THE RESEARCH OF MATHEMATICS

INTRODUCTION
The nature of mathematics consist of formal mathematics/axiomatic mathematics/pure mathematics, applied mathematics, school mathematics/ concret mathematics/real 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 world, in optic, mechanic, astronomy, engineering, etc. For example we use angle in compass if going to sea, mountain,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.
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.
PREFACE
In this time, I try to take a research of formal mathematics, especially about number theory. Formal mathematics builds on formal logic. It reduces mathematical relationships to questions of set membership. The only undefined primitive object in formal mathematics is the empty set that contains nothing at all.
BACKGROUND
There is a lot of mathematical knowledge. This knowledge is mainly stored in books and in the heads of mathematicians and other scientists. Mathematics should be more readily available to be used by others. In this respect, a positive thing about mathematical knowledge is that it has a rather formal, which makes it easier to formalize.
A number is a mathematical object used in counting and measuring. There a lot of kinds of number. If we study mathemathics, we can not without number, whereas there are many kinds of number. So, in this time, i try take a research about formal mathematics, especially about number theory.
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. To be able to productively develop mathematical theories with all theorems and proofs formal
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
-by hermeunitika method, etc
DISCUSSION
Number theory is one of the oldest branches of pure mathematics, and one of the largest. Of course, it concerns questions about numbers, usually meaning whole numbers or rational numbers (fractions).
Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated, such us: Elementary number theory, analytic number theory, algebraic number theory, geometry of number, combinatorial number theory, computational number theory, Arithmetic dynamics, Modular forms, etc.
Elementary number theory involves divisibility among integers. ", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets Z/nZ as commutative rings), including Fermat's little theorem and Euler's theorem extending it. But the term "elementary" is usually used in this setting only to mean that no advanced tools from other areas are used not that the results themselves are simple.
In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold.
The geometry of numbers incorporates some basic geometric concepts, such as lattices, into number-theoretic questions. It starts with Minkowski's theorem about lattice points in convex sets, and leads to basic proofs of the finiteness of the class number and Dirichlet's unit theorem, two fundamental theorems in algebraic number theory.
Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers.
Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.
CONCLUSION
From the above we have field has application in mathematics. Number theory has been applied to: physics, chemistry, biology, computing, engineering, coding and cryptography, random number generation, acoustics, communications, graphic design and even music and business. Most of common people can not applicate the number theory in the dailyy life.








REFERENCES
Encarta
http://en.wikipedia.org/wiki/Number_theory(23-12-09)
http://www.math.niu.edu/~rusin/known-math/index/11-XX.html923-12-09)
http://www.mtnmath.com/whatth/node21.html(23-12-09)

1. Find solution of differential equation y’= єy – αy3 , є>0 and α>0 !
Answer:
y’= єy – αy3 , є>0 and α>0
Rewriting equation in the standard from
y’= єy – αy3 ,…………….(1) (left and right side divided by y3)
» y’/ y3= єy/ y3 – αy3 /y3………….(2)
» (1/ y3)dy/dt = є/ y2 – α ……………………..(3)
Suppose: u = 1/ y2 = y-2
du=-2/ y3 so, -1/2 du = dy/ y3
substitution dy/ y3 and y-2 in equation (3), we have
(du/-2dt) – є u = -α…………….(4)
(-1/2) u’ – є u = -α ……………..(5)
To find this solution,we first compute µ(t)
Left and right side of (5) multiply by -2 µ(t), we have
µ(t) u’ + 2єu µ(t) = 2αµ(t) ………..(6)
µ(t) u’ + µ’(t) u = (µ(t)u)’ ………..(7)
so that,equation (6) =(7)
µ’(t)u = 2єu µ(t)
µ’(t) = 2є µ(t)
µ(t) = e ∫2єdt
µ(t) = e 2єt+ c ,e c= k, constant
we have µ(t) = e2єt
substitution µ(t) = in equation (7), we obtain
e2єtu’ + u e2єt = (e2єtu)’ ……………(8)
and therefor (e2єtu)’ = ∫ 2α e2єtdt
so we have u = (α + c є e-2єt) / є
where u = 1/ y2, it follow that
y2 = 1/ u
y = ± √(1/u)
y = ± √(є/(α + c є e-2єt))
The solution becomes y = ± √(є/(α + c є e-2єt))

2. Prove if {an} konvergen and {bn} divergen , { an + bn }divergen!
Answer:
{an} konvergen, suppose : {an} = 1/n
{bn} divergen, suppose {bn} = n, n2
a 1-5 , where n = 1-5
b 1-5 ,where n = 1-5
a 1-5 = 1/1 , ½, 1/3, ¼,1/5,……..,1/n (1)
b 1-5 = 1, 2, 3, 4, 5, ………….,n (2)
____________________________________ _____ +
2, 5/2, 7/3, 17/4, 26/5,……, (n + 1/n)
{a 1-5 + b 1-5} = 2, 5/2, 7/3, 17/4, 26/5
So, , { an + bn }divergen if {an} konvergen and {bn} divergen
3. In triangle the measure of side is 30, 17, and 26. Determine measure of each angles !
Answer:
C



A B
The three of angles which will counted measure is α, β, γ.
AC = 17, AB = 30, BC = 26
α is angle of CAB, β is angle of ABC, γ is angle of BCA
a. For determine measures α, using cosine
BC2 = AC2 + AB2 – 2 AC.BC cos α
262 = 172 + 302 – 2.17.30 cos α
Cos α = 0,5029
Using calculator or table , value of α = 59,800
b. For determine measure of β, using sine
BC /sin α = AC / sin β
26/ sin 59,80 = 17/ sin β
Sin β = (17. Sin 59, 800)/26
Sin β = 0,5623
β = 34,210
c. Counting measure of γ angle
γ = 1800- (α+β)
γ = 1800- (59,800+ 34,210)
γ = 85,990

4. Prove if C bisector of PAQ angle and AP =AQ . So, APC angle congruent with AQC angle!
Answer:
Triangle APQ, AP ≈ (congruent) AQ
AP = line segment af AP
Angle APC ≈ angle AQC
• AP ≈ AQ (measure of AP = AQ)
• Э C € PQ ( exsistence one point in line segment )
• Э AC ( exsistence line segment from two point)
• Э < QAC and < PAC ( existence of angles)
• Э ! AC
< QAC ≈ < PAC ( existence bisector of angle)
• ∆ QAC and ∆ PAC (existence teiangle)

AP ≈ AQ (given)
AC = AC ( reflektif)
< QAC ≈ < PAC
• ∆ QAC ≈ ∆ PAC (existence side – angle - side)
• Because ∆ QAC ≈ ∆ PAC,
So, PC ≈ QC
< PCA ≈ < QCA
< APC ≈ < AQC
So, we can prove that APC angle congruent with AQC angle

Assigment 5

1.Prove if C bisector of PAQ angle and AP =AQ . So, APC angle congruent with AQC angle!
Answer:
Triangle APQ, AP ≈ (congruent) AQ
AP = line segment af AP
Angle APC ≈ angle AQC
• AP ≈ AQ (measure of AP = AQ)
• Э C € PQ ( exsistence one point in line segment )
• Э AC ( exsistence line segment from two point)
• Э < QAC and < PAC ( existence of angles)
• Э ! AC
< QAC ≈ < PAC ( existence bisector of angle)
• ∆ QAC and ∆ PAC (existence teiangle)

AP ≈ AQ (given)
AC = AC ( reflektif)
< QAC ≈ < PAC
• ∆ QAC ≈ ∆ PAC (existence side – angle - side)
• Because ∆ QAC ≈ ∆ PAC,
So, PC ≈ QC
< PCA ≈ < QCA
< APC ≈ < AQC
So, we can prove that APC angle congruent with AQC angle

I am aware English is very IMPORTANT for communicating Mathematics at International Level

Now, I am Aware that English is very Important for Communicating Mathematics at International Level
English is considered to be the most important and common language of the world today. A great number of people understand and use English in every part of the world. Because of its importance I have chosen English as a second language after Indonesian.

English is the most useful language. Being good at English, we can travel to any place or any country we like. We shall not find it hard to make others understand what we wish to say.

In Fact, English has become the international language. Everywhere people talk English. Because of his popularity, innumerable books have been written in English. The English help to spread ideas and knowledge to all the corners of the world. There is no subject that cannot be learned in the English language. So one of my ways to improve my English is read books on variety of subjects: economics, philosophy, physiology, mostly books which linking to Mathematics such geometry, calculus,algebra, history of mathematics,etc.

To acquire correct and fluent English I should learn basic rules of English Grammar and practice speaking English. Like idiom:"Practice makes perfect". I should frequently listen to native speakers through videos, television, and radio. Taking action is better than waiting, even if it's just a small step.

Now I am aware that English is important. Being aware of the preeminence of the English language in every aspect of man's activity, I have been learning it great zeal and avidity. I remember the phrases: If you dream it, you can achieve it.

Now I am study in Mathematics study program. That's mean I should have communicated English for Mathematics. As we know, English has been used in all important meeting and conferences at International level. As a person who know English, easily get more knowledge from many parts of the world. It is for all these reasons that I want to improve my English. So...What will I do? The most frequently heard question in my life maybe "What will I do?". My answer to this question can start changes in my life. Then I am describing the how and why of what I will doing.

I will study more diligent to improve my English for Mathematics. That's the way to enter International level. Positive attitude and action will lead me to better life through Mathematics because Mathematics not isolated from part of our life.

Book Review "Mathematic for Junior High School Years IX" By Marsigit

Preface

First of all, reviewrs be grateful ot Allah SWT because we still given ability to review Mathematic for Junior High School IX by Marsigit. Thank’s to Mr. Marsigit who give us opportunity to review his book.We feel happy can review your’s book.
We get some benefit because review book. For examples,we know content of book,we know scheme of book, quality of book, method of writing, information and we get more knowledge.
We hope this review book useful. Especially to student and teachers of Junior High School before buy mathematic book.The book Mathematic for Junior High School by Marsigit is good,proper to use. The book completed two language (bilingual),Indonesian and English language. The content is complete,that use language don’t difficult to understand.
To release of the review book has been made possible due to the assistance and contributions of various people who cannot mention one by one. To all who involved in this preparation of this review book, I wouldlike to express my high appreciation.
Comment and suggestion,we always welcome. Thank you.





Scheme Of Book

My name is Ika Indriyati. I will review the book “Mathematic for Junior High School Years IX” by Mr. Marsigit from the aspect of Scheme of book. The scheme of book that is good. The book is attractive because of bilingual, Indonesian and English Language, so the student not only learn mathematic but also about English language in mathematic. Cover of this book, we can see a building that can let you know that the content of the book is in English. The book consist of five chapters. In each chapter served very interest,such us presented with full colour, so the students will be easier to understand and remember that content. Images that also provided good, even print the picture is also very clear and no picture is not clear or blur. The content of this book is presented detailedly and in face chapter accompanied by pictures. That picture is application of content it. In every chapter of this book is given basic competence. The scheme of book divided into definition, questions problem and problem solving is given in every chapter. And there are some examples of final exam, national exam. Then, in this book completed by more exercise,that’s variously.Finally,the book “Mathematic for Junior High School Years IX” by Mr. Marsigit is good, proper used by students and teachers in Junior High School Years IX.

By : IKA INDRIYATI
NIM :08305141020

English1

1.Factoring (memfaktorkan)
If a polynomial is written as a produc of other polynomials, then each of the latter polynomials is called a factor of original polinomial. The process of finding such a product is called factoring. Since x2-1 = (x + 1) (x -1), we know that x+1 and x-1 are factor of x2-1. The concept of factor can be extended to general algebraic expressions.Our main use for factoring,however,is in the simplification of expressions which are made up of polynomials.
Example : factor 6x2 -7x-3
Solution :If we write 6x2 -7x-3 = (ax + b)(cx + d), the product of a and c is 6, whereas the product of b and d is -3. Trying various possibilities, we arrive at the factorization 6x2 -7x-3 = (2x -3)(3x + 1)


2. Solution (penyelesaian)
Solution is a product or answer of questions.
Example : find solutions of 2x -5 = 3
Solution : 2x – 5 = 3
2x = 8
x = 4
Solution x = 4


3. Solution Set (himpunan solusi)
Solution set is a set with solutions. Symbol: HS = {….}.
Example : Find HS of 2x + 6 = 0 , x € R
Solution : 2x + 6 = 0
2x = -6
X = -3
HS = {-3}


4. System of Linier Equation (Sistem Persamaan Linier)
Linier equations with n variable x1, x2,…..,xn is a equations form:
a1x1 + a2x2 + …..+anxn = b,
with a1, a2, ….., an , b € R
System of linier equation with n variable x1, x2, ….., xn and m equations, is equations form:
a11x1 + a12x2 +…..a1jxj + …..+ a1nxn = b1
a21x1 + a22x2 +…..a2jxj + …..+ a2nxn = b2
am1x1 + am2x2 +…..amjxj + …+ amnxn = bm
with aij € R, i =1,2,….,m dan j = 1,2,….,n
Example :2x + 7y = 8
2x + 3y = 6


5. Cuadratic Form (bentuk form)
Quardratic form is a number degreed two (a2), a € R..
Example : Find quadratic form of 16
Solution :162 = 256


6. Tangent (garis singgung lingkaran)
If one distinct point are selected on the circumference of a circle is called tangent.
. tangent


7. Ekstrim of Point
General definition of the quadratic function is
f(x) = ax2 + bx + c
Ekstrim of point (-b/2a, -[b2- 4ac]/4a)


8. Base (bidang alas)
If we have a form, side in above is called base.
Example : base of conical is circle.


9. Completing the square….(melengkapkan kuadrat sempurna)
Completing the square is a way to solve equation system. We know that ax2 + bx + c = 0
May be written : a(x2 + b/a x+ c = 0
a(x + b/2a)2 + c – b2/4a = 0

11. Prove ( akan dibuktikan)
If we have theorm or something we must prove that.
Example : Theorm. For all a, b, c € R, a(b-c) = ab-ac
Proof : a(b-c) = a[b + (-c)] definition of subtraction
= ab + a(-c) distributive law
=ab – ac definition of subtraction


12. Inequality ( Pertidaksamaan)
If p and q are algebraic expressionsin variable x, the a statement of the form p < q or p > q, is called an inequality. If a true statement is obtained when x is replaced by a real number a,then a is called a solution of the inequality. Suppose p < q is an inequality, when p and q are algebraic expressions in a variable x. If r is another algebraic expressions in x, then
• The inequality p < q is equivalent to the inequality p + r < q + r
• If the value of r is positive for all value values of x , then the inequality p < q is equivalent to p r < q r
• If the value of r is negative for all value values of x , then the inequality p < q is equivalent to p r > q r
Example : solve 4x – 3 < 2x + 5
Solution :4x – 3 < 2x + 5
(4x – 3)+(3 – 2x)< (2x + 5)+(3 – 2x)
2x < 8
x < 4
The solution set HS = {x € R I x < 4}


13. Determinant (determinan)
Associated with each square matrix A is a number called the d eterminant of A. Determinants of square matrices can be use to solve system of linier equations when the number of equations is the same as the number of variables.
The determinant of matric a11 a12
a21 a22
|A| = a11 a22 - a12 a21

Example : Find |A| of 2 -1
4 -3
Solution |A| = (2)(-3) – (4)(-1) = -2


14. Polynomials (Suku banyak)
A polynomial is any sum of monomials. A polynomials in a variable x (with real coefficiens) as any sum of the form axk , where a is a real number and k is non negative integer. Thus any such polynomial can be written:
anxn + an-1xn-1+ ……+ a1x + a0
where n is a nonnegative integer and the coefficient a0, a1,….., an are real number. Each akxk is called a term of the polynomials.
Example : x4 + 16x3 – 3x2 + 19x + 90


15. Have any Solutions (memiliki banyak penyelesaian)
In system of linier equations,that have any solution if each linier equations have a proportional of element same.
ax + by = c
px + q y = r
Have any solution if a/p = b/q = c/r


17. Permutation (Permutasi)
If we have a collection (set) of object, then each different ordering or arrangement which can be obtained by taking some or all of the objects is called permutations. Consider the three letter A, B, C. All permutations of these letter which can be obtained by taking two time appear in the following list:
(A,B), (B,A), (C,A)
(A,C), (B,C), (C,B)

18. Complement of…(Sekawan dari….)
Example: complement of x is –x

19. Hipotenusa (sisi miring)
In triangle,hipotenusa is longest line.

21. Truncated cone (kerucut terpancung)
If conical cross section of two segment called is truncated cone.

22. Space diagonal (diagonal ruang)

23. Root of….(akar dari…)
Example: Root of 16 is 4.
24. The biggest Factor ( FPB)
Example: FPB of 6 and 8 is 2.
25. Titik potong (cutting point)

1. Penyelesaian: Solution
2. Sistem Persamaan linier: linier equation system
3. Bentuk kuadrat:cuadratic form
4. Garis singgung (lingkaran): tangent
5. Bidang alas : base (base of......)
6. Akan dibuktikan: proof
7. Determinan: determinan
8. Suku banyak: polinom
9. Memiliki solusi banyak: have any solution
10. Himpunan solusi: solution set
11. Turunan fungsi: derivatif
12. Permutasi: permutation
    
13. Luas permukaan: surface area of.........
14. Kerucut: conical
15. Diagonal ruang: space diagonal
16. Akar dari: root of........
17. Berlainan pihak: in other side
18. Sudut berseberangan: cross angle