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.

History of Math-Janos Bolyia

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 ever 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. 

Sources

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

What is Math?

What is math? When I first read the question I thought, "Well duh. Math is ..." Then I stopped. I realize that the question is not nearly as simple as it sounds. I first started off by thinking that it was the study of numbers. That is only one component of math. Math includes a lot more than that and a lot more than most people think. Looking back on my four years here at Grand Valley, I realize that I have studied many components of math and there are still many that I have yet to come across. So to me, math is not just one topic. It is a bunch of components ranging anywhere from algebra to geometry, trig to calculus, and from numbers to everything in between. I also believe that math is not just simply doing whatever these components ask of you to do but more importantly proving the existent of them.

There are so many important discovers to happen within mathematics that it is difficult to pick the top discoveries. A lot of discoveries are important to different areas of math. This makes it hard to rank any of them in level of importance. In my opinion, some of the top discoveries include

• Calculus
• Pythagorean Theorem
• Fibonacci
• Non-Euclidean Geometry
• Algebra

I feel like out of these five discoveries, algebra is the most important. I believe this because algebra is the foundation for all of the other discoveries. For examples, without algebra we would not have calculus. If you talk to other people, they would most likely believe that calculus is an important discovery. Thanks to algebra, this discovery could happen.