The Research of Mathematics

The Research of Mathematics

Created by:

Rochmania Septikasari

08305141005

FACULTY OF MATHEMATICS AND SCIENCE

STATE UNIVERSITY OF YOGYAKARTA

2009

A.   Preface

Thanks to Allah SWT for help the writer finished this paper. For the opportunity, health and times have given to writer because without them writer can’t finish this paper well.

This paper arranged to fill English assignment by Mr.Marsigit. It discussed about the research of mathematics. Consist of nature of mathematics, what is the research of mathematics, purpose, structure of the research of mathematics and discussion about the applied mathematics as example of the research of mathematics. The research of mathematics is very important to examine and develop our knowledge. This paper takes from gleanings (book and internet).

Although this paper still has many mistakes, the writer hopes this paper can develop our perception about the research of mathematics, elementary matrix and method that using to seek A^-1. Thank you.

B.   Abstract

Mathematics is one of the important knowledge. It’s the basics of other knowledge. But the problems in mathematics develop widely from day to day. Research of mathematics is needed to answer the problems or develop mathematics knowledge. in this paper I will explain about the research of mathematics. Consist of the nature of mathematics (formal mathematics, applied mathematics, school mathematics), what is the research of mathematics, purpose the research of mathematics, discussion a problem and conclusion.

C.   Analysis

a)      Nature of Mathematics

Before we study about research of mathematics, it’s very important to know about the nature of mathematics. The natures of mathematics consist of formal mathematics/axiomatic mathematics/pure mathematics, applied mathematics, school mathematics/concrete mathematics/real mathematics.

1.      Formal mathematics/axiomatic mathematics/pure mathematics.

There are different opinions to define mathematics. Mr.Marsigit said that mathematics is deductive system consist of definitions, axioms and theorem in which there is no contradiction in side. It’s very easy to establish mathematics system. For example make a definition that it can be proved and trusted, use theorem or axioms, and proof some of theorem.  The content of formal mathematics consists of number theory, group theory, ring theory, field theory, Euclidian geometry, and non-Euclidian geometry.

2.      Applied mathematics

Now the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptology), applications of mathematics  in engineering and computing, economy, and etc. (http://en.wikipedia.org/wiki/Applied_mathematics). The general example of applied mathematics is use the Pythagoras theorem to count shortest distance between 2 locations or else.

3.      School mathematics

School mathematics is different with formal mathematics. In school mathematics students must intention awareness mathematics phenomenon.  We must transform the mathematics phenomenon with abstraction and idealization to the student’s mindset. So, the student gets size and form the object in his mind. For example, if given a cube (three dimensional object), in school mathematics we just need to transform about the form and the length sides of cube in the student’s mindset.  The important think in school mathematics is about how to transform the real object to the abstraction in the student’s mindset. According to Ebbute Starker (1995), school mathematics is about:

·         Pattern/relationship

·         Problem solving

·         Investigation

·         Communication

to identify mathematics problem we need mathematics knowledge, mathematics system, and mathematics characteristic. The three aspects can we get if we have a will, attitude, knowledge, skill, and experience.

b)      Research of mathematics

Mathematics research is the long-term, open-ended exploration of a set of related mathematics questions whose answers connect to and build upon each other. Problems are open-ended because mathematicians continually come up new questions to ask based on their observations.( Http://www2.edc.org/making math/handbook/teacher/introduction/introduction.asp#wimr )

c)      Purpose of Research Mathematics

According Alan H. Schoenfeld, Research in mathematics has two main purposes, there are :

·         Pure (basic science): to understand the nature of mathematical thinking teaching and learning.

·         Applied mathematics: to use such understandings to improve mathematics instruction.

Whereas, according to Mr.marsigit, the purpose of mathematics research is to examine and develop mathematics.

d)     The structure of mathematics research

a.       Identity problem

Supporting factor to identity problem with read the history of mathematics, the work of mathematics, and the philosophy of mathematics and to in-depth study of mathematics.

b.      Background

The stabilities of criteria finding a solution of problem and explain why we do the research.

c.       Formulate the aim of research

Explain specific purpose of research and what we want to reach with do the research of mathematics.

d.      Develop method

For example: analyze, synthetic, deductive, phenomenology, harmonistic

e.       Discussion

study the problem which is focuses in mathematics research systematically.

f.       Conclusion

Explain what we result after we do our mathematics research.

D.   Discussion

In this part, I will focus the research of mathematics in applied mathematics. Applied mathematics is very complex because mathematics applied in many field in our life and our activity.

I will give an example the application in first order linear differential equation about determination of the time of death.

In the investigation of a homicide or accidental death it is often important to estimate the time of death. We need the mathematical way to approach this problem.

From the experimental observations it is known that, to accuracy satisfactory in many circumstances, the surface temperature of the object changes at a rate proportional to the difference between the temperature of the object and that of the surrounding environment (the ambient temperature). This is known as Newton’s law of cooling. Thus, if Ɵ(t) is the temperature of the object at time t, and T is the constant ambient temperature, then Ɵ must satisfy the linear differential equation

                                  ………………                    (1)

Where k > 0 is a constant of proportionality. The minus sign equation (1) above is due to the fact that if the object is warmer than its surrounding (Ɵ>T), then it will become cooler with time. Thus .

Now suppose that at time t = 0 a corpse is discovered, and that its temperature is measured to be. We assume that at the time of death  the body temperature  had the normal value of F for C. If we assume that equation (1) is valid in this situation, then our task is determined.

The solution of Equation (1) subject to the initial condition  is

                                 ……………………                (2)

However, the cooling rate k that appears in this expression is as yet unknown. We can determine k by making a second measurement of the body’s temperature at some later time ; suppose that . By substituting these value in equation (2) we find that

 

Hence                                                             ……………          (3)
where   are known quantities.

Finally, to determine td we substitute  and  in equation(ii) and solve for td. We obtain

                                                           …………..…..        (4)                
Where k is given by equation (3).

For example, suppose that the temperature of the corpse is F when discovered and F two hours later, and that the ambient temperature is F. Then, from equation (3)

And fro equation (4)

Thus we conclude that the body was discovered approximately 1hr, 8min after death.

E.   Conclusion

From the example of mathematics research about the applied mathematics application in first order linear differential equation be a model to determination of the time of death, we know that is very useful in our life. So needed to examine and develop mathematics which is the aim of research mathematics more and more.

F.    References

1.      E.B.William, and C.D.Richard.1997. Elementary Differential Equations and Boundary Value Problems. New York : John Wiley & Sons, Inc

2.     Http://en.wikipedia.org/wiki/Applied_mathematics, accesses on Monday, 3 January 2010

3.     Http://www2.edc.org/makingmath/handbook/teacher/Introduction/Introduction.asp#wimr, accesses on Monday, 3 January 2010

4.     Mr.Marsigit opinion, at Wednesday, 16 December 2009