THE RESEARCH OF MATHEMATICS

INTRODUCTION

A. PURPOSE
The aim is to examine and develop mathematics.

B. BACKGROUND
The research of mathematics is very important in the development of our mathematical abilities. In mathematical research, we can do some stage. The first stage we have to determine which objects will be thorough. Besides, we also have to write down goals in the research and also write the reason why we chose to write in the research subject's. We also need to determine the background of why we chose it to be our research. After we write the background, then we collect data to complete our research. This data that we get to our analysis and we are the order to produce good research. After that, we can get conclusion the research of mathematics.

















DISCUSSION

The research of mathematics is important to student of mathematics. To get the research of mathematics we must have a good ways. One of the ways is to search the references about the mathematics such as, a browsing internet, search the related books, etc.
We should have knowledge about it to improve our knowledge about mathematics and to know the procedure or the steps to found the new formula, axioms, and theorems in mathematics. Theorem in which there is no contradiction in side.

The nature of mathematics:
• Formal mathematics
• Applied mathematics
• School mathematics

It is very easy to establish mathematics system. Let X₁, X₂, X₃, ..........are real members.
The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system

1. Numbers Theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general.

2. Group Theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

3. Ring Theory
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).(This definition take from Wikipedia)
4. Field Theory
Field theory is a branch of mathematics which studies the properties of fields.
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.(This definition take from wikipedia)
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.(This definition take from wikipedia)
7. Number system
In mathematics, a 'number system' is a set of numbers, together with one or more operations, such as addition or multiplication.
Application of mathematics a lot in our lives. Such applications in the field of mathematical economics. Mathematics is used in many economic areas. In economic calculation, math is very useful. Besides that, mathematics is used in the field of architecture. In the field of architecture used math a lot. Pythagoras theorems such as that used to make buildings. Information technology refers to the use of computers and software to convert, store, protect process, transmit, and retrieve information. Computational theory, algorithm analysis, formal methods and data representation are just some computing techniques that require the use of mathematics.
School mathematics
• Awareness
• Intention
• Abstraction
• Idealization
Eburt and Stracker (1995)
• Pattern/relationship
• Problem solving
• Investigation
• Communication
Mathematics phenomena
• To identify problem: skill, attitude, knowledge, experience.
• Mathematics knowledge
• Mathematics system
• Mathematics character


Supporting factor:
• history of mathematics
• the philosophy of mathematics
• the works of mathematics
• to in depth study of mathematics
For example we choose the Pythagorean Theorem.
1. Pythagorean
Mathematics is very important to our life. Mathematics is an exact science. The branch of mathematics is Pythagorean. In many building design cannot be separated from the Pythagorean Theorem. Pythagorean Theorem is very helpful in measuring the height of a building.
2. Pythagorean theorem
A theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a that is, a2 + b2 = c2.
3. Proof


We can proof the The Pythagorean theorem from the this graph
From this graph we can get any side:
The square has the side c, a, and (a-b)
So, c2 = (a-b)2 + 4. ½ (ab)
= (a-b)2+ 2(ab)
= a2 + b2 – 2ab +2ab
= a2 + b2
From the research of mathematics about Pythagorean we can get the conclusion:
• Right-angled side is the side that forms a right angle.
While is the side before the right angel.
• Square of the hypotenuse equals the sum of squares square is called Pythagoreans theorem.












CONCLUTION

The nature of mathematics:
• Formal mathematics
• Applied mathematics
• School mathematics

The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system
Application of mathematics a lot in our lives for example:
• in economics
• in architecture
• in civil engineering
School mathematics
• Awareness
• Intention
• Abstraction
• Idealization