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

english

1. There is a differential equation :dy/dx = (x^2+ 3y^2)/2xy
2. How to solve the differential equation above and find the solution of the differential equation ?
There is an equation :
Z = 2e^3y cos⁡2x and point P( π/3, 0, -1 )
Determine the tangent plane equation toward the surface of the point above !
3. In a mathematics test, each students can choose 8 questions from 10 equations. If number 4 and 9 should be solve (do), how many choice questions that students can do?
4. Determine mac laurin series for function f(x) = cos (x2) !
5. Obserb whether these line and plane parallel or not ?
h≡(■(x,y,z))=(■(0,4,0))+t(■(5,-2,-2))α≡(■(x,y,z))=(■(2,1,0))+a(■(-1,2,0))+b(■(2,0,-1))

Solving :
1. dy/dx = (x^2+ 3y^2)/2xy the form of homogeneous equation suggest that it may be simplified by introducing a new variable, which will we denote by v, to represent the ratio of y to x. Thus : y = xv ↔ v = y/x
And y = xv becomes dy/dx = x dv/dx+ v
Writing dy/dx = (x^2+ 3y^2)/2xy as dy/dx = (x^2 (1+ 3 y^2/x^2 ))/(x^2 (2 y/x) )
dy/dx = ((1+ 3 y^2/x^2 ))/((2 y/x) )
It shows that this equation is homogenous. Then given equation is neither linear, nor separable, nor exact. So we can substitute dy/dx = x dv/dx+ v into dy/dx = ((1+ 3 y^2/x^2 ))/((2 y/x) ) we obtain :
x dv/dx+ v = ((1+ 3v^2 ))/((2v) )
x dv/dx = ((1+ 3v^2 ))/((2v) )-v
x dv/dx = (1+3v^2-2v^2)/2v
x dv/dx = (1+v^2)/2v
if v ≠ 0, then equation above can be written as 2v/(1+v^2 ) dv= 1/x dx
integrating both sides, assumption : u=1+v^2 and du=2v dv
so , ∫▒〖u^(-1) du= ∫▒1/x〗 dx
〖 ln〗⁡|u|+ln|c_1 |=ln|x|+ln|c_2 |
〖 ln〗⁡|1+v^2 |+ln|c_1 |=ln|x|+ln|c_2 | ↔ 〖 ln〗⁡|1+v^2 |= ln|x|+ln|c|
Where c is an orbitrary constant. Hence, combining the logarithms and taking the exponential of both sides, we obtain :
e^〖 ln〗⁡|1+v^2 | =e^(ln|x|+ln|c| )
1+v^2=cx
v^2=cx-1
Finally, substituting for v in terms of y gives the solution of dy/dx = (x^2+ 3y^2)/2xy in the form :
y^2/x^2 =cx-1
y^2=cx^3-x^2
So, y=√(cx^3-x^2 )

2. Z = 2e^3y cos⁡2x and point P( π/3, 0, -1 )
Assumption : f(x,y)=2e^3y cos⁡2x then,
∇f(x,y)=-4e^3y cos⁡〖2xi+6e^3y cos⁡2xj 〗
Substitution (π/3,0) to equation above
∇f(x,y)=-4e^3.0 cos⁡〖2(π/3)i+6e^3.0 cos⁡2(π/3)j 〗
=-2√3 i-3j
In accordance with theorem the tangent plane equation at ( π/3, 0, -1 ) is
z+1=-2√3 (x-π/3)-3(y-0)
z+1=-2√3 x+2π/√3-3y
2√3 x+3y+z=2π/√3-1
3. Each students can choose 8 questions from 10 questions. Number 4 and 9 should be solve, so each students can choose 6 questions. With used combination formula, we obtain :
C_(8-2)^(10-2)=C_6^8=8!/(8-6)!6!= 8.7.6!/2.6!= 28
So, many ways to choose the questions are 28 ways.

4. f(x)=cos⁡〖(x^2 〗)→f(0)=1
f^' (x)=-2 sin⁡(x^2 )→f^' (0)=0
f^'' (x)=-4 cos⁡〖(x^2 〗) →f^'' (0)=-4
f^''' (x)=8sin⁡(x^2 ) →f^''' (0)=0
f^IV (x)=16cos⁡〖(x^2 〗) →f^IV (0)=16
f^v (x)=-32 sin⁡(x^2 ) →f^v (0)=0
f^VI (x)=-64 cos⁡〖(x^2 〗) → f^VI (0)=-64,etc
So, mac laurin series for f(x)=cos⁡〖(x^2 〗) is
cos⁡〖(x^2 〗)=f(0)+f^' (0)x+(f"(0)x^2)/2!+(f"'(0)x^3)/3!+ …..
cos⁡〖(x^2 〗)=1+0.x+(-4x^2)/2!+(0.x^3)/3!+(16.x^4)/4!+(0.x^5)/5!+(-64.x^6)/6!+⋯
cos⁡〖(x^2 〗)=1-2x^2+(2.x^4)/3-(4.x^6)/15+…..

5. h≡(■(x,y,z))=(■(0,4,0))+t(■(5,-2,-2))
α≡(■(x,y,z))=(■(2,1,0))+a(■(-1,2,0))+b(■(2,0,-1))
Gradient of h and α form become matriks
Assumption=A=[■(5,-2,-2),(-1,2,0),(2,0,-1)]
Then determine determinant A
Det(A) = (■((5,-2,-2),(-1,2,0),(2,0,-1)│■((5,-2,-2),(-1,2,0))=0
Because determinant A equals to zero, so h lines parallel with α plane.

Review of Book

Quality

Viewed in terms of quality, over all the book “Mathematics for Junior High School VIII” by Marsigit included in high quality. Both in terms of physical and contents. The book use two language, they are in English and Indonesia which consists of 240 pages.
Viewd in terms of cover, quality of paper which used is good, with an interesting image and the colour not too bright. the paper which used in that book has high quality and not too thin so that any post in the next side not visible. The output of the book bending are tidy and solid. The ink which used also has good quality, so the written appears clearly, the font of the alphabet of the written is not too small. The book is printed in various kind of colour which is very clearly and with gradiation colour, so it attracts attention from the students to read that book
the book is completed with curve and scheme picture which is use different colour and clearly visible, so it can help student understand subject matter. The design of each capture has a picture which is related to the theme of each capture in the book.
I think that is all my oppinions about that book.

These are the definitions, examples, and explanations of the terms in the rom I made by Heni Ayu Pertiwi, which I write in the English language as a result of a dictionary, thesaurus, learn the English language books, internet download, opened the marsigit’s blog, and also follow Mr.marsigit’s lesson in the class.

1. Akar kuadrat: Square root.

· Square root: A divisor of a quantity that when squared gives the quantity.

For example, the square roots of 25 are 5 and -5 because 5 × 5 = 25 and (-5) × (-5) = 25.

· Square root (Noun): A number that when multiplied by itself gives a given number. The square roots of 4 are 2 and -2 .

2. Nilai mutlak: Absolute value.

• The numerical value of a real number without regard to its sign.

For example, the absolute value of -4 (written -4) is 4. Also called numerical value.

• The modulus of a complex number, equal to the square root of the sum of the squares of the real and imaginary parts of the number.

3.Diintegralkan: Integraled

· A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed.

• A definite integral.
• An indefinite integral.

the sum of a large number of minute quantities, summed either between stated limits ([definite integral]) or in the absence of limits ([indefinite integral])

4. Hasil bagi: quotient

• The number obtained by dividing one quantity by another.

For example, In 45 ÷ 3 = 15, 15 is the quotient.

• The result of the division of one number or quantity by another [Latin quotiens how often]
• The number that results when one number is divided by another. If 6 is divided by 3, the quotient can be represented as 2, or as 6 ÷ 3, or as the fraction ⁶/₃.

5. Kongruen : Congruence.

As an abstract term, congruence means similarity between objects. Congruence, as opposed to equivalence or approximation, is a relation which implies a kind of equivalence, though not complete equivalence.

In mathematics, congruence is formally represented by a tilde over an equal sign , whereas approximation is jointed by the double-tilde . In both cases, the single tilde (~) may sometimes be used as a substitute (which may cause confusion due to its usage in logic and computer programming languages for representing negation). Congruence modulo n is usually denoted "≡" (see congruence relation for an example).

• In arithmetic, algebra and encryption, congruence refers to a congruence relation.
• In euclidean geometry, congruence refers to the equivalence of shapes. See congruence (geometry).
• In the theory of smooth manifolds, especially in the context of general relativity, congruence refers to the integral curves defined by a vector field.
• In psychology and NLP, congruence could be defined as rapport within oneself, or internal and external consistency, perceived by others as sincerity or certainty.
• An alternative definition of congruence in the work of Virginia Satir, one of the sources of NLP, is the balance between self, other and context.
• In cladistics, congruence is a test of homology, or shared, derived character states, in which the distributions of supposed homologies among taxa are compared for consistency.
• horizontal and vertical congruence, two related principles of grouping

6. Sejajar: Parallel.

Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate.

Euclidean Parallelism

As shown by the tick marks, lines a and b are parallel. We can prove this because the transversal t produces congruent angles.

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

1. Every point on line m is located exactly the same minimum distance from line l ('equidistant lines', not including the degenerate case where m = l).
2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction).
3. Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are equal.

In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space. A parallel combination of a line and a plane may be located in the same three-dimensional space. Lines parallel to each other have the same gradient. Compare to perpendicular.

7. Kelipatan : multiple.

1. Having, relating to, or consisting of more than one individual, element, part, or other component; manifold.
2. A number that may be divided by another number with no remainder.For example, 4, 6, and 12 are multiples of 2.
3. the product of a quantity by an integer; "36 is a multiple of 9"
4. mathematical product, product - a quantity obtained by multiplication; "the product of 2 and 3 is 6"

8. Koordinat:coordinate

1. One that is equal in importance, rank, or degree.
2. Mathematics Any of a set of two or more numbers used to determine the position of a point, line, curve, or plane in a space of a given dimension with respect to a system of lines or other fixed references.

Noun1.coordinate - a number that identifies a position relative to an axis

co-ordinate

a) Cartesian coordinate - one of the coordinates in a system of coordinates that locates a point on a plane or in space by its distance from two lines or three planes respectively; the two lines or the intersections of the three planes are the coordinate axes

b) polar coordinate - either of two values that locate a point on a plane by its distance from a fixed pole and its angle from a fixed line passing through the pole

number - a concept of quantity involving zero and units; "every number has a unique position in the sequence"

9. Logika dan Himpunan : logic and collection

Logic and collection: The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.

10. Tegak lurus : Perpendicular.

1. Mathematics, Intersecting at or forming right angles.
2. Being at right angles to the horizontal; vertical. See Synonyms at vertical.

1. at right angles to a given line or surface

2. upright; vertical

Noun1.perpendicular - a straight line at right angles to another line

straight line - a line traced by a point traveling in a constant direction; a line of zero curvature; "the shortest distance between two points is a straight lin

11. Bilangan kuadrat : Square number

Square number : a natural number that can be shown as a result multiple two same number.

For example, 9 is the square number of 3.

12. Luas : Area, wide

Area: size of a surface

For example, a wide rectangle is 16 m

13. Lebih dari sama dengan : more than equal

More than equal: use to indicate a number greater or equal.

For example, x more than equal from y.

14. kurang dari : less than…

Less than : Used to indicate a number is smaller than the other.

For example, x less than 3.denoted by x < y

15. Titik belok : point convolution

Point where changes occur in the concavity of a sunken kurva up to become a concave bottom, or vice versa.For example, point convolution function f (x) = X3 +1 at the point (0, 1).

16. Nilai Ekstrim : Ekstrim value

Ekstrim value : used to determine the value of the maximum curve.

17. Pemisalan : assumption, premise, taking an example

assumption - a statement that is assumed to be true and from which a conclusion can be drawn; "on the assumption that he has been injured we can infer that he will not to play"

18. Saling bersinggungan : mutual touch,jog

19. Saling berpotongan : Have the look of

when two lines meet at one point.For example, lines A and B each have the look in C.

20. Bilangan Ganjil dan Genap : odd number & even number

• Odd number : A natural number not divisible by 2.

For example, 1,3,5,7 ... including the odd number

• Even number: A natural number divisible by 2.

Example the even number is 2,4,6,8….

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