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.
Minggu, 24 Mei 2009
Small Dictionary of Mathematics
Although it's about english of mathematics, but for the first time I want to write about vocabularies which often we used. We often get difficulties to translate or know the meaning of them. Pay attention!!
we know that the meaning of break is "patah", but.. please read it!
break away : lolos break down : memerinci
break in : mendobrak break open : membuka, membongkar
broken down : rusak (ks) broken hearted : patah hati
broken with : memutuskan hubungan
Come
come about : terjadi come across : menemukan
come after : menjemput come along : berjalan
come round : mampir come before : menghadap
come between : memisahkan come down : datang
come in for : mendapat come into : mewarisi
come through : lewat come within : tercakup
go
go for : menyerbu go down : turun
go by : melewati go through : mengalami
take
take about : membawa take for : mengira
take down : melepas take of : keberangkatan (plane)
Yeah, it's my new vocabularies. from one word but can be made more meaning, depend the contex.
Now it is my small dictionary of mathematics
segitiga siku-siku : right triangle
One of the angle of right triangle is ninety degree.
segitiga sebarang : scalene triangle
The sum of the angle of scalene triangle is one hundred and eighty degree.
segitiga samakaki : isosceles triangle
The isosceles triangle have two same sides.
lancip : acute
the acute angle has measure more than zero but less than ninety degree.
FPB : the greatest common divisor (GCD)
the GCD of two and four is two.
KPK : the least common multiple (LCM)
The LCM of two and four is four.
keliling : circumference, perimeter.
If we have a picture,the length is 20 cm and its width is 10 cm. so the perimeter of the picture is 60 cm.
Bilangan asli : natural number
the natural number is started by one.
Garis bilangan : number line
for early study about integer we can use number line to helping us find the solution/answer.
pengetian : concept
Before we study, we should know its concept.
Jari-jari : radius
the radius of the circle is 7cm, then the area of circle is 154 cm^2.
busur derajat : protactor
what is protactor, what's for?
sejajar : parralel
the parallel line don't have common point.
berimpit : superimpose
the superimpose line have infinite solution because it has infinite common point.
himpunan kosong : empty
The set of cows which have five legs is empty.
The above is small dictionary of mathematics (use sentences) and now, it is small dictionary without sentences
suku sejenis : like term
gabungan himpunan : union
berpotongan : intersection
titik persekutuan : common point
segiempat : quadrilateals
garis bagi : bisector
garis berat : median
garis sumbu : perpendicular bisector
pembilang : numerator
penyebut : denominator
titik singgung : tangent
titik kuasa : power of point
garis potong : secant
sudut pusat : central angle
talibusur : chord
sebanding : proporsional
bola : sphere
kongruen : congruent
jajargenjang : parallelogram
alas :base
trapesium : trapezoid
sebangun : similar
bersilangan : cross
aneh : strange
sering muncul : modus
nama samaran : nickname
terkejut : surprised
sama dan sebangun : congruent
menjiplak : plagiarize
pembulatan : rounded
bingung : confused
kalem : calm
semboyan : maxim
pola : pattern
hakikat matematika : natural of mathematics
bangun datar : plane
bangun ruang : solid
acak : random
kabur : blur
pasrah : give up
clean up : menyelesaikan
varnish : menghapus
turmoil : kerusuhan
fate : nasib
rata-rata: mean, average
jiwa : soul
kecewa : dissapointed
menghindari : avoid
sempurna : perfect
garis lengkung : curve
barisan : sequence
permukaan : surface
cabang : branch
uang kertas :note
uang logam : coin
sebarang : scalene
konsisten : consistent
persegi : square
persegi panjang : rectangle
lingkaran :circle
tabung : sylinder
prisma : prism
penaksiran : estimation
mengungkap : uncover
anggapan : assumsed
we know that the meaning of break is "patah", but.. please read it!
break away : lolos break down : memerinci
break in : mendobrak break open : membuka, membongkar
broken down : rusak (ks) broken hearted : patah hati
broken with : memutuskan hubungan
Come
come about : terjadi come across : menemukan
come after : menjemput come along : berjalan
come round : mampir come before : menghadap
come between : memisahkan come down : datang
come in for : mendapat come into : mewarisi
come through : lewat come within : tercakup
go
go for : menyerbu go down : turun
go by : melewati go through : mengalami
take
take about : membawa take for : mengira
take down : melepas take of : keberangkatan (plane)
Yeah, it's my new vocabularies. from one word but can be made more meaning, depend the contex.
Now it is my small dictionary of mathematics
segitiga siku-siku : right triangle
One of the angle of right triangle is ninety degree.
segitiga sebarang : scalene triangle
The sum of the angle of scalene triangle is one hundred and eighty degree.
segitiga samakaki : isosceles triangle
The isosceles triangle have two same sides.
lancip : acute
the acute angle has measure more than zero but less than ninety degree.
FPB : the greatest common divisor (GCD)
the GCD of two and four is two.
KPK : the least common multiple (LCM)
The LCM of two and four is four.
keliling : circumference, perimeter.
If we have a picture,the length is 20 cm and its width is 10 cm. so the perimeter of the picture is 60 cm.
Bilangan asli : natural number
the natural number is started by one.
Garis bilangan : number line
for early study about integer we can use number line to helping us find the solution/answer.
pengetian : concept
Before we study, we should know its concept.
Jari-jari : radius
the radius of the circle is 7cm, then the area of circle is 154 cm^2.
busur derajat : protactor
what is protactor, what's for?
sejajar : parralel
the parallel line don't have common point.
berimpit : superimpose
the superimpose line have infinite solution because it has infinite common point.
himpunan kosong : empty
The set of cows which have five legs is empty.
The above is small dictionary of mathematics (use sentences) and now, it is small dictionary without sentences
suku sejenis : like term
gabungan himpunan : union
berpotongan : intersection
titik persekutuan : common point
segiempat : quadrilateals
garis bagi : bisector
garis berat : median
garis sumbu : perpendicular bisector
pembilang : numerator
penyebut : denominator
titik singgung : tangent
titik kuasa : power of point
garis potong : secant
sudut pusat : central angle
talibusur : chord
sebanding : proporsional
bola : sphere
kongruen : congruent
jajargenjang : parallelogram
alas :base
trapesium : trapezoid
sebangun : similar
bersilangan : cross
aneh : strange
sering muncul : modus
nama samaran : nickname
terkejut : surprised
sama dan sebangun : congruent
menjiplak : plagiarize
pembulatan : rounded
bingung : confused
kalem : calm
semboyan : maxim
pola : pattern
hakikat matematika : natural of mathematics
bangun datar : plane
bangun ruang : solid
acak : random
kabur : blur
pasrah : give up
clean up : menyelesaikan
varnish : menghapus
turmoil : kerusuhan
fate : nasib
rata-rata: mean, average
jiwa : soul
kecewa : dissapointed
menghindari : avoid
sempurna : perfect
garis lengkung : curve
barisan : sequence
permukaan : surface
cabang : branch
uang kertas :note
uang logam : coin
sebarang : scalene
konsisten : consistent
persegi : square
persegi panjang : rectangle
lingkaran :circle
tabung : sylinder
prisma : prism
penaksiran : estimation
mengungkap : uncover
anggapan : assumsed
Senin, 13 April 2009
Test of English
1. How we get phi?
Circumference of a circle is a limit of polygon on its circle. Circumference of circle noticed by “s”. It has relationship with radius (r) , namely s divided two r equals phi (s:2r = phi)
Proportion s : 2r of circumference of a circle with radius is equal of all circle. Phi equals twenty two divided by seven or three point one four.
2. How we get abc formula?
Formula of quadratic equation is a square x plus bx plus c equals zero
Change koefisien x square to be x square plus bx divided by a plus c divided by a equals zero.
Miss the konstanta on left, so we get x square plus bx divided by a equals negative c divided by a.
Add both of the sides right and left with quadratic of a half koefisien, we get x square plus bx divided by a plus the square of b divided by 2a equals negative c divided by a plus the square of b divided by 2a.
Change side of left be perfect quadratic, so
the square of x plus b divided by 2a equals negative 4ac plus b square divided by four a square then x plus b divided by 2a equals the square root of negative four ac plus b square divided by four a square, after that we get x. x equals negative b divided by 2a plus minus one 2a the root of b square minus 4ac, finally, x equals negative b plus minus the square root of b square minus 4ac divided by 2a.
3. Determine the area which limited y equals x square and y equals x plus two
To find that solution, we can draw it but for the first, we should make table. We’ll use scalene number to convert x.
For example, if x is zero, y is zero
if x is plus minus one, y is one
If x is plus minus two, y is four
And if x is plus minus three, y is nine, etc
For the second equation, y equals x plus two
I’ll choose x is zero so that y is two. (zero,two)
If y is zero, x is negative two. (negative two, zero)
The next step is we search the intersection point,
x square equals x plus two, so we get x equals negative one or x equals two
The last step is we have to integral from negative one until two on x plus two minus x square. By counting the integral we get the area which limited by way equals x square and y equals x plus two
4. Determine the volume of cone
Most of the three dimension have a formula volume namely, base area multiplied by height. For example the formula of cube volume is cubic sides. It happened because the base of cube is square and it has congruen sides so the volume of cube id cubic sides. Meanwhile the volume of cube equals one third limas volume. Limas volume has similar with cone volume, it just difference in base. So the cone volume is one third area base multiplied by height of cone. Or one third multiplied by circle area multiplied by height of cone.
5. Prove that the sum of angle of scalene triangle is one hundred and eighty degree.
Triangle have three angles, and sum of the angle is one hundred and eighty degree. On Element’s Euclid book 1, propositions 1 27 through 1 32 establish the theory of parallels and prove that the sum of the angle of a triangle is equal to two right angles.
If we have the picture of triangle and we cut the three of angles. After that we arrange the piece of them, we’ll get straight line or the measure of angle is one hundred and eighty degree.
6. Determine the probability of sum number more than 6 from two dices by throwing once.
A dice has six sides namely, one , two, three, four, five, and six. There are two dices so there are thirty six probability. The question how many probability come sum of number more than six.
The probability are (one, six), (two,five),(two, six),(three,four), (three, five), (three, six), (four, three), (four, four), (four, five), (four, six), (five, two), (five, three), (five, four), (five, five), (five,six),(six,one), (six, two), (six, three), (six, four), (six, five), (six,six)
There are twenty one, so the probability come sum of number more than six is 21/36.
7. How we determine the equation which on ( ten, zero) and have segment of circle x square plus y square equals nine?
To solving this question, we use equation segment formula which follow point (a,b), namely x index once x plus y index once y equals the square of radius. So the equation can be searched by entering elements which known. x index one is ten, y index one is zero and the square of radius is nine.
By accounting the equation we can give solution that the equation is ten x minus nine equals zero.
8. How we prove the phytagoras ?
Imagine that!
I have drawn two pictures, the pictures were square. For the first square. It’s divided into six pieces, namely two squares and four right triangles congruent. The second square is divided into five pieces namely the square on the hypotenuse and four right triangles congruent. a, b, c are sides of square. c means hypotenuse. By subtracting equals from equals. It follows that the square on the hypotenuse is equal to the sum of the squares on the legs. It’s one of the proof to make sure that it’s right.
9. Determine the sum of two hundred odd number first!
Odd number is started from one, three, five, seven,…
The sum of odd number is one plus three plus five plus seven,…
Let’s we find the sum of the odd number with formula.
one plus three plus five equivalent with five plus three plus one, if we sum of them, the result is six plus six plus six, or six multiplied by n
So we can conclude that one plus three plus five equals (six plus six plus six) multiplied by two =nx6/2=a half multiplied by n(one plus four)=a half multiplied by n (a+un)
n means the many of odd number
a means first term
un means last term
Our duty is finding the solution for the sum of the odd number, two hundred first
so one plus three plus five plus…plus one hundred and ninety nine equals…
the difference from 1 to three is two, therefore we can find the value of n
un= one hundred and ninety nine
one hundred and ninety nine =a+(n-1)b
one hundred and ninety nine =1+(2n-2)
one hundred and ninety nine =2n-1
two hundred=2n
n=one hundred
The sum is a half multiplied by one hundred multiplied by (one plus one hundred and ninety nine)
a half multiplied one hundred multiplied by two hundred equals ten thousand
10. the step to draw the cube ( the element of cube known)
Before, let’s know about the elements of cube
· frontal plane means every parallel plane with drawing plane.
· Frontal line is every line or segment which located on frontal plane.
· Orthogonal line is every line which located perpendicular on frontal plane.
· Surut angle means the angle which located between horizontal frontal segment to right and orthogonal segment to behind.
· The compare of projection or compare of orthogonal means the propotion between length of the map with the real length.
For example:
Draw ABCDEFGH cube. Given, the length of side is 3 cm, ABFE is frontal plane, sudut simpang 30’ and proportion of projection is one third.
First, make AB segment which length 3 cm. At point A make angle of 30’,so there is AD segment the length of AD is one third multiplied by 3 cm equals one cm. Because the projection of ABCD parallelogram, then the base can be done. The middle of side is vertical segment, therefore E, F, G and H can be drawn.
Circumference of a circle is a limit of polygon on its circle. Circumference of circle noticed by “s”. It has relationship with radius (r) , namely s divided two r equals phi (s:2r = phi)
Proportion s : 2r of circumference of a circle with radius is equal of all circle. Phi equals twenty two divided by seven or three point one four.
2. How we get abc formula?
Formula of quadratic equation is a square x plus bx plus c equals zero
Change koefisien x square to be x square plus bx divided by a plus c divided by a equals zero.
Miss the konstanta on left, so we get x square plus bx divided by a equals negative c divided by a.
Add both of the sides right and left with quadratic of a half koefisien, we get x square plus bx divided by a plus the square of b divided by 2a equals negative c divided by a plus the square of b divided by 2a.
Change side of left be perfect quadratic, so
the square of x plus b divided by 2a equals negative 4ac plus b square divided by four a square then x plus b divided by 2a equals the square root of negative four ac plus b square divided by four a square, after that we get x. x equals negative b divided by 2a plus minus one 2a the root of b square minus 4ac, finally, x equals negative b plus minus the square root of b square minus 4ac divided by 2a.
3. Determine the area which limited y equals x square and y equals x plus two
To find that solution, we can draw it but for the first, we should make table. We’ll use scalene number to convert x.
For example, if x is zero, y is zero
if x is plus minus one, y is one
If x is plus minus two, y is four
And if x is plus minus three, y is nine, etc
For the second equation, y equals x plus two
I’ll choose x is zero so that y is two. (zero,two)
If y is zero, x is negative two. (negative two, zero)
The next step is we search the intersection point,
x square equals x plus two, so we get x equals negative one or x equals two
The last step is we have to integral from negative one until two on x plus two minus x square. By counting the integral we get the area which limited by way equals x square and y equals x plus two
4. Determine the volume of cone
Most of the three dimension have a formula volume namely, base area multiplied by height. For example the formula of cube volume is cubic sides. It happened because the base of cube is square and it has congruen sides so the volume of cube id cubic sides. Meanwhile the volume of cube equals one third limas volume. Limas volume has similar with cone volume, it just difference in base. So the cone volume is one third area base multiplied by height of cone. Or one third multiplied by circle area multiplied by height of cone.
5. Prove that the sum of angle of scalene triangle is one hundred and eighty degree.
Triangle have three angles, and sum of the angle is one hundred and eighty degree. On Element’s Euclid book 1, propositions 1 27 through 1 32 establish the theory of parallels and prove that the sum of the angle of a triangle is equal to two right angles.
If we have the picture of triangle and we cut the three of angles. After that we arrange the piece of them, we’ll get straight line or the measure of angle is one hundred and eighty degree.
6. Determine the probability of sum number more than 6 from two dices by throwing once.
A dice has six sides namely, one , two, three, four, five, and six. There are two dices so there are thirty six probability. The question how many probability come sum of number more than six.
The probability are (one, six), (two,five),(two, six),(three,four), (three, five), (three, six), (four, three), (four, four), (four, five), (four, six), (five, two), (five, three), (five, four), (five, five), (five,six),(six,one), (six, two), (six, three), (six, four), (six, five), (six,six)
There are twenty one, so the probability come sum of number more than six is 21/36.
7. How we determine the equation which on ( ten, zero) and have segment of circle x square plus y square equals nine?
To solving this question, we use equation segment formula which follow point (a,b), namely x index once x plus y index once y equals the square of radius. So the equation can be searched by entering elements which known. x index one is ten, y index one is zero and the square of radius is nine.
By accounting the equation we can give solution that the equation is ten x minus nine equals zero.
8. How we prove the phytagoras ?
Imagine that!
I have drawn two pictures, the pictures were square. For the first square. It’s divided into six pieces, namely two squares and four right triangles congruent. The second square is divided into five pieces namely the square on the hypotenuse and four right triangles congruent. a, b, c are sides of square. c means hypotenuse. By subtracting equals from equals. It follows that the square on the hypotenuse is equal to the sum of the squares on the legs. It’s one of the proof to make sure that it’s right.
9. Determine the sum of two hundred odd number first!
Odd number is started from one, three, five, seven,…
The sum of odd number is one plus three plus five plus seven,…
Let’s we find the sum of the odd number with formula.
one plus three plus five equivalent with five plus three plus one, if we sum of them, the result is six plus six plus six, or six multiplied by n
So we can conclude that one plus three plus five equals (six plus six plus six) multiplied by two =nx6/2=a half multiplied by n(one plus four)=a half multiplied by n (a+un)
n means the many of odd number
a means first term
un means last term
Our duty is finding the solution for the sum of the odd number, two hundred first
so one plus three plus five plus…plus one hundred and ninety nine equals…
the difference from 1 to three is two, therefore we can find the value of n
un= one hundred and ninety nine
one hundred and ninety nine =a+(n-1)b
one hundred and ninety nine =1+(2n-2)
one hundred and ninety nine =2n-1
two hundred=2n
n=one hundred
The sum is a half multiplied by one hundred multiplied by (one plus one hundred and ninety nine)
a half multiplied one hundred multiplied by two hundred equals ten thousand
10. the step to draw the cube ( the element of cube known)
Before, let’s know about the elements of cube
· frontal plane means every parallel plane with drawing plane.
· Frontal line is every line or segment which located on frontal plane.
· Orthogonal line is every line which located perpendicular on frontal plane.
· Surut angle means the angle which located between horizontal frontal segment to right and orthogonal segment to behind.
· The compare of projection or compare of orthogonal means the propotion between length of the map with the real length.
For example:
Draw ABCDEFGH cube. Given, the length of side is 3 cm, ABFE is frontal plane, sudut simpang 30’ and proportion of projection is one third.
First, make AB segment which length 3 cm. At point A make angle of 30’,so there is AD segment the length of AD is one third multiplied by 3 cm equals one cm. Because the projection of ABCD parallelogram, then the base can be done. The middle of side is vertical segment, therefore E, F, G and H can be drawn.
Minggu, 12 April 2009
Nature of mathematics
Nature of mathematics
Nature of mathematic consist of something, they are relationship, problem solving, research and tool of communication.1. Relationship
In mathematics must has a relationship. It is impossible single, something of mathematics has relationship with other. For instances, there is two dimensionIt can be relationship with square, triangle, circle, etc
2. Problem Solving
Mathematic is problem solving. Many difficulties in our life can be finish by mathematics. Every activities use mathematics.For example:I have one thousand rupiah, I want to buy five candies, the price of each candy is one hundred rupiah.With mathematic, we know, we should pay five hundred rupiahs.
3. Research
Mathematics can definitied by research. Because the importantance of mathematic is finding something use logical. We observate something, give definition, teorema and make a concliution.
4. Tool of communicationMathematics is one of tool of communication. So mathematic is very important.For example: ihave many books, magazines, story books, novels and comics. Other people know we have them if we communicate with other. Other peoples will asked how many book you have . we answer their question use mathematic, for example I answer that I have ten story books, eight comics, twenty magazines, etc
Wat to say it
Way to say it
in mathematics we should know how we say the sentence of mathematics. for example in addition, subtractions, fractions, etc.
FractionsFractions or in bahasa Indonesia known as pecahan. Are divided two variant; ordinal fractions and decimal fractions. On ordinal fractions can be divided again to be three kinds, namely proper fractions, improper fractions and mixed fractions. But how to read?
Pay close attension to the following examples:
Ordinal fraction
1/2= one second or a half
1/8= one eight or an eight
2/3=two thirds
1 5/9=one and five ninths
Decimal fractions
0,5 = point five or zero point five
3,25 = three point two fivePower
8^2 = 64
The second power of 8 is 64
Or
eight to the second power is 64
Or eight square is 64
The second exponent of 8 is 64
Rootv144 = 12
The square root of 144 is 12
?27 = 3The cubic of 27 is 3, or
The cube root of 27 is 3v16 =….
What is the square root of 16?
Additions
2 + 2 = 4
Two and two are four. It is in small additions but if in larger additions and in more formal style additions, we use plus or added For example:
712 + 145 = 857
Seven hundred twelve plus one hundred forty- five is or equals eight hundred and fifty-seven.
Subtractions
In conversational style, for the operation7-3 = 4, we say:Three from seven leaves / is four
Or
seven takes away three leaves / is four.
In a more formal style, we say minus for…. And equals.For instance:
619-428 = 191, six hundred and nineteen minus for hundred and twenty eight equals one hundred and ninety one.
Multiplication.
3 x 4 = 12Three fours are twelve.
6 x 7 = 42 Six sevens are forty two
3 x 71 = 213Three times seventy one makes two hundred and thirteen
Or three multiplied by seventy one equals two hundred and thirteen.
Division :
261 : 9 = 29Two hundred and sixty one divided by nine equals twenty-nine or two hundred and sixty one into nine goes twenty nine.
The form of quadratic equation is ax^2+bx+c=0
the find the root or solution from the quadratic equation, we can use :
# factor
# complete perfect quadratic
# abc formula
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