INTRODUCTION
A. PURPOSE
The aim is to examine and develop mathematics.
B. BACKGROUND
The research of mathematics is very important in the development of our mathematical abilities. In mathematical research, we can do some stage. The first stage we have to determine which objects will be thorough. Besides, we also have to write down goals in the research and also write the reason why we chose to write in the research subject's. We also need to determine the background of why we chose it to be our research. After we write the background, then we collect data to complete our research. This data that we get to our analysis and we are the order to produce good research. After that, we can get conclusion the research of mathematics.
DISCUSSION
The research of mathematics is important to student of mathematics. To get the research of mathematics we must have a good ways. One of the ways is to search the references about the mathematics such as, a browsing internet, search the related books, etc.
We should have knowledge about it to improve our knowledge about mathematics and to know the procedure or the steps to found the new formula, axioms, and theorems in mathematics. Theorem in which there is no contradiction in side.
The nature of mathematics:
• Formal mathematics
• Applied mathematics
• School mathematics
It is very easy to establish mathematics system. Let X₁, X₂, X₃, ..........are real members.
The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system
1. Numbers Theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general.
2. Group Theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
3. Ring Theory
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).(This definition take from Wikipedia)
4. Field Theory
Field theory is a branch of mathematics which studies the properties of fields.
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.(This definition take from wikipedia)
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.(This definition take from wikipedia)
7. Number system
In mathematics, a 'number system' is a set of numbers, together with one or more operations, such as addition or multiplication.
Application of mathematics a lot in our lives. Such applications in the field of mathematical economics. Mathematics is used in many economic areas. In economic calculation, math is very useful. Besides that, mathematics is used in the field of architecture. In the field of architecture used math a lot. Pythagoras theorems such as that used to make buildings. Information technology refers to the use of computers and software to convert, store, protect process, transmit, and retrieve information. Computational theory, algorithm analysis, formal methods and data representation are just some computing techniques that require the use of mathematics.
School mathematics
• Awareness
• Intention
• Abstraction
• Idealization
Eburt and Stracker (1995)
• Pattern/relationship
• Problem solving
• Investigation
• Communication
Mathematics phenomena
• To identify problem: skill, attitude, knowledge, experience.
• Mathematics knowledge
• Mathematics system
• Mathematics character
Supporting factor:
• history of mathematics
• the philosophy of mathematics
• the works of mathematics
• to in depth study of mathematics
For example we choose the Pythagorean Theorem.
1. Pythagorean
Mathematics is very important to our life. Mathematics is an exact science. The branch of mathematics is Pythagorean. In many building design cannot be separated from the Pythagorean Theorem. Pythagorean Theorem is very helpful in measuring the height of a building.
2. Pythagorean theorem
A theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a that is, a2 + b2 = c2.
3. Proof
We can proof the The Pythagorean theorem from the this graph
From this graph we can get any side:
The square has the side c, a, and (a-b)
So, c2 = (a-b)2 + 4. ½ (ab)
= (a-b)2+ 2(ab)
= a2 + b2 – 2ab +2ab
= a2 + b2
From the research of mathematics about Pythagorean we can get the conclusion:
• Right-angled side is the side that forms a right angle.
While is the side before the right angel.
• Square of the hypotenuse equals the sum of squares square is called Pythagoreans theorem.
CONCLUTION
The nature of mathematics:
• Formal mathematics
• Applied mathematics
• School mathematics
The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system
Application of mathematics a lot in our lives for example:
• in economics
• in architecture
• in civil engineering
School mathematics
• Awareness
• Intention
• Abstraction
• Idealization
Rabu, 20 Januari 2010
Selasa, 29 Desember 2009
THE RESEARCH OF MATHEMATICS
The research of mathematics is important to student of mathematics. To get the research of mathematics we must have a good ways. One of the ways is to search the references about the mathematics such as, a browsing internet, search the related books, etc.
We should have knowledge about it to improve our knowledge about mathematics and to know the procedure or the steps to found the new formula, axioms, and theorems in mathematics. Theorem in which there is no contradiction in side.
It is very easy to establish mathematics system. Let X₁, X₂, X₃, ..........are real members.
The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system
ABSTRACT
Mathematics is an exact science, where in all the calculations can
accountability. The nature of mathematics are deductive axiomatic,
means to learn mathematics depart from the base of understanding,
statement of the base, then axiom. Approach from the base of understanding,
statements and the axiom base can result in a theorem that
becomes the basis for finding a solution of a problem, then comes
theorem-Theorem others. Many branches of mathematics, one
only is the field of geometry. The usefulness of the science of geometry in
design, measurement of height, and other applications.
There are several fundamental theorem in the science of geometry, one of them
is the Pythagorean theorem. This theorem was discovered by a mathematician
named Pythagoras. In this case the author tried to analyze through
Pythagorean theorem and properties derived from the Pythagorean theorem
can answer the problem of measurement segment that was not possible
manually.
INTRODUCTION
A. Background
The nature of mathematics are deductive axiomatic, that is to
study of mathematics starting from understanding the base, the base statement
the definition agreed upon, then the axiom is a statement
accepted without proof. The approach of understanding the base, the base statement and axioms can come up with a theorem that is the basis of find solution.
Many branches of mathematics, one of which is the field
geometry. The geometry of the usefulness of science in engineering,
measurement of height, and other applications. There are several fundamental theorems is the science of geometry, one is the Pythagorean theorem only. This theorem was discovered by mathematician named Phytagoras. Pythagoras was born around the year 582 BC on the island of Samos, Greece. Pythagoras discovered a geometrical formula
simply about the relationship side of the triangle. This formula later known as the Pythagorean Theorem. In many building design cannot be separated from the Pythagorean theorem, in the physical sciences Pythagorean theorem is very helpful in measuring the height of a building, in mathematics itself Pythagorean theorem is
assist in making a line that can not be measured with a ruler.
B. Purpose
Based on the above problem formulation, the goal in This research is proving that the Pythagorean theorem and properties derived from the Pythagorean theorem can be answered a problem of measurement segment that was not possible manually
DISCUSSION
1. Pythagorean
In mathematics, the Pythagorean Theorem is a relationship in Euclidean geometry among the three sides of a right triangle. This theorem is called by a Greek philosopher and mathematician, 6th century BC, Pythagoras. Pythagoras is often regarded as the inventor of this theorem, although the actual facts of this theorem was known by mathematicians of India (in Sulbasutra Baudhayana and Katyayana), Greek, and Babylonian Tionghoa long before Pythagoras was born. Pythagoras get credit because it was he who first proved the universal truth of this theorem by mathematical proofs.
2. Pythagorean Theorem
Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a2 + b2 = c2.
While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers, implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.
3. Proof
Now we start with four copies of the same triangle. Three of these have been rotated 90°, 180°, and 270°, respectively. Each has area ab/2. Let's put them together without additional rotations so that they form a square with side c.
The square has a square hole with the side (a - b). Summing up its area (a - b)² and 2ab, the area of the four triangles (4•ab/2), we get
c² = (a - b)² + 2ab
= a² - 2ab + b² + 2ab
= a² + b²
CONCLUTION
• Right-angled side is the side that forms a right angle.
While is the side before the right angle.
• Square of the hypotenuse equals the sum of squares square is called Pythagoreans theorem.
We should have knowledge about it to improve our knowledge about mathematics and to know the procedure or the steps to found the new formula, axioms, and theorems in mathematics. Theorem in which there is no contradiction in side.
It is very easy to establish mathematics system. Let X₁, X₂, X₃, ..........are real members.
The formal mathematics:
1. Numbers theory
2. Group theory
3. Ring theory
4. Field theory
5. Euclidean geometry
6. Non. Euclidean geometry
7. Number system
ABSTRACT
Mathematics is an exact science, where in all the calculations can
accountability. The nature of mathematics are deductive axiomatic,
means to learn mathematics depart from the base of understanding,
statement of the base, then axiom. Approach from the base of understanding,
statements and the axiom base can result in a theorem that
becomes the basis for finding a solution of a problem, then comes
theorem-Theorem others. Many branches of mathematics, one
only is the field of geometry. The usefulness of the science of geometry in
design, measurement of height, and other applications.
There are several fundamental theorem in the science of geometry, one of them
is the Pythagorean theorem. This theorem was discovered by a mathematician
named Pythagoras. In this case the author tried to analyze through
Pythagorean theorem and properties derived from the Pythagorean theorem
can answer the problem of measurement segment that was not possible
manually.
INTRODUCTION
A. Background
The nature of mathematics are deductive axiomatic, that is to
study of mathematics starting from understanding the base, the base statement
the definition agreed upon, then the axiom is a statement
accepted without proof. The approach of understanding the base, the base statement and axioms can come up with a theorem that is the basis of find solution.
Many branches of mathematics, one of which is the field
geometry. The geometry of the usefulness of science in engineering,
measurement of height, and other applications. There are several fundamental theorems is the science of geometry, one is the Pythagorean theorem only. This theorem was discovered by mathematician named Phytagoras. Pythagoras was born around the year 582 BC on the island of Samos, Greece. Pythagoras discovered a geometrical formula
simply about the relationship side of the triangle. This formula later known as the Pythagorean Theorem. In many building design cannot be separated from the Pythagorean theorem, in the physical sciences Pythagorean theorem is very helpful in measuring the height of a building, in mathematics itself Pythagorean theorem is
assist in making a line that can not be measured with a ruler.
B. Purpose
Based on the above problem formulation, the goal in This research is proving that the Pythagorean theorem and properties derived from the Pythagorean theorem can be answered a problem of measurement segment that was not possible manually
DISCUSSION
1. Pythagorean
In mathematics, the Pythagorean Theorem is a relationship in Euclidean geometry among the three sides of a right triangle. This theorem is called by a Greek philosopher and mathematician, 6th century BC, Pythagoras. Pythagoras is often regarded as the inventor of this theorem, although the actual facts of this theorem was known by mathematicians of India (in Sulbasutra Baudhayana and Katyayana), Greek, and Babylonian Tionghoa long before Pythagoras was born. Pythagoras get credit because it was he who first proved the universal truth of this theorem by mathematical proofs.
2. Pythagorean Theorem
Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a2 + b2 = c2.
While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers, implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.
3. Proof
Now we start with four copies of the same triangle. Three of these have been rotated 90°, 180°, and 270°, respectively. Each has area ab/2. Let's put them together without additional rotations so that they form a square with side c.
The square has a square hole with the side (a - b). Summing up its area (a - b)² and 2ab, the area of the four triangles (4•ab/2), we get
c² = (a - b)² + 2ab
= a² - 2ab + b² + 2ab
= a² + b²
CONCLUTION
• Right-angled side is the side that forms a right angle.
While is the side before the right angle.
• Square of the hypotenuse equals the sum of squares square is called Pythagoreans theorem.
Jumat, 04 Desember 2009
1.Determine the integration factors of the following differential equation and the find the solution
(3x2y + 2xy + y3) dx + (x2+ Y2) dy =0
2.Known : sin -1 x =
∫_0^x▒1/√((1-t2)) dt
Determine the first four tribes taknol in Mac Laurin series for sin -1
3.Known :
g =((█(x@y@z))= (█(1@-3@5))+ a (█(-6@4@7)) ) a parameter
Determine the line g and h has an intersection
4.Chanted a coin in a row as much as 6 times, how many chances at least one comes face?
5.Consider the following data for 120 mathematics at the college concerning the languages French, German, Russian
65 study French
45 study German
42 study Russian
20 study French and German
15 study German and Russian
8 study all three languages
Let F, G, and R denote the sets of students studying French, German, and Russian, respectively. We wish to find the number of students who study at least one of the three languages, and to fill the correct number of students in each of the eight regions
Answers:
1.(3x2y + 2xy + y3) dx + (x2+ Y2) dy =0
M(x,y) = 3x2y + 2xy + y3
N(x,y) = x2+ Y2
∂M(x,y) / ∂y = 3x2+ 2x + 3y2
∂N(x,y)/ ∂x = 2x
Formula to get the integrations factors is :
(1/N(x,y)) [(∂M(x,y)/∂y)] - [(∂N(x,y)/∂x)]
Substitute the equation 1 and 2 to formulas;
1/(x2 + y2) ((3x2+ 2x + 3y2) – (2x))
1/(x2 + y2) (3x2 + 3y2) =3
You can get μ(x) = e∫3 dx = e3x
Multiply e3x to M (x,y) and M (x,y) be M* (x,y)
M* (x,y) = 3x2ye3x + 2xy e3x + y3e3x
Differential M* (x,y) to y, so:
∂M*(x,y) / ∂y = 3x2e3x + 2xe3x + 3y2e3x
And so N(x,y) multiply with e3x
N*(x,y) = x2e3x + y2e3x
Differential N*(x,y) to x so:
∂N*(x,y)/∂x = 2xe3x + 3e3xx2 + 3y2e3x
Cause the different equation is exact differential equation, so:
∂F/∂x=M* AND ( ∂F)/∂y=N*
To get the solution we must be :
F (x,y) = ∫(3x2ye3x + 2xye3x + y3e3x) dx + Q (y)
= yx2e + 1/3 y3 e3x + Q(y)
∂F(x,y) /∂y = x2e3x + y2 e3x +Q’(y)
∂F/∂y = N*
e2 e3x + y2 e3x = x2e3x + y2e3x + Q’(y)
Q’(y) = 0
Q(y) = c
So the solution from this problem is
F(x,y) = yx2e + 1/3 y3 e3x + c
2. Known : sin -1 x =
∫_0^x▒1/√((1-t2)) dt
We have a formula :
(1 + X)P = 1 + (█(p@1))x + (█(p@2))x2 +…
Where : + (█(p@k)) =( p (p-1) (p-2))/k!
So we can apply that
(1-t2)-1/2 = 1 + (-1/2) (-t2) + (-1/2) (-3/2) (-t2)2 + (-1/2) (-t2) (-5/2) (-t2)3+…
= 1 + t2/2 + (3/8) t4 + (15/48) t6 +…
= t2/2 + (3/8) t4 + (5/16) t6 +…
Sin -1 x = ∫_0^x▒〖1 〗+ t2/2 + (3/8)t4 + (5/16)t6 +…
= x + x3/6 + (3/40)x5 + (5/112)x7
3.Known :
g = (█(x@y@z)) = (█(1@-3@5))+ a(█(-6@4@7)) a parameter
x = -2 -9b
h = y = 8-3b b parameter (2)
z = -6 +4b
to prof gnh = N
N € g => coordinate N must on the g
N € h => coordinate N must on the h
x = 1 – 6a
y = -3 + 4a
z = 5 + 7a xN= 1 – 6a
substitute coordinate N to (1) => yN = -3 +4a (3)
zN = 5 + 7a
substitute coordinate N to (2) => xN = -2 -9b
h = yN = 8-3b b parameter (4)
zN = -6 +4b
left segment of (3) equals left segment of (4)
right segment of (3) equals right segment of (4)
1 – 6a = -2 – 9b 3 = 6a – 9b
1 = 2a – 3b …………. (5)
-3 + 4a = 8-3b 4a + 3b =11……………(6)
5 + 7a = -6 + 4b 7a – 4b = -11……….(7)
So:
2a – 3b = 1
(8) 4a + 3b=11
7a – 4b = -11
Eliminate (5) and (6) => b = 3
Substitute b to (5) => a = 5
Substitute value a and b to (7)
7a – 4b = -11
7.5 – 4.3 = -11
35 - 12 ≠ -11
If that no similar gnh = empty seT
N if (8) have a solution
4.Suppose that E happened at least once come home. Sample space S contains the sample is 26 = 64 points. Because each reflection can produce two kinds of results (fronts or near), we know that P (E) = 1 – P (E’). if E’ represents the event that no face appeared. This will only happen in a way, which is all behind chanting produces. Be P(E’) = 1/64 so P(E)=1 - 1/64 = 63/64
5.(F U G U R) = n(F) + n(G) + n(R) –n(F slices G) – n(F slices R) - n( G slices R)
+ n(F sliceS G slices R)
= 65+45+42-20-25-15+8 = 100
That is, n(F U G U R)=100 students at least one of the three languages
Now, we use this result to fill in the venn diagram. We have 8 study all the three languages
20 – 8 =12 study French and german but not Russian
25 – 8 = 17 study French and Russian but not german
15 – 8 = 7 study german and Russian but not French
65 – 12 – 8 -17 =28 study only French
45 – 12 – 8 – 7 =18 study only german
42 – 17 – 8 -7 = 10 study only Russian
120 – 100 =20 do not study at the languages
Observe that 28 + 18 + 10 = 56 students only one of the languages
Problems solving created by salindri murbarani 08305141034
Solutions the problems created by Narita yuri
(3x2y + 2xy + y3) dx + (x2+ Y2) dy =0
2.Known : sin -1 x =
∫_0^x▒1/√((1-t2)) dt
Determine the first four tribes taknol in Mac Laurin series for sin -1
3.Known :
g =((█(x@y@z))= (█(1@-3@5))+ a (█(-6@4@7)) ) a parameter
Determine the line g and h has an intersection
4.Chanted a coin in a row as much as 6 times, how many chances at least one comes face?
5.Consider the following data for 120 mathematics at the college concerning the languages French, German, Russian
65 study French
45 study German
42 study Russian
20 study French and German
15 study German and Russian
8 study all three languages
Let F, G, and R denote the sets of students studying French, German, and Russian, respectively. We wish to find the number of students who study at least one of the three languages, and to fill the correct number of students in each of the eight regions
Answers:
1.(3x2y + 2xy + y3) dx + (x2+ Y2) dy =0
M(x,y) = 3x2y + 2xy + y3
N(x,y) = x2+ Y2
∂M(x,y) / ∂y = 3x2+ 2x + 3y2
∂N(x,y)/ ∂x = 2x
Formula to get the integrations factors is :
(1/N(x,y)) [(∂M(x,y)/∂y)] - [(∂N(x,y)/∂x)]
Substitute the equation 1 and 2 to formulas;
1/(x2 + y2) ((3x2+ 2x + 3y2) – (2x))
1/(x2 + y2) (3x2 + 3y2) =3
You can get μ(x) = e∫3 dx = e3x
Multiply e3x to M (x,y) and M (x,y) be M* (x,y)
M* (x,y) = 3x2ye3x + 2xy e3x + y3e3x
Differential M* (x,y) to y, so:
∂M*(x,y) / ∂y = 3x2e3x + 2xe3x + 3y2e3x
And so N(x,y) multiply with e3x
N*(x,y) = x2e3x + y2e3x
Differential N*(x,y) to x so:
∂N*(x,y)/∂x = 2xe3x + 3e3xx2 + 3y2e3x
Cause the different equation is exact differential equation, so:
∂F/∂x=M* AND ( ∂F)/∂y=N*
To get the solution we must be :
F (x,y) = ∫(3x2ye3x + 2xye3x + y3e3x) dx + Q (y)
= yx2e + 1/3 y3 e3x + Q(y)
∂F(x,y) /∂y = x2e3x + y2 e3x +Q’(y)
∂F/∂y = N*
e2 e3x + y2 e3x = x2e3x + y2e3x + Q’(y)
Q’(y) = 0
Q(y) = c
So the solution from this problem is
F(x,y) = yx2e + 1/3 y3 e3x + c
2. Known : sin -1 x =
∫_0^x▒1/√((1-t2)) dt
We have a formula :
(1 + X)P = 1 + (█(p@1))x + (█(p@2))x2 +…
Where : + (█(p@k)) =( p (p-1) (p-2))/k!
So we can apply that
(1-t2)-1/2 = 1 + (-1/2) (-t2) + (-1/2) (-3/2) (-t2)2 + (-1/2) (-t2) (-5/2) (-t2)3+…
= 1 + t2/2 + (3/8) t4 + (15/48) t6 +…
= t2/2 + (3/8) t4 + (5/16) t6 +…
Sin -1 x = ∫_0^x▒〖1 〗+ t2/2 + (3/8)t4 + (5/16)t6 +…
= x + x3/6 + (3/40)x5 + (5/112)x7
3.Known :
g = (█(x@y@z)) = (█(1@-3@5))+ a(█(-6@4@7)) a parameter
x = -2 -9b
h = y = 8-3b b parameter (2)
z = -6 +4b
to prof gnh = N
N € g => coordinate N must on the g
N € h => coordinate N must on the h
x = 1 – 6a
y = -3 + 4a
z = 5 + 7a xN= 1 – 6a
substitute coordinate N to (1) => yN = -3 +4a (3)
zN = 5 + 7a
substitute coordinate N to (2) => xN = -2 -9b
h = yN = 8-3b b parameter (4)
zN = -6 +4b
left segment of (3) equals left segment of (4)
right segment of (3) equals right segment of (4)
1 – 6a = -2 – 9b 3 = 6a – 9b
1 = 2a – 3b …………. (5)
-3 + 4a = 8-3b 4a + 3b =11……………(6)
5 + 7a = -6 + 4b 7a – 4b = -11……….(7)
So:
2a – 3b = 1
(8) 4a + 3b=11
7a – 4b = -11
Eliminate (5) and (6) => b = 3
Substitute b to (5) => a = 5
Substitute value a and b to (7)
7a – 4b = -11
7.5 – 4.3 = -11
35 - 12 ≠ -11
If that no similar gnh = empty seT
N if (8) have a solution
4.Suppose that E happened at least once come home. Sample space S contains the sample is 26 = 64 points. Because each reflection can produce two kinds of results (fronts or near), we know that P (E) = 1 – P (E’). if E’ represents the event that no face appeared. This will only happen in a way, which is all behind chanting produces. Be P(E’) = 1/64 so P(E)=1 - 1/64 = 63/64
5.(F U G U R) = n(F) + n(G) + n(R) –n(F slices G) – n(F slices R) - n( G slices R)
+ n(F sliceS G slices R)
= 65+45+42-20-25-15+8 = 100
That is, n(F U G U R)=100 students at least one of the three languages
Now, we use this result to fill in the venn diagram. We have 8 study all the three languages
20 – 8 =12 study French and german but not Russian
25 – 8 = 17 study French and Russian but not german
15 – 8 = 7 study german and Russian but not French
65 – 12 – 8 -17 =28 study only French
45 – 12 – 8 – 7 =18 study only german
42 – 17 – 8 -7 = 10 study only Russian
120 – 100 =20 do not study at the languages
Observe that 28 + 18 + 10 = 56 students only one of the languages
Problems solving created by salindri murbarani 08305141034
Solutions the problems created by Narita yuri
Senin, 04 Mei 2009
Technology
Technology
The making of this book use modern technology. This can be indicated as
Follows:
1. This book use good design an interesting design. So that readers feel happy when reading the book. The design use modern technology that is compounding two modern software.
2. When seen from the results of printing the book showed that the used modern printing machine and high quality printing machine.
3. The book use interesting color and no stain in the book shows that these books use the modern ink and high quality.
4. Book bindery use of modern technology because it looks neat and the book does not fall off easily.
5. The layout of this book takes from internet. So this layout uses modern picture and the latest pictures.
6. Right interpretation and good pictures show that use sophisticated technology and modern
Minggu, 15 Maret 2009
TRANSLATE
TRANSLATE
1. Pembuktian matematika : mathematical verification
2. Induksi matematika : mathematical induction
3. Perhitungan : A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.
http://en.wikipedia.org/wiki/Calculation
3. Pembelajaran matematika : Learning MATH
4. Hitung : arithmetic.
Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic[1] when referring to number theory, but this should not be confused with elementary arithmetic.
http://en.wikipedia.org/wiki/Arithmetic
5. Begitu sulit belajar matematika : so difficult to learn mathematics.
6. kesejajaran : parallelism.
7. Kekongruenan :congruenceIn geometry, two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).(http://en.wikipedia.org)
8. Sebidang : coplanar.
In geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. For example, three distinct points are always coplanar; but four points in space are usually not coplanar.
Points can be shown to be coplanar by determining that the scalar product of a vector that is normal to the plane and a vector from any point on the plane to the point being tested is 0. To put this another way, if you have a set of points which you want to determine are coplanar, first construct a vector for each point to one of the other points (by using the distance formula, for example). Secondly, construct a vector which is perpendicular (normal) to the plane to test (for example, by computing the cross product of two of the vectors from the first step). Finally, compute the dot product (which is the same as the scalar product) of this vector with each of the vectors you created in the first step. If the result of each dot product is 0, then all the points are coplanar.
http://en.wikipedia.org/wiki/Coplanar
9. Sepihak : unilateral.
10. Sudut dalam sepihak : angle in the unilateral
11. Penerapan matematika : the application of mathematics.
12. Himpunan penyelesaian : collective settlement.
13. Sudut dalam berseberangan : Same-side interior angles Interior angles on the same side of a transversal.(http://www.ovec.org)
14. Pembanding : benchmark.
In electronics, a comparator is a device which compares two voltages or currents and switches its output to indicate which is larger.A dedicated voltage comparator will generally be faster than a general-purpose op-amp pressed into service as a comparator. A dedicated voltage comparator may also contain additional features such as an accurate, internal voltage reference, an adjustable hysteresis and a clock gated input.
15. Segi banyak : polygonA closed plane figure that is formed by joining three or more line segments at their endpoints. (http://www.ovec.org)
16. Titik pusat : center of point.
17. Ruas garis : line segment.
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).
http://en.wikipedia.org/wiki
18. Juring : segment.
Line segment, part of a line that is bounded by two end points
Circular segment, the area which is "cut off" from the rest of the circle by a secant or a chord
http://en.wikipedia.org/wiki
19. Penyebut : denominator.
20. Pembilang : numerator.
A numeral used to indicate a count, particularly of the equal parts in a fraction. A numerator is the number on top of the fraction. For example, in 3/4, 3 is the numerator and 4 is the denominator.
A person who counts
21. Irisan : slice.
Slice genus, in knot theory
Slice knot, in knot theory
Slice sampling, a Monte Carlo sampling method
Projection-slice theorem, a theorem about Fourier transforms
http://en.wikipedia.org/wiki
22. Keliling : circumference.
23. Trapesium : trapezoid.
in geometry, a trapezoid or trapezium is a quadrilateral with two parallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain. (To add to the confusion, the word “trapezium” is used in North America to refer to a quadrilateral with no parallel sides, while the word “trapezoid” has sometimes been used historically with this same meaning.)
http://en.wikipedia.org/wiki/Trapezoid
1. Pembuktian matematika : mathematical verification
2. Induksi matematika : mathematical induction
3. Perhitungan : A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.
http://en.wikipedia.org/wiki/Calculation
3. Pembelajaran matematika : Learning MATH
4. Hitung : arithmetic.
Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic[1] when referring to number theory, but this should not be confused with elementary arithmetic.
http://en.wikipedia.org/wiki/Arithmetic
5. Begitu sulit belajar matematika : so difficult to learn mathematics.
6. kesejajaran : parallelism.
7. Kekongruenan :congruenceIn geometry, two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).(http://en.wikipedia.org)
8. Sebidang : coplanar.
In geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. For example, three distinct points are always coplanar; but four points in space are usually not coplanar.
Points can be shown to be coplanar by determining that the scalar product of a vector that is normal to the plane and a vector from any point on the plane to the point being tested is 0. To put this another way, if you have a set of points which you want to determine are coplanar, first construct a vector for each point to one of the other points (by using the distance formula, for example). Secondly, construct a vector which is perpendicular (normal) to the plane to test (for example, by computing the cross product of two of the vectors from the first step). Finally, compute the dot product (which is the same as the scalar product) of this vector with each of the vectors you created in the first step. If the result of each dot product is 0, then all the points are coplanar.
http://en.wikipedia.org/wiki/Coplanar
9. Sepihak : unilateral.
10. Sudut dalam sepihak : angle in the unilateral
11. Penerapan matematika : the application of mathematics.
12. Himpunan penyelesaian : collective settlement.
13. Sudut dalam berseberangan : Same-side interior angles Interior angles on the same side of a transversal.(http://www.ovec.org)
14. Pembanding : benchmark.
In electronics, a comparator is a device which compares two voltages or currents and switches its output to indicate which is larger.A dedicated voltage comparator will generally be faster than a general-purpose op-amp pressed into service as a comparator. A dedicated voltage comparator may also contain additional features such as an accurate, internal voltage reference, an adjustable hysteresis and a clock gated input.
15. Segi banyak : polygonA closed plane figure that is formed by joining three or more line segments at their endpoints. (http://www.ovec.org)
16. Titik pusat : center of point.
17. Ruas garis : line segment.
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).
http://en.wikipedia.org/wiki
18. Juring : segment.
Line segment, part of a line that is bounded by two end points
Circular segment, the area which is "cut off" from the rest of the circle by a secant or a chord
http://en.wikipedia.org/wiki
19. Penyebut : denominator.
20. Pembilang : numerator.
A numeral used to indicate a count, particularly of the equal parts in a fraction. A numerator is the number on top of the fraction. For example, in 3/4, 3 is the numerator and 4 is the denominator.
A person who counts
21. Irisan : slice.
Slice genus, in knot theory
Slice knot, in knot theory
Slice sampling, a Monte Carlo sampling method
Projection-slice theorem, a theorem about Fourier transforms
http://en.wikipedia.org/wiki
22. Keliling : circumference.
23. Trapesium : trapezoid.
in geometry, a trapezoid or trapezium is a quadrilateral with two parallel sides. The term “trapezoid” is used in North America, while the term “trapezium” is prevalent in Britain. (To add to the confusion, the word “trapezium” is used in North America to refer to a quadrilateral with no parallel sides, while the word “trapezoid” has sometimes been used historically with this same meaning.)
http://en.wikipedia.org/wiki/Trapezoid
Senin, 23 Februari 2009
ENGLISH I
ASSALAMUALIKUM.WR.WB
Hi…., I will tell you about myself. My full name is Salindri Murbarani. my nick name is Sali. I was born in Banyumas. I live in Sleman Yogyakarta with my family. I live with my parent and my brother. I am university student in UNY. In there, I have many friends. They come from many cities in Indonesia. Example they come from Lampung, Yogyakarta, Purbalingga, Kebumen, and many more. I love my friends very much. I hope they become my friends forever.
I will tell you about my English I Lesson. I found out my English lesson on semester II. My university-level instructor of English 1 is Mr. Marsigit. Mr. Marsigit is very kind. He has much Experience. And then he shares his experience in our class. Before he shares about his experience, he introduces about his self. His name is Dr. Marsigit. He comes from Kebumen. He is 51 years old. He has 1 wife and 2 children. After he introduces his self, he shares about his experience. He ever came in many countries in the world, example Japan, Australia, etc. He has much experience in the countries. He is very happy came there. He met many different people in the world. Besides, he ever came in many countries; he also likes posting something in his blogs. His blogs are powermathematic.blogspot.com, marsigitenglish.blogspot.com. After he shares about his experience, and then he ask one by one student where we come from.
He orders his student to make blog and follower his blog. And then, he explains how to make blog and follower in his blog.
1. We should have email account in gmail.com
2. Register in blogger.com
3. Follower in marsigitenglish.blogspot.com.
Fourth, he explains how to communicate math in English?
1. Not teach 100% because the time not enough
2. Not plagiarist.
3. Take a test, and study about sentences, grammar, etc.
After then, he explains 3 steps to develop our competent.
1. We must have high spirit. Spirit is very important to achieve the purpose. We must have high spirit to achieve the purpose.
2. Behavior
The highest motivation is praying god.
3. Knowledge
Use your knowledge to others. Knowledge Communication are to talk, to write, to discuss, to understand, to translate, to comment, etc.
Adult must develop our skill:
a. Have responsibility.
b. Independent learner.
c. Collaborate with others.
d. Able to find out the different.
WASS.WR.WB.
English I Salindri Murbarani Mat R 08
Langganan:
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