Monday, January 20, 2014

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. 

Janos Bolyai Biography-

Sunday, January 12, 2014

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.