The Research of Mathematics

Introduction

As we know, the nature of mathematics consist of formal/axiomatic/pure mathematics, applied mathematics, and school mathematics.

1. Formal mathematics/axiomatic mathematics/pure mathematics
Mathematics is a deductive system consists of definitions, axioms, and theorems in which there is no contradiction inside. It is very easy to establish mathematical system step by step like make a definition then use the axiom and theorem, proof the theorem. The substance of formal mathematics such as numbers theory, group theory, ring theory, field theory, Euclidian Geometry, Non Euclidian Geometry, etc.

2. Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. The are a lot of applied mathematics in this word, in optic, mechanic, astronomy, engineering, etc. The most simple example is the use of pythagoras’ theorem to make a right angle (for example when someone wants to make a building,etc).

3. School mathematics/ concret mathematics/ real mathematics.
School mathematics is really different with formal mathematics. In school mathematics we just focus in mathematics phenomenon. We must transform the mathematics phenomenon with its abstraction and idealization to the student’s mindset. So, the student just need to aware about the characteristic of the object. It is need awareness and intention from the student. For example, if given a cube (three dimentional object), in school mathematics we just need to transform about the its shape and the length of its sides in the student’s mindset. We no need to transform about the material that we use to make the cube, the color of the cube, etc. It is not necessary. So, the important think in school mathematics is about how to transform the real object to the abstraction in the student’s mind.
According to Ebbute Straker (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 above can we get easily if we have a will, attitude, knowledge, skill, and experience.
 
Purposes

The aim of the research of mathematics is to examine and develop mathematics. If we want to be a mathematician we must doing the research of mathematics to develop our subject. If we do many research in mathematics, automatically it will improve our knowledge and experience, beside that, we can develop mathematics as well as we can. It is very useful to the future, especially in sciences.

Method

There are a lot of methods that we can use in the research of mathematics, such:
- by analyze the data,
- by collect data/ literature,
- deductive method /syntetic method, etc.
And in this paper, I use some of the methods above that is collect the data (literature) and analyze it.

Discussion

From the three branchs of mathematics above I will focus in more spesific case. I will focus in applied mathematics. According to me, applied mathematics is very useful in the real world. As we know, one of mathematical attitude is applicable. So it means that mathematics be able to be put to practice use (both practical and theoretical aspects).
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis), and applied probability.
In this case, I will give an example about applied mathematics, that is, the application in which first order linear differential equation be a model to 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. So, we need the mathematical way to approach this problem.
From the experimental observations it is known that, to an 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 environtment(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
dƟ/dt=-k(Ɵ-T ) ...........(i)
where k > 0 is a constant of proportionality. The minus sign equation (i) 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 dƟ/dt <> 0.
Now suppose that at time t = 0 a corpse is discovered, and that its temperature is measured to be Ɵ0. We assume that at the time of death td the body temperature Ɵd had the normal value of 98.6oF for 37oC. If we assume that equation (i) is valid in this situation, then our task is determine td.
The solution of equation (i) subject to the initial condition Ɵ(0)= Ɵ0 is
Ɵ(t) = T +( Ɵ0 - T )e-kt ..........(ii)
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 t1; suppose that Ɵ= Ɵ1 when t = t1. By substituting these value in equation (ii) we find that
Ɵ1 = T +( Ɵ0 - T )e-kt1
hence k = - 1/t_1 ln (θ_1- T)/(θ_2- T) .......(iii)
where Ɵ0, Ɵ1 , T, and t1 are known quantities.
Finally, to determine td we substitute t = td and Ɵ= Ɵd in equation(ii) and solve for td. We obtain
td = - 1/k ln (θ_d- T)/(θ_0- T) ............(iv)
where k is given by equation (iii).
For example, suppose that the temperature of the corpse is 85 oF when discovered and 74 oF two hours later, and that the ambient temperature is 68 oF. Then, from equation(iii)
k = - 1/2 ln (74-68)/(85-68)≈0,5207 hr^(-1)
and from equation(iv)
td = - 1/0,5207 ln (98.6-68)/(85-68)≈-1,129hr
Thus we conclude that the body was discovered approximately 1hr, 8min after death.
 
Conclusion

From the example above, we should realize that applied mathematics is very useful in our life. The application in which first order linear differential equation be a model to 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. This method is necessary to predict how long the body death (the time of death). It just one example of applied mathematics. In fact (especially the application of first order differential equation), can also be used to analyze a number of other financial situations, including annuities, mortgages, and automobile loans among others.

References

http://kirk.math.twsu.edu/appliedmath.html
http://en.wikipedia.org/wiki/Applied_mathematics
Microsoft® Encarta® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
E.B.William, and C.D.Richard. Elementary Differential Equations and Boundary Value Problems. New York: John Wiley & Sons,Inc. 1997.