Last week in English class, we discussed about mathematical research. First of all, I want to give the definition. Mathematics is the science of numbers, quantity and space, arithmetic, algebra, trigonometry and geometry are some of the branches of mathematics. Other definition is the study of the relationships among numbers, shapes, and quantities. It uses signs, symbols, and proofs and includes arithmetic, algebra, calculus, geometry, and trigonometry. While mathematical is belonging to, relating to, or used in mathematics. Then the definition of research is methodical investigation into a subject in order to discover facts, to establish or revise a theory, or to develop a plan of action based on the facts discovered. So we can conclude that mathematical research is a method to investigate something/problem and to find the solution use principal of mathematics.
There is relationship between mathematical research and mathematics system. System is a method or set of procedures for achieving something. Mathematics system means a method or set of procedures for achieving something use in mathematics. The following are examples of mathematics system :
1. Group Theory
Group theory is the branch of mathematics that answers the question, such as : "What is symmetry?"
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right.
2. Ring Theory
In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. 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), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
3. Field Theory
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined.
4. Numbers Theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated.
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. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could be fit into a comprehensive deductive and logical system. The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.
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.
7. Number System
In mathematics, a 'number system' is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication. Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.
The aim of mathematical research is to examine and establish new system of mathematics, there are some aspects, namely : definition, axioms, theorems, and rule/law/procedure. Besides that, should be there supports factors in order to the aim can be fact, there are our knowledge of mathematics, our knowledge of the history of mathematics and our knowledge of the mathematician, our knowledge about experiences of developing mathematics, and facultative/ the philosophy of mathematics.
The following, I want to give explanation about the steps of mathematical research:
• Indeepth study references
We should get a lot of references, it is from magazine, newspaper, browsing internet, article and so on.
• Identify formulate the problems
For the first, we should identified it. We should elaborate the WH question, what, who, where, when, how, whom, which, and whose. Find the answer about the topic use WH question.
• Develop method of research
Develop the method of research use principle of mathematics. In order to the outcome more valid.
• Process of research
Be patient, because it need long time and process.
• Publish
After you had finish, publish it.
• Journal
Make a journal about your result.
http://encarta.msn.com/encnet/features/dictionary/DictionaryResults.aspx?lextype=3&search=system
http://en.wikipedia.org/wiki/Group_theory
http://en.wikipedia.org/wiki/Ring_theory
Senin, 28 Desember 2009
Senin, 21 Desember 2009
World Class University
Yogyakarta State of University has a plan to be world class university. It has six faculties, but no all of faculties have bilingual class. Mathematics education and Accounting education department to be first to go international. It is not easy to be world class university, need more time and process. Many procedure has be done in order to Yogyakarta State of University to be world class university, for example have relationship with other country related to academic.
Now, let’s give definition about world class university. World class university is a university which has international level, related to learning system, material/content, learning method and behavior of all person at college, such as : on time, discipline, diligent, responsibility. So that world class university not only use English in learning activities but also has strong character in level international. But master English also play an important rule. Therefore P3B, one of institute of Yogyakarta State of University hold a program In English. It is preparation for world class university, the members of program are student of university who take mathematics education and accounting education department. This program be done during one month from 5 pm until 7 pm. The student of university who join this program were trained by lecturer of English in speaking, grammar, structure and active conversation. But the important thing is all of the element of Yogyakarta State of University ready to go international/become world class university. So all of student should have more motivation to study hard, maybe can changed abroad.
Many steps have done by Yogyakarta State of University to make goal come true. First, strong intention from all of civitas. If all of civitas make unity, it will be good capital to go international, namely world class university. Second, defining world class university (WCU) and continued by socialization. Third, describe definition of world class university on a work program. Fourth, increase the relationship with other country (abroad), for example: exchange student, exchange academic staff, relationship in research and soon. The statements above are some steps in order to Yogyakarta State of University can be world class university, so it has :
1) Excellence in education of their students
2) Research, development and dissemination of knowledge
3) Activities contributing to the cultural, scientific and civic life society.
Now, let’s give definition about world class university. World class university is a university which has international level, related to learning system, material/content, learning method and behavior of all person at college, such as : on time, discipline, diligent, responsibility. So that world class university not only use English in learning activities but also has strong character in level international. But master English also play an important rule. Therefore P3B, one of institute of Yogyakarta State of University hold a program In English. It is preparation for world class university, the members of program are student of university who take mathematics education and accounting education department. This program be done during one month from 5 pm until 7 pm. The student of university who join this program were trained by lecturer of English in speaking, grammar, structure and active conversation. But the important thing is all of the element of Yogyakarta State of University ready to go international/become world class university. So all of student should have more motivation to study hard, maybe can changed abroad.
Many steps have done by Yogyakarta State of University to make goal come true. First, strong intention from all of civitas. If all of civitas make unity, it will be good capital to go international, namely world class university. Second, defining world class university (WCU) and continued by socialization. Third, describe definition of world class university on a work program. Fourth, increase the relationship with other country (abroad), for example: exchange student, exchange academic staff, relationship in research and soon. The statements above are some steps in order to Yogyakarta State of University can be world class university, so it has :
1) Excellence in education of their students
2) Research, development and dissemination of knowledge
3) Activities contributing to the cultural, scientific and civic life society.
Jumat, 04 Desember 2009
Just For Mathematics
Properties of Logarithms
Logarithms is a type of mathematics which related to power and root. That the power of root are used for a basic to learn it. Therefore, we should be familiar and understand abaut that. For example : ten to the second power is one hundreds, the square root of onen hundred and the fourty four is twelve. Let’s start to talk abaut the logarithms and its properties. This explanation I get from browsing in the internet. There is definition Log b X= y
It means that b to the y power equals y. Besides that there is a agreement abaut writing log 10 X. It can be written just log X and log e X equals ln x. It is called as natural logarithms. Then, let’s start to do the exercise to find a solving in logarithms.
If log 10 100 is x. Find the value of x!
• First, make a equation, ten to the x power is one hundred.
• Remember that one hundred is ten to the second power.
• From statement above we can conclude that the value of x is two because ten to the second power is one hunderd. Therefore, two is right answer.
Next question, if log 2 x is three. Find the value of x!
Write that two to the three is x. Therefore x is eight. In logarithms, there are some properties like that:
Log b (M times N) equals log b M and log b N
Log b (M/N) equals log b M minus log b N.
Log b (x^n) equals n times log b x
Let’s practise it. If log 3 (x^3(y+1)/z^3). How find the solution?
First, log 3 (x^3(y+1) minus for log log 3 z.
Second, log 3 x^3 and log 3 (y+1) minus for 3 log 3 z.
Next, 3 log 3 (x^3(y+1)/z^3)= 3 log3 x + log 3 (Y+1) – 3log3 z.
To get more complete explanation and exercise, we can search in the internet or read a bool about logarithms because it just explain the concept or the principle of properties of logarithms
Common Factor and Grouping
( Greatest Common Factors)
I’ m sure that most student know about GCD and LCM. GCD means the greatest common divisor while LCM means the least common multiple. We got them when we studied in Elementary school. In this section, explain of greatest common factors or written GCF in Indonesia “ Faktor Persekutuan Terbesar.
The objective of greatest common factor are :
Find the greatest common factor (GCF) of numbers
Find the GCF of them
Factor out the greatest common facto
Factor a four term expression by grouping and the getting started are product and factor. For example:
20=2x2x5, two and five are prime numbers.
Next, how we got GCF? Finding the greatest common of number are :
The greatest common factor at a list of integers in the largest common factor of the integers in the list.
45=3x3x5=3^2x5
60=2^2x3x5
To find the greatest common factor (GCF), choose prime factors with the smallest exponent and find their product.
For example : find GCF of thirty six, sixty and one hundred and eight.
Thirty six is two square times three square or 36=2^2x3^2
Sixty is two square times three times five or 60=2^2x3x5
One hundred and eight is two square times three cube or 108=2^2x3^2 so the GCF is two square times three equals twelve.
Others way:
2 36 60 108 thirty six, sixty and one hundred and eight divided by
2 18 30 54* two , the result is * then divided by two again, the result
3 9 15 27# is #. After that divided by three.
3 5 9
Function
Based on Oxford Advanced Learner’s dictionary, function is a quantity whose value depends on the varying value of others. In the statement 2x=y, y is a function of x, or functioin is an algebra statement that providers a link between two or more variable . function used to find the value of one variable. If you know the value of the others. y=2x, if you know x, you can find the value of y.
For example x is five so that the value of y is two times five equals ten. One variable appears by itself, on one side of the equation y=3x+4. A function is a codependent relationship between gx’s dan y’s without x, you can’t get y. Function related in which each element of one set is paired with one and only one, element of the second set. Besides that any numerial expression relations one number, or set of numbers, to another. There are two kinds of relations namely equations and inequalities. The sample of equation is 1+3=4 while inequality is 8>5 (eight more than five).
Specific numbers are being related to each other nonspecific numbers. Nonspecific number related to variable, namely x and y. X and y related to variables relations which have expression that contain variables, example: 2x+3=9y.
Equation and inequalities are the kind that can be used to determine just one value for one of the variable. Y=3x+4, when you substitute a value for x, you can calculate just one value for y. Y=3x+4 an equation with one variable by itself on one side, one variable is a function of whatever variable appear on the other side. Function of x, f(x)=y, y=3x+4, 3x+4 is function of x so y=f(x)=3x+4. F(x)=3x+4 is standart form, put functions that are not in standart form into standart form.
Given an equation for a function, you can calculate the function as soon as you know a value of x
g(x)=x^2-3x+2, x is five so that , g(x)=5^2-3.5+2
five to the second power minus three times five plus two is twelve.
Parallelogram
Geometri is branch of mathematics dealing with measurement and relationship of lines, angles, surfaces and solids. In geometri, we know about plane, such us: triangle, rectangle, square, rhombus and parallelogram.
Triangle has three sides and sum of the angle is one hundred and eighty degree. There are some kinds of triangle, namely: right triangle, scalene triangle and isosceles triangle. To more complete, we can browse in the internet or find the geometry book. In this section, we just talk about parallelogram. Before, I want to give definition about quadrilateral. Quadrilateral is space which has four sides. If a quadrilateral is a parallelogram, then opposite sides ara parallel.
Picture parallelogram
If ABCD is a parallelogram, then ABIIDC
ADII BC
Parallelogram has four sides, has four angles and has two pairs of parallel sides. The sum of the angle of parallelogram is three hundred and sixty degree.
Trigonometri Functions
Based on Oxford Advanced Learner’s dictionary, trigonometri is the type of mathematics that deals with the relationship between the sides and angles of triangle. Trigonometri functions only need to know values of sides to find measure of an angle figure out values of all part of a triangle. There are six basic trigonometri functions:
1. Sine
2. Cosine
3. Tangent
4. Cosecant
5. Secant
6. Cotangent
Basic trigonometri functions define six basic based on: 1) sides of a triangle, 2) Angle being measured
To make easily we learnt about trigonometri functions, let’s use a picture. It’s difficult is we don’t use picture to make us understand
OPP
Notes: OPP is side opposite theta
ADJ is side adjacent to theta
HYP is hypotenusa, the following are formula for six basic trigonometri function
Sin=OPP/HYP
Cos=ADJ/HYP
Tan=OPP/ADJ
Csc=HYP/OPP
Sec=HYP/ADJ
Cot=ADJ/HPP
For example: in this picture of triangle , find the value of sin, cos and tan!
Factoring Polynomials
Based on Oxford Advanced Learner’s dictionary, factorial means the result when you multiply a whole number by all the number below. But, in this section we don’t talk about factorial but factoring it’s different. Polynomial have many terms. The form of polynomials is a0+a1x+a2x^2+...
Now let’s learn about Factoring polynomials. Before, I want to review that factor of fiveteen is five and three. Five times equals fiveteen so five and three are the factoring of fiveteen. Next, is x minus three (x-3) a factor of x^3-7x-6? (x cube minus seven x minus six?) or we can write =(x^3-7x-6)/(x-3), how we do that?
x-3 x^3+0x^2-7x-6 x^2+3x+2
x^3-3x^2
3x^2-7x
3x^2-9x
2x-6
2x-6
0
So x-3 (x minus three) is a factor of x^3-7x-6. X^3-7x-6 divided by x-3 equals x^2+3x+2 ( it also a factor of x^3-7x-6). So we can write that x^3-7x-6= (x-3) (x^2+3x+2)
= (x-3) (x+2) (x+1)
X=3 atau x=-2 atau x=-6
The conclution
X cube minus seven x minus x has three roots, for this 3rd degree equation. Quadratic (2nd degreea) equation always have at most two roots. Ath degtee equation would have four or fewer roots. The degree at a polynomial equation always limits the number of roots. Nest, if there are long divisi for a 3rd order polynomial, the steps to find solution are : find a partial equation of x^2, by dividing x into x^3 to get x^2. Multiply x^2 by the divisor and substract the product from the diveident. Repeat the process until you either “clear it out” or reach a remainder.
Logarithms is a type of mathematics which related to power and root. That the power of root are used for a basic to learn it. Therefore, we should be familiar and understand abaut that. For example : ten to the second power is one hundreds, the square root of onen hundred and the fourty four is twelve. Let’s start to talk abaut the logarithms and its properties. This explanation I get from browsing in the internet. There is definition Log b X= y
It means that b to the y power equals y. Besides that there is a agreement abaut writing log 10 X. It can be written just log X and log e X equals ln x. It is called as natural logarithms. Then, let’s start to do the exercise to find a solving in logarithms.
If log 10 100 is x. Find the value of x!
• First, make a equation, ten to the x power is one hundred.
• Remember that one hundred is ten to the second power.
• From statement above we can conclude that the value of x is two because ten to the second power is one hunderd. Therefore, two is right answer.
Next question, if log 2 x is three. Find the value of x!
Write that two to the three is x. Therefore x is eight. In logarithms, there are some properties like that:
Log b (M times N) equals log b M and log b N
Log b (M/N) equals log b M minus log b N.
Log b (x^n) equals n times log b x
Let’s practise it. If log 3 (x^3(y+1)/z^3). How find the solution?
First, log 3 (x^3(y+1) minus for log log 3 z.
Second, log 3 x^3 and log 3 (y+1) minus for 3 log 3 z.
Next, 3 log 3 (x^3(y+1)/z^3)= 3 log3 x + log 3 (Y+1) – 3log3 z.
To get more complete explanation and exercise, we can search in the internet or read a bool about logarithms because it just explain the concept or the principle of properties of logarithms
Common Factor and Grouping
( Greatest Common Factors)
I’ m sure that most student know about GCD and LCM. GCD means the greatest common divisor while LCM means the least common multiple. We got them when we studied in Elementary school. In this section, explain of greatest common factors or written GCF in Indonesia “ Faktor Persekutuan Terbesar.
The objective of greatest common factor are :
Find the greatest common factor (GCF) of numbers
Find the GCF of them
Factor out the greatest common facto
Factor a four term expression by grouping and the getting started are product and factor. For example:
20=2x2x5, two and five are prime numbers.
Next, how we got GCF? Finding the greatest common of number are :
The greatest common factor at a list of integers in the largest common factor of the integers in the list.
45=3x3x5=3^2x5
60=2^2x3x5
To find the greatest common factor (GCF), choose prime factors with the smallest exponent and find their product.
For example : find GCF of thirty six, sixty and one hundred and eight.
Thirty six is two square times three square or 36=2^2x3^2
Sixty is two square times three times five or 60=2^2x3x5
One hundred and eight is two square times three cube or 108=2^2x3^2 so the GCF is two square times three equals twelve.
Others way:
2 36 60 108 thirty six, sixty and one hundred and eight divided by
2 18 30 54* two , the result is * then divided by two again, the result
3 9 15 27# is #. After that divided by three.
3 5 9
Function
Based on Oxford Advanced Learner’s dictionary, function is a quantity whose value depends on the varying value of others. In the statement 2x=y, y is a function of x, or functioin is an algebra statement that providers a link between two or more variable . function used to find the value of one variable. If you know the value of the others. y=2x, if you know x, you can find the value of y.
For example x is five so that the value of y is two times five equals ten. One variable appears by itself, on one side of the equation y=3x+4. A function is a codependent relationship between gx’s dan y’s without x, you can’t get y. Function related in which each element of one set is paired with one and only one, element of the second set. Besides that any numerial expression relations one number, or set of numbers, to another. There are two kinds of relations namely equations and inequalities. The sample of equation is 1+3=4 while inequality is 8>5 (eight more than five).
Specific numbers are being related to each other nonspecific numbers. Nonspecific number related to variable, namely x and y. X and y related to variables relations which have expression that contain variables, example: 2x+3=9y.
Equation and inequalities are the kind that can be used to determine just one value for one of the variable. Y=3x+4, when you substitute a value for x, you can calculate just one value for y. Y=3x+4 an equation with one variable by itself on one side, one variable is a function of whatever variable appear on the other side. Function of x, f(x)=y, y=3x+4, 3x+4 is function of x so y=f(x)=3x+4. F(x)=3x+4 is standart form, put functions that are not in standart form into standart form.
Given an equation for a function, you can calculate the function as soon as you know a value of x
g(x)=x^2-3x+2, x is five so that , g(x)=5^2-3.5+2
five to the second power minus three times five plus two is twelve.
Parallelogram
Geometri is branch of mathematics dealing with measurement and relationship of lines, angles, surfaces and solids. In geometri, we know about plane, such us: triangle, rectangle, square, rhombus and parallelogram.
Triangle has three sides and sum of the angle is one hundred and eighty degree. There are some kinds of triangle, namely: right triangle, scalene triangle and isosceles triangle. To more complete, we can browse in the internet or find the geometry book. In this section, we just talk about parallelogram. Before, I want to give definition about quadrilateral. Quadrilateral is space which has four sides. If a quadrilateral is a parallelogram, then opposite sides ara parallel.
Picture parallelogram
If ABCD is a parallelogram, then ABIIDC
ADII BC
Parallelogram has four sides, has four angles and has two pairs of parallel sides. The sum of the angle of parallelogram is three hundred and sixty degree.
Trigonometri Functions
Based on Oxford Advanced Learner’s dictionary, trigonometri is the type of mathematics that deals with the relationship between the sides and angles of triangle. Trigonometri functions only need to know values of sides to find measure of an angle figure out values of all part of a triangle. There are six basic trigonometri functions:
1. Sine
2. Cosine
3. Tangent
4. Cosecant
5. Secant
6. Cotangent
Basic trigonometri functions define six basic based on: 1) sides of a triangle, 2) Angle being measured
To make easily we learnt about trigonometri functions, let’s use a picture. It’s difficult is we don’t use picture to make us understand
OPP
Notes: OPP is side opposite theta
ADJ is side adjacent to theta
HYP is hypotenusa, the following are formula for six basic trigonometri function
Sin=OPP/HYP
Cos=ADJ/HYP
Tan=OPP/ADJ
Csc=HYP/OPP
Sec=HYP/ADJ
Cot=ADJ/HPP
For example: in this picture of triangle , find the value of sin, cos and tan!
Factoring Polynomials
Based on Oxford Advanced Learner’s dictionary, factorial means the result when you multiply a whole number by all the number below. But, in this section we don’t talk about factorial but factoring it’s different. Polynomial have many terms. The form of polynomials is a0+a1x+a2x^2+...
Now let’s learn about Factoring polynomials. Before, I want to review that factor of fiveteen is five and three. Five times equals fiveteen so five and three are the factoring of fiveteen. Next, is x minus three (x-3) a factor of x^3-7x-6? (x cube minus seven x minus six?) or we can write =(x^3-7x-6)/(x-3), how we do that?
x-3 x^3+0x^2-7x-6 x^2+3x+2
x^3-3x^2
3x^2-7x
3x^2-9x
2x-6
2x-6
0
So x-3 (x minus three) is a factor of x^3-7x-6. X^3-7x-6 divided by x-3 equals x^2+3x+2 ( it also a factor of x^3-7x-6). So we can write that x^3-7x-6= (x-3) (x^2+3x+2)
= (x-3) (x+2) (x+1)
X=3 atau x=-2 atau x=-6
The conclution
X cube minus seven x minus x has three roots, for this 3rd degree equation. Quadratic (2nd degreea) equation always have at most two roots. Ath degtee equation would have four or fewer roots. The degree at a polynomial equation always limits the number of roots. Nest, if there are long divisi for a 3rd order polynomial, the steps to find solution are : find a partial equation of x^2, by dividing x into x^3 to get x^2. Multiply x^2 by the divisor and substract the product from the diveident. Repeat the process until you either “clear it out” or reach a remainder.
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