Mathematical Research

Last week in English class, we discussed about mathematical research. First of all, I want to give the definition. Mathematics is the science of numbers, quantity and space, arithmetic, algebra, trigonometry and geometry are some of the branches of mathematics. Other definition is the study of the relationships among numbers, shapes, and quantities. It uses signs, symbols, and proofs and includes arithmetic, algebra, calculus, geometry, and trigonometry. While mathematical is belonging to, relating to, or used in mathematics. Then the definition of research is methodical investigation into a subject in order to discover facts, to establish or revise a theory, or to develop a plan of action based on the facts discovered. So we can conclude that mathematical research is a method to investigate something/problem and to find the solution use principal of mathematics.
There is relationship between mathematical research and mathematics system. System is a method or set of procedures for achieving something. Mathematics system means a method or set of procedures for achieving something use in mathematics. The following are examples of mathematics system :
1. Group Theory
Group theory is the branch of mathematics that answers the question, such as : "What is symmetry?"
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right.
2. Ring Theory
In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
3. Field Theory
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined.
4. Numbers Theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated.
5. Euclidean Geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could be fit into a comprehensive deductive and logical system. The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.
6. Non Euclidean Geometry
A non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor—it is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry.
7. Number System
In mathematics, a 'number system' is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication. Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.

The aim of mathematical research is to examine and establish new system of mathematics, there are some aspects, namely : definition, axioms, theorems, and rule/law/procedure. Besides that, should be there supports factors in order to the aim can be fact, there are our knowledge of mathematics, our knowledge of the history of mathematics and our knowledge of the mathematician, our knowledge about experiences of developing mathematics, and facultative/ the philosophy of mathematics.
The following, I want to give explanation about the steps of mathematical research:
• Indeepth study references
We should get a lot of references, it is from magazine, newspaper, browsing internet, article and so on.
• Identify formulate the problems
For the first, we should identified it. We should elaborate the WH question, what, who, where, when, how, whom, which, and whose. Find the answer about the topic use WH question.
• Develop method of research
Develop the method of research use principle of mathematics. In order to the outcome more valid.
• Process of research
Be patient, because it need long time and process.
• Publish
After you had finish, publish it.
• Journal
Make a journal about your result.

http://encarta.msn.com/encnet/features/dictionary/DictionaryResults.aspx?lextype=3&search=system
http://en.wikipedia.org/wiki/Group_theory
http://en.wikipedia.org/wiki/Ring_theory