MATHEMATICS, COMPUTING AND COMPUTER
1. Mathematics: What is it?
According to the Dictionary of Indonesian Language, mathematics is "the science of numbers, relationships among numbers, and operational procedures used in solving the problem of numbers". This sentence is not the exact formula, though can be said to be sufficient for inclusion in the dictionary. There is a branch of mathematics that are not directly dealing with numbers, such as geometry, topology, graph theory and logic.
In this manuscript is defined as the science of mathematical statements, as well as the conditions necessary for a statement is true. In a statement the truth of mathematics tested, researched the meaning or implications of every word contained therein, as well as trying developed other statements related. The statement can be about anything, which by the mathematician chosen as objects that deserve investigation and scrutiny. One of the objects of interest is, of course, a number. Apparently there are a variety of numbers, both native, and non-native. Apparently there are a number that have special characteristics, which are then called prime numbers. Study about it is very important not only in mathematics, but also in computer engineering.
There are other interesting objects in geometry, such as points, lines, areas, and shapes that may arise therefrom, such as triangles, circles, ellipses, cubes, spheres, cones, pyramids, and others. It is well-known statement of Pythagoras, that in a right-angled triangle, the square on the hypotenuse wide (hypotenuse) is equal to the number of squares wide either side of the other.
In the study of shapes in geometry, appears naturally concepts such as length, area, contents, weight and center of gravity. Developing the concept of a unit thereof, the amount and function.
Object of interest for mathematicians is set, and the relationship between the various sets. The study of this case has produced a variety of practical uses, as well as directing mathematicians to various fundamental problems, often abstract, about mathematics and the foundations upon which there is a building which is now called mathematics. Whatever is learned and performed, all back to attempt to make at least a statement of accountability.
The statement will be accepted as a true statement, if it can be given to him (at least) a convincing evidence, that argument or a row of neat sentences, coherent and plausible, that thereof there is no longer any doubt about the truth of the theorem are discussed.
Statement whose truth was never in doubt, but was never given the evidence, called axioms. Statement whose truth was guaranteed at least by a convincing proof of the theorem or lemma called. The difference between the theorem and lemma need not be discussed here. For example, at the beginning of his classic book entitled "Grundlage der Geometrie", David Hilbert wrote axioms about points, lines and areas. Of axioms that he was able to prove all the important theorems in geometry. - That is a concrete example of mathematics.
Mathematics and logic. Math is usually placed in the same category with the logic, which is a branch of philosophy. Philosophy itself in the Indonesian dictionary is described as "the knowledge and the investigation by reason of the nature of all that exists, the cause, origin and legal" - scientia rerum per causas Ultimas. According to Bertrand Russell, the logic is the future sister-school of mathematics.
Mathematics and physics. Mathematics is also usually placed in the same category with the physics, the childhood called cosmology, which is a branch of philosophy as well. In physics studied natural objects (objects), and above it a try made object statements is true, which is now commonly called the laws of nature or the laws of physics. (In this paper the chemical is treated as an integral part of physics).
Is there a significant difference between the theorem (in mathematics) and laws (in physics)? One difference is in the proof of the truth. Theorem proved by argument, while the laws of physics proved by experiment or observation.
Another difference is evident in the nature of research conducted over the theorems and laws. Studied law, and try to be redefined, in the context of the phenomena of nature are realized, or at least suspected, to contain new facts, so that the law can be reformulated into a new include the fact that as well as having a universal characteristic, which is applicable anywhere, anytime . Research generally studied in the pattern of inductive thinking.
Theorem was studied with similar goals. But the theorem is often expressed by n assumption: "if n assumptions A1, A2, ... An this is true, then the statement P is true". Does chopped assumptions can be reduced, so that the theorem can be formulated without assuming (for example)? If the theorem is based on the assumption that a very tight, strict assumptions it can be softened, without changing the material in a theorem? - Research studies generally deductive and directed toward the establishment of a complete system of mathematics: logical, consistent and efisen (useful).
2. Mathematics: modeling tool for engineers
In this text is meant by the technicians are those who live by doing activities in the field of engineering sciences, that their attention is to attempt a solution to the real problems in society. The issue may be the planning of the building, or the provision of telecommunications infrastructure that spans a number of residents in an area. One real problem today is that the value of the rupiah. Appropriate measures are to be done to the national economy is not mired in a prolonged crisis? This is certainly a problem for the technicians in the field of economics. Another current issue of large scale today is the creation of a set of smoke that worry in most of the forests of Sumatra and Borneo, which has given rise to a variety of real problems in the areas of health, social, economic, legal, domestic politics and international relations.
For engineers in certain fields (eg, law enforcement), the math may be no clear benefit in solving the problem of large scale. But for engineers in other fields (eg, toward the preservation of environmental quality control) math definitely helps. In this last case, just be dictum or advice, that "mathematics is a substitute to thinking". Mathematics serves as a knife to make a sharp analysis leads to the desired problem-solving. In that context, as expressed by Professor Adhi Susanto, math fill analytics aspect engineering sciences.
Faced with the real issues in the community, whether that can be done by the technicians? It seems inconceivable that the technicians will formulate a model for the problem to be solved. Called the model is in fact also be a statement about the problems faced. This was done, among others, by looking at the scope of the problem, create categories based on field studies and science, as well as sorting on variable parameters which are primary and which is secondary, as well as setting a model, which is considered to have simple enough for further analysis, but at the same time realistic enough to describe the situation in the real world. "Great engineering is simple engineering". In this whole process of mathematical modeling (logic) helped pave the way for the formulation of the desired model.
There are three points to note:
1. The laws of nature that apply;
2. The information and experience in the field;
3. The final target to be achieved.
Given the laws of nature are generally disclosed in the statement that is certainly as well as containing no doubts would cause and effect, the model obtained therefrom is also deterministic. This model is often in the form of mathematical equations derived from the principles of conservation of energy, mass and momentum. This model is deterministic. In contrast, the insufficiencies of information on certain aspects of reality, or the unavailability of an adequate formula for stating the laws of nature that apply to the realities often encourage the formation of a model of non-deterministic. Models of this type are often developed using the concept of chance or probability, but it is also possible that the model purely heuristic or ad-hoc.
The model can also be classified as an equation-based, if the equation was derived from the laws of nature, or obtained in the form of the regression equation is lifted from observation and intensive measurements in the field. This model is objective. Instead classified as information-based models, if developed through a process of adaptive reasoning, using the concept of neural networks and utilizing Fuzzy logic paradigm. Information-based models often contain elements that are subjective, at least to some degree, because it depends on the maturity of insight and thought the model development technician. Models of this type are now becoming more and more popular, partly because of its practical, "the end justifies the means". Equation-based models for the same problem can not have a solution or provide a solution that is not acceptable (for example, should give a positive answer, but the figure turned out to be negative).
Furthermore, also known as model-based operations. This type of model is a paradigm example utilizes a queue that is contained in the various processes in nature and in everyday activities. Model-based operations directly or indirectly enter the time as an important element in the assessment of the problems faced. In practice commonly used paradigm and the concept of input-output transfer function ("transfer function"). In short, trying to simulate the operational model of the process that is expected to occur.
In contrast to the operational model, known also planning model, which utilizes optimization paradigm in the use of resources. The main target of the settlement that has unique characteristics in terms of the assessment of costs and benefits.
The facts:
1. The model only reflects the accumulated experience of the developers of the model will be in the context of the problems faced in real-world situations. In this sense the model is just another formulation of the data and information in its possession.
2. The issue itself is often open-ended, with chopped answers, solutions or settlement can be more than one, even if the solution is unique (one and only one solution) desired.
3. The model is often implicit, - information hidden behind the relationships contained in the model. Expressed in other words, the models that have been developed, the desired information can be nothing in it.
4. Therefore, in solving real problems often have to do the process of iteration, the iterative process with the input of a cycle is determined by the output of the previous cycle, and eventually gained a promising model of the desired settlement.
3. Computing: tools, methods and theories
Computing is the activity of the settlement or solution to the problem stated in the mathematical model. Mathematically the model generally takes the form
f (x) = y,
with x = the set of hidden information in the model, such as quantities whose value should be set so that the real issues can be solved, y = the set of available data, such as quantities whose values are known, and f (.) = the operator of the mathematical models . Briefly in the given computing f (.) As well as the numerical value of y, do activities to obtain the numerical value of x, so that f (x) = y met.
Mathematically, x is obtained through inverse operation on y. Concretely,
x = f-1 (y),
with f-1 mathematical operators to implement the inverse operation is intended. The main problem: in practice not many operators f by f-1 are known or can be determined directly with ease. It is therefore often have to go through the process of computing the indirect path.
Engineering is the science of computing device (usually a computer), a method (called an algorithm) and theoretical (mathematical proof that the computation give the correct result) required to carry out the computation. Meanwhile, in the conduct of computing to solve a problem, a technician should pay attention to the interaction of the tool (the computer used), the method (ie owned programs), and the unique nature of the matter at hand, because in practice the questions difficulty level different: there is relatively very easy matter, nothing difficult, but there is also a very difficult matter.
4. Numbers, size and function: from KBBI
Numbers: "abstract ideas are not a symbol or emblem, which gives information about the many mem- bers set". Next in KBBI described a variety of numbers in mathematics and physics.
Figures: "signs or symbols instead of numbers". (Later in KBBI also given a lot of entries about the various kinds of numbers.)
Magnitude: "tree, the leaves can be used as medicine, and is used for food silkworm". (Obviously not intended in this text).
Modifiers: "the symbol used to denote elements that are not necessarily in a set".
Function: "scale-related, if the magnitude of the change, the other size is also changed".
comment:
1. A layman who wants mengkonsultasi KBBI to get clarity on these terms is definitely hard to understand.
2. Creating a dictionary, let alone KBBI, not an easy job.
5. Align the term: Numbers
Numbers: "abstract ideas that are used to provide information about a quantity (eg mince, amount, how many, how much, ...).
Known in various ways to group a number. There are grouping on the basis of positive numbers and negative numbers. There is also a grouping on the basis of an integer (integer), real numbers, exponential numbers, and complex numbers.
Numbers are called round (integer) if it does not contain a decimal point. Numbers are called real if it contains a decimal point. Then the "7" are integers, but "7", "7.0" and "-17 453" is a real number. An example is the exponential number "-0.17453102", which is the expression in exponential numbers above "-17 453". In the print-outs often this number is written "-0.17453E + 2", for reasons that are now clear.
Numbers are called complex if it consists of two parts, namely (1) the real part, and (2) the imaginary part (complex). Example: "-3 + 4i", the real part is "-3", the imaginary part (complex) is "4". Surely complex numbers "+ 2.71-5.38i" consists of a real part "+2.71" and the imaginary part is "-5.38".
Both the real part and an imaginary part (complex) is of the type of real numbers or a month. Imaginary part is distinguished from the estate under the symbol "i", written in advance or behind. Here i =. Of course applies the property that i2 = -1, i3 = -i, and i4 = +1.
In mathematics taught, that an integer can be represented as a point in a line of integers I, a real number can be represented as a point in the line of real numbers R, and a complex number represented as a point in the complex field C.
6. Align the term: Magnitude
The following formulation is considered closer to the truth. Magnitude: "inherent in the nature of an object or objects (concrete or abstract), the nature of which is contained in, or that can not be separated from, the object or objects so that they can be understood as one of the characteristics, attributes or identity of the object or objects".
On an object an observer may note a number scale. For example, if the object was a student, some of the quantities that are, among others, (1) the name of the student, (2) place of birth, (3) date of birth, (4) which is derived from the high school, (5) with NEM how, (6) enrolled in the study do, (7) is now already in the semester keberapa, (8) how many credits have been collected, (9) cumulative grade index, (10) home address, - and so on.
Magnitude attached to an object can be categorized into two types, namely (1) numerical magnitude, and (2) the amount of non-numerical. Age, cumulative grade index, registered in the semester to how, a few examples of numerical magnitude. Magnitude-called numerical if on the magnitude of the observer (in one way or another, usually by measurement) can provide an appropriate number possessed by the object. In everyday language it is said that on these quantities is given a value that describes the amount of objec on the object in question. For the complete set of information is added, which is a standard or basic used to provide that value. That way doubts can be reduced as small as possible, especially if used in standard units. So over the life of a student is given a value of 20 years, and over the life of a chicken is given a value (for example) 81.7 days. It is definitely different from the age of 20 013 years for another student, and the age of 81.6 days over the other chickens. Even the difference between 20 years and 20 013 years is uncertain, ie 0013 years.
Above a non-numerical scale, by definition, an observer can not give a number as a numerical value on the scale. Which can be provided on a non-numerical scale is an assessment, which can be subjective. A typical example is the amount of color (flowers). Top of this magnitude can only give a label a proper adjective describing the state of these quantities, such as "green". Assessment "green" is called subjective because other observers barangkalai labeled "slightly reddish green". What's different about "green" to "slightly reddish green", can not easily be explained.
Another example is the magnitude of the achievement index. The labels can be given to him is "A", "B", "C" and "D", respectively to the nature of "very good", "good", "pretty" and "ugly". That there is a label for the achievement index "E" and "F", no consideration.
On this occasion it should be noted that the allocation of marks (for example) on the amount of 2.74 cumulative grade index is based on the assumption that "A" is equal to 4, "B" with 3 "C" 2 and "D" to 1. That assumption comes other assuming that the "E" is equal to 0. the rationale for why such is not mentioned here. But it should be asked whether the unit is used to index the feat?
7. Function: is it?
The function is a mathematical concept that is intended to describe succinctly ("tight aos", Javanese) relationship between two numerical quantities. Of course, usually it is based on the assumption that a unique relationship exists and, until proven otherwise.
Often the function f is expressed also as a mapping ("mapping") between a value in the magnitude of the target value on the other scale. The range of values in the magnitude of the form "domain" of the function D, while the range of the target value on the size of the other so-called "range" of the function R, and written
f = D R.
The mapping is called one-on-one ("one-to-one mapping"), if for a value of x in the "domain" D is only one and only one value of y in the "range" R of the function. Therefore written
y = f (x).
Related to this notion is the idea, if x is known, then the operation f (.) Over x produces y. This formulation is explicit.
The mathematical model
f (x) = y
display different ideas. Operation f (.) Over x produces y. Mathematical models normally be such that the information on f (.) As well as the values of y are known or possessed, and with the help of a model that an analyst wants to set the value of x that satisfies the condition that y = f (x). The formulation of the mathematical model is always implicit.
Back to the problem of searching for the value of. This issue is explicitly stated as a problem in the set y = x0.5, with x: = 10. (Here the sign ": =" should read "rated equal to"). In contrast, the implicit formulation to search x that x2 = 10. This means that there was an operation f (x) x2 over x produces y. For y: = 10; assign the value of x.
In this implicit formulation basically want to find the inverse function f-1 (.), Such that x = f-1 (y). It has been shown upfront, that basically f-1 (.) Can be expressed as a result of the operation explicitly g (.) Is carried out repeatedly. Concretely,
f-1 (x) = g (g (g (... (g (x) ...))),
with g (x) (x + y / x) / 2 and y here appears as a parameter in the function g (.) and its initial value is known, ie y: = 10. Here, x is the value of the root is sought. Because the value of x is not known, then at the beginning it was only made estimates only, and the theory of truth guarantees results.
Furthermore, what happens now? Mathematical problem is implicit f (x) = y initially formulated as a problem set inverse function x = f-1 (y) (which is explicit), but then it turns out that the function (inverse nature) explicit f-1 (.) it can be expressed as a result of repeated application of an explicit function g (.) which is unique.
The algorithm is the standard term for the process of repeated computing to solve problems in the real world that are explicit mathematical formulation. Each step in the computing operations are explicitly computing operations. In practice, the algorithm is still to be written in a program (in a computer language) for input to the computer to be carried out in. Basically, the algorithm must have a sequence of statements that intentionally communicated to the computer as the idea of solving an issue, as a defender delivered an oration = row in a court statement as an argument to justify his client. The algorithm is called true if the row that statement does reveal the true idea for solving problems in the real world. It can also be said that the algorithm is a means for someone to communicate ideas to the computer, so the computer help people in solving problems in the real world. Here, the computer acts as a helper agent problem-solving and real here also benefits the study of algorithms and the study of language (in particular computer language).
An algorithm to determine the appropriate root x of a positive real number a is as follows:
Input a;
State x: = a;
If not convergent, continue doing that contained the following:
y: = (x + a / x) / 2; x: = y;
Show x;
Done.
But it may be asked what is meant by "not converge"? One of the criteria of convergence is x - y 0.000001. So if that condition is not fulfilled, the two explicit operation y: = (x + a / x) / 2 and x: = y of the above must be repeated. The algorithm is complete below:
Input a;
State x: = a; y: = 0;
If x - y 0.000001, do:
y: = (x + a / x) / 2; x: = y;
Show x;
Done.
1. Mathematics: What is it?
According to the Dictionary of Indonesian Language, mathematics is "the science of numbers, relationships among numbers, and operational procedures used in solving the problem of numbers". This sentence is not the exact formula, though can be said to be sufficient for inclusion in the dictionary. There is a branch of mathematics that are not directly dealing with numbers, such as geometry, topology, graph theory and logic.
In this manuscript is defined as the science of mathematical statements, as well as the conditions necessary for a statement is true. In a statement the truth of mathematics tested, researched the meaning or implications of every word contained therein, as well as trying developed other statements related. The statement can be about anything, which by the mathematician chosen as objects that deserve investigation and scrutiny. One of the objects of interest is, of course, a number. Apparently there are a variety of numbers, both native, and non-native. Apparently there are a number that have special characteristics, which are then called prime numbers. Study about it is very important not only in mathematics, but also in computer engineering.
There are other interesting objects in geometry, such as points, lines, areas, and shapes that may arise therefrom, such as triangles, circles, ellipses, cubes, spheres, cones, pyramids, and others. It is well-known statement of Pythagoras, that in a right-angled triangle, the square on the hypotenuse wide (hypotenuse) is equal to the number of squares wide either side of the other.
In the study of shapes in geometry, appears naturally concepts such as length, area, contents, weight and center of gravity. Developing the concept of a unit thereof, the amount and function.
Object of interest for mathematicians is set, and the relationship between the various sets. The study of this case has produced a variety of practical uses, as well as directing mathematicians to various fundamental problems, often abstract, about mathematics and the foundations upon which there is a building which is now called mathematics. Whatever is learned and performed, all back to attempt to make at least a statement of accountability.
The statement will be accepted as a true statement, if it can be given to him (at least) a convincing evidence, that argument or a row of neat sentences, coherent and plausible, that thereof there is no longer any doubt about the truth of the theorem are discussed.
Statement whose truth was never in doubt, but was never given the evidence, called axioms. Statement whose truth was guaranteed at least by a convincing proof of the theorem or lemma called. The difference between the theorem and lemma need not be discussed here. For example, at the beginning of his classic book entitled "Grundlage der Geometrie", David Hilbert wrote axioms about points, lines and areas. Of axioms that he was able to prove all the important theorems in geometry. - That is a concrete example of mathematics.
Mathematics and logic. Math is usually placed in the same category with the logic, which is a branch of philosophy. Philosophy itself in the Indonesian dictionary is described as "the knowledge and the investigation by reason of the nature of all that exists, the cause, origin and legal" - scientia rerum per causas Ultimas. According to Bertrand Russell, the logic is the future sister-school of mathematics.
Mathematics and physics. Mathematics is also usually placed in the same category with the physics, the childhood called cosmology, which is a branch of philosophy as well. In physics studied natural objects (objects), and above it a try made object statements is true, which is now commonly called the laws of nature or the laws of physics. (In this paper the chemical is treated as an integral part of physics).
Is there a significant difference between the theorem (in mathematics) and laws (in physics)? One difference is in the proof of the truth. Theorem proved by argument, while the laws of physics proved by experiment or observation.
Another difference is evident in the nature of research conducted over the theorems and laws. Studied law, and try to be redefined, in the context of the phenomena of nature are realized, or at least suspected, to contain new facts, so that the law can be reformulated into a new include the fact that as well as having a universal characteristic, which is applicable anywhere, anytime . Research generally studied in the pattern of inductive thinking.
Theorem was studied with similar goals. But the theorem is often expressed by n assumption: "if n assumptions A1, A2, ... An this is true, then the statement P is true". Does chopped assumptions can be reduced, so that the theorem can be formulated without assuming (for example)? If the theorem is based on the assumption that a very tight, strict assumptions it can be softened, without changing the material in a theorem? - Research studies generally deductive and directed toward the establishment of a complete system of mathematics: logical, consistent and efisen (useful).
2. Mathematics: modeling tool for engineers
In this text is meant by the technicians are those who live by doing activities in the field of engineering sciences, that their attention is to attempt a solution to the real problems in society. The issue may be the planning of the building, or the provision of telecommunications infrastructure that spans a number of residents in an area. One real problem today is that the value of the rupiah. Appropriate measures are to be done to the national economy is not mired in a prolonged crisis? This is certainly a problem for the technicians in the field of economics. Another current issue of large scale today is the creation of a set of smoke that worry in most of the forests of Sumatra and Borneo, which has given rise to a variety of real problems in the areas of health, social, economic, legal, domestic politics and international relations.
For engineers in certain fields (eg, law enforcement), the math may be no clear benefit in solving the problem of large scale. But for engineers in other fields (eg, toward the preservation of environmental quality control) math definitely helps. In this last case, just be dictum or advice, that "mathematics is a substitute to thinking". Mathematics serves as a knife to make a sharp analysis leads to the desired problem-solving. In that context, as expressed by Professor Adhi Susanto, math fill analytics aspect engineering sciences.
Faced with the real issues in the community, whether that can be done by the technicians? It seems inconceivable that the technicians will formulate a model for the problem to be solved. Called the model is in fact also be a statement about the problems faced. This was done, among others, by looking at the scope of the problem, create categories based on field studies and science, as well as sorting on variable parameters which are primary and which is secondary, as well as setting a model, which is considered to have simple enough for further analysis, but at the same time realistic enough to describe the situation in the real world. "Great engineering is simple engineering". In this whole process of mathematical modeling (logic) helped pave the way for the formulation of the desired model.
There are three points to note:
1. The laws of nature that apply;
2. The information and experience in the field;
3. The final target to be achieved.
Given the laws of nature are generally disclosed in the statement that is certainly as well as containing no doubts would cause and effect, the model obtained therefrom is also deterministic. This model is often in the form of mathematical equations derived from the principles of conservation of energy, mass and momentum. This model is deterministic. In contrast, the insufficiencies of information on certain aspects of reality, or the unavailability of an adequate formula for stating the laws of nature that apply to the realities often encourage the formation of a model of non-deterministic. Models of this type are often developed using the concept of chance or probability, but it is also possible that the model purely heuristic or ad-hoc.
The model can also be classified as an equation-based, if the equation was derived from the laws of nature, or obtained in the form of the regression equation is lifted from observation and intensive measurements in the field. This model is objective. Instead classified as information-based models, if developed through a process of adaptive reasoning, using the concept of neural networks and utilizing Fuzzy logic paradigm. Information-based models often contain elements that are subjective, at least to some degree, because it depends on the maturity of insight and thought the model development technician. Models of this type are now becoming more and more popular, partly because of its practical, "the end justifies the means". Equation-based models for the same problem can not have a solution or provide a solution that is not acceptable (for example, should give a positive answer, but the figure turned out to be negative).
Furthermore, also known as model-based operations. This type of model is a paradigm example utilizes a queue that is contained in the various processes in nature and in everyday activities. Model-based operations directly or indirectly enter the time as an important element in the assessment of the problems faced. In practice commonly used paradigm and the concept of input-output transfer function ("transfer function"). In short, trying to simulate the operational model of the process that is expected to occur.
In contrast to the operational model, known also planning model, which utilizes optimization paradigm in the use of resources. The main target of the settlement that has unique characteristics in terms of the assessment of costs and benefits.
The facts:
1. The model only reflects the accumulated experience of the developers of the model will be in the context of the problems faced in real-world situations. In this sense the model is just another formulation of the data and information in its possession.
2. The issue itself is often open-ended, with chopped answers, solutions or settlement can be more than one, even if the solution is unique (one and only one solution) desired.
3. The model is often implicit, - information hidden behind the relationships contained in the model. Expressed in other words, the models that have been developed, the desired information can be nothing in it.
4. Therefore, in solving real problems often have to do the process of iteration, the iterative process with the input of a cycle is determined by the output of the previous cycle, and eventually gained a promising model of the desired settlement.
3. Computing: tools, methods and theories
Computing is the activity of the settlement or solution to the problem stated in the mathematical model. Mathematically the model generally takes the form
f (x) = y,
with x = the set of hidden information in the model, such as quantities whose value should be set so that the real issues can be solved, y = the set of available data, such as quantities whose values are known, and f (.) = the operator of the mathematical models . Briefly in the given computing f (.) As well as the numerical value of y, do activities to obtain the numerical value of x, so that f (x) = y met.
Mathematically, x is obtained through inverse operation on y. Concretely,
x = f-1 (y),
with f-1 mathematical operators to implement the inverse operation is intended. The main problem: in practice not many operators f by f-1 are known or can be determined directly with ease. It is therefore often have to go through the process of computing the indirect path.
Engineering is the science of computing device (usually a computer), a method (called an algorithm) and theoretical (mathematical proof that the computation give the correct result) required to carry out the computation. Meanwhile, in the conduct of computing to solve a problem, a technician should pay attention to the interaction of the tool (the computer used), the method (ie owned programs), and the unique nature of the matter at hand, because in practice the questions difficulty level different: there is relatively very easy matter, nothing difficult, but there is also a very difficult matter.
4. Numbers, size and function: from KBBI
Numbers: "abstract ideas are not a symbol or emblem, which gives information about the many mem- bers set". Next in KBBI described a variety of numbers in mathematics and physics.
Figures: "signs or symbols instead of numbers". (Later in KBBI also given a lot of entries about the various kinds of numbers.)
Magnitude: "tree, the leaves can be used as medicine, and is used for food silkworm". (Obviously not intended in this text).
Modifiers: "the symbol used to denote elements that are not necessarily in a set".
Function: "scale-related, if the magnitude of the change, the other size is also changed".
comment:
1. A layman who wants mengkonsultasi KBBI to get clarity on these terms is definitely hard to understand.
2. Creating a dictionary, let alone KBBI, not an easy job.
5. Align the term: Numbers
Numbers: "abstract ideas that are used to provide information about a quantity (eg mince, amount, how many, how much, ...).
Known in various ways to group a number. There are grouping on the basis of positive numbers and negative numbers. There is also a grouping on the basis of an integer (integer), real numbers, exponential numbers, and complex numbers.
Numbers are called round (integer) if it does not contain a decimal point. Numbers are called real if it contains a decimal point. Then the "7" are integers, but "7", "7.0" and "-17 453" is a real number. An example is the exponential number "-0.17453102", which is the expression in exponential numbers above "-17 453". In the print-outs often this number is written "-0.17453E + 2", for reasons that are now clear.
Numbers are called complex if it consists of two parts, namely (1) the real part, and (2) the imaginary part (complex). Example: "-3 + 4i", the real part is "-3", the imaginary part (complex) is "4". Surely complex numbers "+ 2.71-5.38i" consists of a real part "+2.71" and the imaginary part is "-5.38".
Both the real part and an imaginary part (complex) is of the type of real numbers or a month. Imaginary part is distinguished from the estate under the symbol "i", written in advance or behind. Here i =. Of course applies the property that i2 = -1, i3 = -i, and i4 = +1.
In mathematics taught, that an integer can be represented as a point in a line of integers I, a real number can be represented as a point in the line of real numbers R, and a complex number represented as a point in the complex field C.
6. Align the term: Magnitude
The following formulation is considered closer to the truth. Magnitude: "inherent in the nature of an object or objects (concrete or abstract), the nature of which is contained in, or that can not be separated from, the object or objects so that they can be understood as one of the characteristics, attributes or identity of the object or objects".
On an object an observer may note a number scale. For example, if the object was a student, some of the quantities that are, among others, (1) the name of the student, (2) place of birth, (3) date of birth, (4) which is derived from the high school, (5) with NEM how, (6) enrolled in the study do, (7) is now already in the semester keberapa, (8) how many credits have been collected, (9) cumulative grade index, (10) home address, - and so on.
Magnitude attached to an object can be categorized into two types, namely (1) numerical magnitude, and (2) the amount of non-numerical. Age, cumulative grade index, registered in the semester to how, a few examples of numerical magnitude. Magnitude-called numerical if on the magnitude of the observer (in one way or another, usually by measurement) can provide an appropriate number possessed by the object. In everyday language it is said that on these quantities is given a value that describes the amount of objec on the object in question. For the complete set of information is added, which is a standard or basic used to provide that value. That way doubts can be reduced as small as possible, especially if used in standard units. So over the life of a student is given a value of 20 years, and over the life of a chicken is given a value (for example) 81.7 days. It is definitely different from the age of 20 013 years for another student, and the age of 81.6 days over the other chickens. Even the difference between 20 years and 20 013 years is uncertain, ie 0013 years.
Above a non-numerical scale, by definition, an observer can not give a number as a numerical value on the scale. Which can be provided on a non-numerical scale is an assessment, which can be subjective. A typical example is the amount of color (flowers). Top of this magnitude can only give a label a proper adjective describing the state of these quantities, such as "green". Assessment "green" is called subjective because other observers barangkalai labeled "slightly reddish green". What's different about "green" to "slightly reddish green", can not easily be explained.
Another example is the magnitude of the achievement index. The labels can be given to him is "A", "B", "C" and "D", respectively to the nature of "very good", "good", "pretty" and "ugly". That there is a label for the achievement index "E" and "F", no consideration.
On this occasion it should be noted that the allocation of marks (for example) on the amount of 2.74 cumulative grade index is based on the assumption that "A" is equal to 4, "B" with 3 "C" 2 and "D" to 1. That assumption comes other assuming that the "E" is equal to 0. the rationale for why such is not mentioned here. But it should be asked whether the unit is used to index the feat?
7. Function: is it?
The function is a mathematical concept that is intended to describe succinctly ("tight aos", Javanese) relationship between two numerical quantities. Of course, usually it is based on the assumption that a unique relationship exists and, until proven otherwise.
Often the function f is expressed also as a mapping ("mapping") between a value in the magnitude of the target value on the other scale. The range of values in the magnitude of the form "domain" of the function D, while the range of the target value on the size of the other so-called "range" of the function R, and written
f = D R.
The mapping is called one-on-one ("one-to-one mapping"), if for a value of x in the "domain" D is only one and only one value of y in the "range" R of the function. Therefore written
y = f (x).
Related to this notion is the idea, if x is known, then the operation f (.) Over x produces y. This formulation is explicit.
The mathematical model
f (x) = y
display different ideas. Operation f (.) Over x produces y. Mathematical models normally be such that the information on f (.) As well as the values of y are known or possessed, and with the help of a model that an analyst wants to set the value of x that satisfies the condition that y = f (x). The formulation of the mathematical model is always implicit.
Back to the problem of searching for the value of. This issue is explicitly stated as a problem in the set y = x0.5, with x: = 10. (Here the sign ": =" should read "rated equal to"). In contrast, the implicit formulation to search x that x2 = 10. This means that there was an operation f (x) x2 over x produces y. For y: = 10; assign the value of x.
In this implicit formulation basically want to find the inverse function f-1 (.), Such that x = f-1 (y). It has been shown upfront, that basically f-1 (.) Can be expressed as a result of the operation explicitly g (.) Is carried out repeatedly. Concretely,
f-1 (x) = g (g (g (... (g (x) ...))),
with g (x) (x + y / x) / 2 and y here appears as a parameter in the function g (.) and its initial value is known, ie y: = 10. Here, x is the value of the root is sought. Because the value of x is not known, then at the beginning it was only made estimates only, and the theory of truth guarantees results.
Furthermore, what happens now? Mathematical problem is implicit f (x) = y initially formulated as a problem set inverse function x = f-1 (y) (which is explicit), but then it turns out that the function (inverse nature) explicit f-1 (.) it can be expressed as a result of repeated application of an explicit function g (.) which is unique.
The algorithm is the standard term for the process of repeated computing to solve problems in the real world that are explicit mathematical formulation. Each step in the computing operations are explicitly computing operations. In practice, the algorithm is still to be written in a program (in a computer language) for input to the computer to be carried out in. Basically, the algorithm must have a sequence of statements that intentionally communicated to the computer as the idea of solving an issue, as a defender delivered an oration = row in a court statement as an argument to justify his client. The algorithm is called true if the row that statement does reveal the true idea for solving problems in the real world. It can also be said that the algorithm is a means for someone to communicate ideas to the computer, so the computer help people in solving problems in the real world. Here, the computer acts as a helper agent problem-solving and real here also benefits the study of algorithms and the study of language (in particular computer language).
An algorithm to determine the appropriate root x of a positive real number a is as follows:
Input a;
State x: = a;
If not convergent, continue doing that contained the following:
y: = (x + a / x) / 2; x: = y;
Show x;
Done.
But it may be asked what is meant by "not converge"? One of the criteria of convergence is x - y 0.000001. So if that condition is not fulfilled, the two explicit operation y: = (x + a / x) / 2 and x: = y of the above must be repeated. The algorithm is complete below:
Input a;
State x: = a; y: = 0;
If x - y 0.000001, do:
y: = (x + a / x) / 2; x: = y;
Show x;
Done.
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