Sunday, January 19, 2014

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.

Sunday, January 12, 2014

What is math?

What is math? The answer to this question depends on how submersed an individual has been in mathematics. For some people, math is adding, subtracting, multiplying and dividing. For others, math is just a hard subject in school that “they will never use”. My answer to this question has developed greatly over the last three semesters. To me, math is thinking critically about different kinds of systems. These systems can range from the number system in algebra and calculus, to a system of shapes, such as geometry, to a system of rings as we saw in modern algebra. Each of the components of math consist of some sort of system that follows a set theorems, or rules, to draw conclusions about the specific field that an individual has chosen to study.

Biggest moments in math:

I think that one of the biggest math moments was the number system. I think that this is important because the number system is the basis for most of the math that people do today.

I think another big math moment was the invention of the computer. Computers allow for certain mathematical processes to be done quicker and with more precision. Computers also allow for us to get a visual representation of certain things with programs like sketchpad.

Another big math moment is the understanding of different formulas. I think this is important because it meant that we can generalize certain aspects of math, such as volume or surface area. These formulas also allow us to see things like patterns, which we can use to find different relationships between two mathematical aspects. One example of this is the relationship between volume and scaling.

I think that algebra is a specific field of math that is important because people use algebra in everyday life.

Lastly, I think that geometry is important because geometry breaks away from the usual idea that math is just computing numbers.