. Check! It still works. I kept scrolling and found that there were more patterns than I could even wrap my head around.

What is Pascal's triangle, you might be wondering. As you have probably already gathered, it is a triangle in which numbers are placed in each row. The point of the triangle contains the number one, and is row n = 0. The next row contains two numbers that are staggered under the first, the third row three numbers staggered under the row above, and so on. This triangle can go on forever. To construct the triangle, each number is the sum of the two numbers directly above it. Below I will show the beginning of the construction of the triangle:

 

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

 

You can see that this pattern will continue for as high as one can count. Although this triangle is named after mathematician Blaise Pascal, the set of numbers were known about before Pascal was even around. The triangle was named after Pascal because he was the first to organize the numbers in this way and he also discovered many different ways that the information could be used. The actual idea of these numbers is thought to have first been known through studies of binomial expansion and combinatorics by Indian mathematicians as well as the Greeks. A number of different mathematicians used these numbers before Pascal. But in 1665, Pascal published Traité du triangle arithmétique, which was his version of the triangle. He used this to solve many probability theory problems. The Name Pascal's Triangle was given Pierre Raymond de Montmort, with the name Table de M. Pascal pour les combinaisons (Table of Mr. Pascal for combinations) and by Abraham de Moivre with the name Triangulum Arithmeticum PASCALIANUM (Pascal's Arithmetic Triangle). As we know, the latter is most commonly used now in Western culture.

 

While I can't even begin to describe all the different ways that the triangle can be used, I will touch on a few that I find particularly interesting and important.

 

 

The first use that I find fascinating is binomial expansions. Each row of the triangle contains the coefficients that are found in binomial expansions. For example, when we expand (x + y)^2 we get the following

x^2+2xy+y^2

 

Notice that the coefficients are 1,2,1 which is row n = 2 on the triangle above. Essentially, this is the binomial theorem in a nutshell. 

 

 

The second use for Pascal's Triangle that I find interesting is the calculation of combinations. For two numbers n and k, you can find n choose k by looking at the kth element in the nth row. To put in simpler terms, if you have n objects and you are taking k objects at one time, the number of different combinations that are possible is n choose k.

 

The last feature that I am going to touch on is the many different patterns that are found I the triangle. Each of these can be checked and tested in your free time. (I've spent hours doing so already! :) )

 

- The sum of every row is twice that of he preceding row.

 

- The sum of the square of the elements in the nth row is the middle element of the row 2n.

 

- The diagonals second from the outside contain the natural numbers in order.

 

- The diagonals next to the natural numbers contain the triangular numbers.

 

- Next to the triangular numbers are the tetrahedral numbers, and then the pentatope numbers.

 

- The triangle is symmetrical.

 

These are just a few of many different patterns that can be found in Pascal's triangle. Further investigating will surely amaze you at all the this set of numbers contains. It sure amazed me.

Second Book Reveiw

The Math Book:

Although I didn't read this book through thoroughly from back to front I did enjoy the time I did spend with it. This book was a great book to have handy for reading when lengthy reading times are not available. One thing that I really liked about this book is that I could read this book for 5 minutes or five hours and not worry that I left off at an important part. It seemed more to me like a math magazine because it just had tidbits of information on each page rather than having to read each page in order to understand it. I think that the pictures were a great idea for a book like this as well. The pictures were a good way to connect a visual to the material.

While it did seem to have more of a magazine style set-up, I don't think this book could be compared to just a simple math magazine. Some of the pages were pretty simple to read and I had already known the information. On the other hand, of the pages talked about things I had never heard of, and I would still probably say I had never heard of it if I heard it again.

I probably would not recommend this book to a high school classroom because I thin that just about every one of the pages would have been to advanced for this level of students. I would recommend this book for a classroom of college level math majors because it gives a broad idea of a plethora of math topics. I think it is a fantastic book for someone who is interested in math but doesn't know where to begin to look for research. This book gives enough information to get an individual interested, but leaves the reader. or at least it left me, wanting to look further into a lot of the ideas that were covered.

Overall, I really enjoyed this book. I think that this is a book that I would like to read in further depth sometime that I could further look into many of the subjects that are talked about.

Book Review

Visions of Infinity

By Ian Stewart

 

 

The book Visions of Infinity talked about some of the very important discoveries made my mathematicians in the past. One aspect of this book that I enjoyed was the position that was taken on presenting these discoveries. This book stayed away from long, confusing proofs of mathematical discoveries. Instead of focusing on the actual mathematics of certain people, this book focused more on the thought process behind the math. Throughout the book, many different theorems and conjectures are discussed, as well as how each of these was thought about. Some of the problems that are talked about are the millennial problems. These problems offer a hefty reward of $1,000,000 if they are solved. While there have been numerous attempts to solve these problems, only one of the problems has been solved so far. Visions of Infinity covers a spectrum of topics. These topics range from simple things like prime number and the four color theorem to the Mass Gap Hypothesis.

                This book does a good job of making the topics readable to an audience that is not an expert in mathematics. However, I don’t think that this book is a good read for everyone. Even though Visions of Infinity does a good job putting much of the information in simpler terms, there were a few parts that went clear over my head. I know that I am no master mathematician by any means, but I would like to think that I have more experience and a better understanding of math than the general population.

                Overall I think that this is a great book to read if you are interested in the mathematical discoveries throughout history, and have at least some knowledge of upper level math. I really enjoyed reading about how each of these discoveries has progressed over time and the contributions of the many mathematicians over the years. I also liked seeing that each of these discoveries required the work of so many different people. I would recommend this book to any undergrad math student with the advice to take the time to really this book, as it is not a quick read if you want to really understand it.



Nature of Math

Does our number system have anything to do with what mathematics is?

I think that this question is can be answered with yes and no depending on how you interpret the question. I would say yes, our number system has everything to do with what mathematics because there is no way to represent what we know about math without using our number system. For example, if we did not adopt our number system, how would we be able to understand just about anything? Would we still understand or discover things like calculus, algebra and physics? How would we quantify anything? How would we make change at a grocery store? How would we add, subtract or multiply?

On the other hand, I would say no, our number system does not have anything to do with mathematics. Our number system doesn’t determine any truth about mathematics. What we know is true about mathematics now will always be true. Our number system is just a way to represent what we know. If we didn’t have our number system, we would more than likely have another representation of what we have discovered about mathematics. To answer the above questions, the concept of calculus, algebra and physics would still remain. Whether it was discovered or represented with our number system would be a different story. We would have found another way to quantify things. Who knows, maybe we would still be using roman numerals. The use of the words “one, two, three… etc.” are essentially just words we agreed to use to represent those numbers. Our number system could go something like “popcorn, carpet, pillow dinner” if that is what we use to represent what we now call “one, two, three, four”. The number system itself does not determine anything about math. It is simply a way for people to represent what they know that is universally accepted so that information can be shared with others.

Doing Math

Creating a Tessellation

A tessellation is the tiling of a plane using on ore more geometric shapes, called tiles, with no overlaps and no gaps. Tessellations can be simple or intricate and can be categorized into smaller categories which include the following categorizations:

Regular: This type of tessellation is made of the same shape.

Semi-Regular: Tiles of more than one shape.

Aperiodic: Tiling that cannot form a repeated pattern.

Below is an example of a more simple tessellation that I have created.

History of Math:Contributions of Nikolai Lobachevski

It’s no wonder to me that Nikolai is the name of both a mathematician, and a vodka. After reading some of the work of Nikolai Lobachevski, I understand why someone would need a glass, or four, of Nikolai to relax. Among other accomplishments, Lobachevski’s greatest work was discovering hyperbolic geometry, also known as Lobachevskian geometry. During the time of this discovery, many other mathematicians were trying to prove Euclid’s fifth postulate, the parallel line postulate, from other axioms. Lobachevski, however, took a different approach to this geometry. Instead of proving the existence of the parallel line postulate, Lobachevski found a geometry in which the parallel postulate was not true, which is where we get hyperbolic geometry. This geometry goes against everything that most people believe to be true about geometry. That is because most people, if they are familiar with any type of geometry, are only familiar with Euclidean geometry. Hyperbolic geometry offers an entirely different set of axioms and theorems. The most shocking of these is that of parallel lines. In hyperbolic geometry, there exists more than one line through any given point such that the lines are parallel to another line not contained by the point. Another consequence of hyperbolic geometry was that every triangle has an angle sum measure less than 180. These discoveries paved the road for other types of geometries, such as differentiable geometry. As E. T. Bell stated “the boldness of his challenge and his successful outcome have inspired mathematicians and scientists in general to challenge other `axioms’ or `accepted truths’”. Nikolai Lobachevski has been referred to as the Copernicus of Geometry for his work in this discipline of math. Along with his work in hyperbolic geometry, Lobachevski was also known to have worked with physics and analysis and is thought to have been the first to make a distinction between continuous and differentiable curves.