III. Заполните пропуски подходящими по смыслу словами.
English for mathematicians
Практикум
Автор: Т.И. Боровик
Редактор: В авторской редакции
Данный практикум представляет собой систематизированный материал для чтения профессионально ориентированных текстов по специальности «Математические методы в экономике». В практикуме представлены следующие разделы: “What is mathematics”, “Unsolved mathematical problems”, “Greek schools of mathematics”, “Descartes’s and P.Fermat’s coordinate geometry”, “Analysis Incarnate – Leonard Euler”. К текстам практикума разработаны предтекстовые и послетекстовые лексико-грамматические упражнения. Практикум содержит также тексты по домашнему чтению, лексический минимум и грамматический справочник. Цель практикума: развитие навыков чтения и перевода текстов по специальности на базе оригинального материала. Для студентов неязыковых специальностей.
| Part I | ||||
| WHAT IS MATHEMATICS? | ||||
| UNIT I | ||||
I. Найдите в тексте интернациональные слова, переведите их. II. Выберите в колонке В эквиваленты к словам колонки А.
|
III. Заполните пропуски подходящими по смыслу словами.
1) The largest branch is that which builds on the ordinary whole numbers, …, and irrational numbers, or what, collectively, is called the real number system.
a) fractions b) calculus
c) differential equations d) areas
2) These concepts must verify explicitly stated … .
a) theorems b) axioms
c) equations c) calculus
3) The certain concept of geometry is point, line and … .
a) area b) triangle
c) fraction d) right angle
4) Some of axioms of the mathematics of … are the associative, commutative, and distributive properties.
a) quantitative b) arithmetic
c) simple d) number
5) Some of the axioms of geometry are that two points determine a … all right angles are equal, etc.
a) angle b) triangle
c) line d) right angle
Text I
The students of mathematics may wonder where the word "mathematics "comes from. Mathematics is a Greek word, and, by origin or etymologically, it means "something that must be learnt or understood", perhaps “acquired knowledge" or "knowledge acquirable by learning" or “general knowledge". The word "mathematics'' is a contraction of all these phrases. The celebrated Pythagorean school in ancient Greece had both regular and incidental members. The incidental members were called "auditors"; the regular members were named "mathematicians" as a general class and not because they specialized in mathematics; for them mathematics was a mental discipline of science of learning. What is mathematics in the modern sense of the term, its implications and connotations? There is no neat, simple, general and unique answer to this question.
Mathematics as a science, viewed as a whole, is a collection of branches. The largest branch is that which builds on the ordinary whole numbers, fractions, and irrational numbers, or what, collectively, is called the real number system. Arithmetic, algebra, the study of functions, the calculus differential, equations, and various other subjects which follow the calculus in logical order, are all developments of the real number system. This part of mathematics is termed the mathematics of number. A second branch is geometry consisting of several geometries. Mathematics contains many more divisions. Each branch has the same logical structure: it begins with certain concepts, such as the whole numbers or integers in the mathematics of number, and such as point, line and triangle in geometry. These concepts must verify explicitly stated axioms. Some of the axioms of the mathematics of number are the associative, commutative, and distributive properties and the axioms about equalities. Some of the axioms of geometry are that two points determine a line, all right angles are equal, etc. From the concepts and axioms theorems are deduced. Hence, from the standpoint of structure, the concepts, axioms and theorems are the essential components of any compartment of mathematics. We must break down mathematics into separately taught subjects, but this compartmentalization taken as a necessity, must be compensated for as much as possible. Students must see the interrelationships of the various areas and the importance of mathematics for other domains. Knowledge is not additive but an organic whole and mathematics is an inseparable part of that whole. The full significance of mathematics can be seen and taught only in terms of its intimate relationships to other fields of knowledge. If mathematics is isolated-from other provinces, it loses importance.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Where does the word “mathematics” come from?
a) Greece b) England
c) Russia d) Alexandria
2. What does the word “mathematics” mean by origin or etymologically?
a) “acquired knowledge” b) “logical construction”
c) “scientific knowledge” d)“knowledge about nature”
3. What is mathematics as a science?
a) a real number system b) a collection of branches
c) a calculus in logical order
d) a calculus, differential equations, and functions
4. What is the largest branch of mathematics?
a) geometry b) differential equations
c) the whole number system d) the real number system
5. What is the certain concept of mathematics of number?
a) whole numbers or integers b) points, lines, and triangles
c) differential equations d)fractions and irrational umbers
6. What is deduced from the concepts and axioms?
a) structures b) theorems
c) calculus d) equations
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Geometry
b. Mathematics of number
c. Mathematics as a science
d. The Pythagorean school
Text II
The basic concepts of the main branches of mathematics are abstractions from experience, implied by their obvious physical counterparts. But it is noteworthy, that many more concepts are introduced which are, in essence, creations of the human mind with or without any help of experience. Irrational numbers, negative numbers and so forth are not wholly abstracted from the physical practice, for the man's mind must create the notion of entirely new types of numbers to which operations such as addition, multiplication, and the like can be applied. The notion of a variable that represents the quantitative values of some changing physical phenomena, such as temperature and time, is also at least one mental step beyond the mere observation of change. The concept of a function, relationship between variables, is almost totally a mental creation.
The more we study mathematics the more we see that the ideas and conceptions involved become more divorced and remote from experience, and the role played by the mind of the mathematician becomes larger and larger. The gradual introduction of new concepts which more and more depart from forms of experience finds its parallel in geometry and many of the specific geometrical terms are mental creations.
As mathematicians nowadays working in any given branch discover new concepts which are less and less drawn from experience and more and more from human mind the development of concepts is progressive and later concepts are built on earlier notions. These facts have unpleasant consequences. Because the more advanced ideas are purely mental creations rather than abstractions from physical experience and because they are defined in terms, of prior concepts it is more difficult to understand them and illustrate their meanings even for a specialist in some other province of mathematics. Nevertheless, the current introduction of new concepts in any field enables mathematics to grow rapidly. Indeed, the growth of modern mathematics is, in part, due to the introduction of new concepts and new systems of axioms.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What are the basic concepts of the main branches of mathematics?
a) abstractions from experience b) experience
c) physical abstractions d) quantitative values
2. What does the notion of a variable represent?
a) the quantitative values of some constant physical phenomena.
b) the quantitative values of some changing physical phenomena.
c) mathematical concepts.
d) functions.
3. Where does the gradual introduction of new mathematical concepts find its parallel in?
a) physics b) geometry
c) experience d) physical practice
4. Where are the new concepts remoted from?
a) experience b) human mind
c) geometry d) mathematics
5. Why are the new concepts more difficult to understand?
a) they are not defined in terms.
b) they are defined in terms of prior concepts.
c) they are defined in terms of physical concepts.
d) they involve many difficult notions.
6. What do new concepts enable mathematics to do?
a)to grow rapidly.
b) to grow slowly.
c) to stop the development.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The basic and new concepts c. Modern mathematicians
b. The basic concepts d. Irrational numbers
Text III
Axioms constitute the second major component of any branch of mathematics. Up to the XIX century axioms were considered as basic self-evident truths about the concepts involved. We know now that this view ought to be given up. The objective of mathematical activity consists of the theorems deduced from a set of axioms. The amount of information that can be deduced from some sets of axioms is almost incredible. The axioms of number give rise to the results of algebra, properties of functions, the theorems of the calculus, the solutions of various types of dif0,ferential equations. Mathematical theorems mist be deductively established and proved. Much of the scientific knowledge is produced by deductive reasoning; new theorems are proved constantly, even in such old subjects as aglebra and geometry and the current developments are as important as the older results.
Growth of mathematics is possible in still another way. Mathematicians are sure now that sets of axioms which have no bearing on the physical world should be explored. Accordingly, mathematicians nowadays investigate algebras and geometries with no immediate applications. There is, however, some disagreement among mathematicians as to the way they answer the question: Do the concepts, axioms, and theorems exist in some objective world and are merely detected by man or are they entirely human creations? In ancient times the axioms and theorems were regarded as necessary truths about the universe already incorporated in the design of the world. Hence each new theorem was a discovery, a disclosure of what already existed. The contrary view holds that mathematics, its concepts, and theorems are created by man. Man distinguishes objects in the physical world and invents numbers and number names to represent one aspect of experience. Axioms are man's generalizations of certain fundamental facts and theorems may very logically follow from the axioms. Mathematics, according to this view-point, is a human creation in every respect. Some mathematicians claim that pure mathematics is the most original creation of the human mind.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What is the second major component of any branch of mathematics?
a) axiom b) concept
c) notion d) theorem
2. What was considered as the basic self-evident truth about concepts up to the 19th century?
a) proofs b) axioms
c) calculus d) theorems
3. What may very logically follow from the axioms?
a) functions b) axioms
c) theorems d) set of notions
4. How is scientific knowledge produced?
a) by deductive reasoning b) by experience
c) from a set of axioms d) by mathematical calculus
5. How do mathematicians nowadays investigate algebra and geometry?
a) with immediate application b) with no immediate application
c) without any application d) with one application
6. What was regarded as necessary truths about the universe in ancient times?
a) hypotheses of future knowledge b) experience
c) the axioms and theorems
d) conclusions of physical processes
7. What may very logically follow from the axioms?
a) notions b) calculus
c) theorems d) concepts
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Mathematical theorems
b. Axioms as the second major component of any branch of mathematics
c. Mathematical concepts
d. Geometry as the second branch of mathematics
Text I
Who of us cannot be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals can there be which the leading mathematical minds of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought can the new centuries disclose?
History teaches the continuity of the development ofscience. We know that every age has its own problems, which the following either solves or casts aside as worthless and replaces by new ones. If we could obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass in our minds and consider the problems which the science of today sets and whose solution we expect from the future. To such a review of present-day problems, raised at the meeting of the centuries, I wish to turn your attention. For the close of a great epoch of the 19th century not only invites us to look back into the past but also directs our thought to the unknown future.
The deep significance of certain problems for the advance of mathematical science, in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long it is alive, a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking seeks after certain objects, so also mathematical research requires its problems. It is by the solution of problems that the researcher tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What does every age have?
a) its own problems b) its own secrets
c) its own purposes d) its own ideas.
2. How long is the branch of science alive?
a) As long as a branch of science offers an abundance of problems, so long it is alive.
b) The branch of science is alive only when all its problems are solved.
c) The branch of science is alive as long as it is interesting for people.
d) The branch of science is always alive.
3. What foreshadows a lack of problems?
a) extinction or the cessation of independent development.
b) Extinction or the cessation of dependent development.
c) The revival of independent development.
d) The disappearance of independent development.
4. What does mathematical research require?
a) the solution of its problems.
b) it requires new investigations.
c) it requires the disappearance of its problems.
d) it requires new methods and ideas.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Connection between mathematical problems in past and in future
b. Mathematical problem in past
c. Mathematical problems in future
d. Mathematical problems of our days
Text II
It is difficult often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless, we can ask whether there are general criteria which mark and label a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered completed until you made it so clear that you can explain it to the first-man whom you meet in the street". This clearness and ease of understanding, here claimed for a mathematical theory, I should still more demand for a mathematical problem that it ought to be perfect; for what is clear and easily 'understandable attracts, while the complicated repels us. Moreover, a mathematical problem should be difficult in order to appeal to us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths and ultimately a reminder of our pleasure in the successful solution.
The mathematicians of past centuries were accustomed to devoting themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of quickest descent" proposed by J. Bernoulli, of Fermat's assertion ; xn + yn =zn, (x, y, z integers) which is unsolvable except in certain self-evident cases. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems. The attempt to prove the impossibility of Fermat's theorem offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. I can remind you as well of the Problem of Three Bodies. The fruitful methods and the far-reaching principles which Poincare brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the fact that he sought to treat anew that difficult problem and to come nearer to its solution.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Whom do the following words belong to? “A mathematical theory is not to be considered completed until you made it so clear that you can explain it to the first man whom you meet in the street”.
a) to an old French mathematician b) to the Greek mathematician
c) to an Alexandrian mathematician d) to the Swiss mathematician
2. What were the mathematicians of past centuries accustomed to?
a) To devoting themselves to the solution of easy problems without passionate zeal.
b) To devoting themselves to the solution of difficult particular problems with passionate zeal.
c) To devoting themselves to finding the new mathematical problems.
d) To devoting themselves to proving mathematical theorems.
3. What mathematical problem was proposed by J. Bernoulli?
a) “The problem of Three Bodies”.
b) “Problem of quickest descent”.
c) “Problem of the straight line”.
d) “The general problem of boundary values”.
4. Whom does the next assertion (xn + yn = zn) belong to?
a) J. Bernoulli b) Fermat
c) Cantor
5. What is it often difficult to do?
a) to judge the value of a problem correctly in advance.
b) to solve the mathematical problem.
c) to find the correct answer to the mathematical question.
d) to prove any mathematical assertion.
6. What did mathematicians of the past know.
a) they knew the solution of problems.
b) they knew the value of difficult problems.
c) they knew the general criteria which mark and label a good mathematical problem.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. A general criteria which mark and label a good mathematical problem
b. The mathematicians of past centuries
c. The clearness and ease of understanding a mathematical theory
d. Fermat’s assertion xn + yn = zn
Text III
But it often happens also that the same special problem finds application in the most diverse and unrelated branches of mathematics. So for example, the problem of the shortest line plays a chief and historically important part in the foundations of Geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And F. Klein convincingly pictured, in his work on the icosahedron, the significance which is attached to the problem of the regular polyhedra in elementary Geometry, in group theory, in the theory of equations and in the theory of linear differential equations.
After referring to the general importance of problems in mathematics, let us return to the question from what sources this science derives its problems. Surely, the first and oldest problems in every field of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with natural numbers were discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first unsolved problems of antiquity, such as the duplication of the cube, the squaring of the circle. Also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential to say nothing of the abundance of problems properly belonging to mechanics, astronomy and physics.
But, in the further development of the special domain of mathematics, the human mind, encouraged by the success of its solutions become convinced of: its independence. It evolves from itself alone, often without appreciable influence from outside by means of logical combination, generalization, specialization, by separating and collecting ideas in elegant ways, by new and fruitful problems and the mind appears then as the real questioner and the source of the new problems. Thus arose the problem of prime numbers and the other unsolved problems of number theory, Galois' theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed, almost all the nicer problems of modern arithmetic and function theory arose in this way.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. In what branches of mathematics can the same special problem find application?
a) in the related branches.
b) in the most diverse and unrelated branches of mathematics.
c) in the connected branches of mathematics.
d) in the different branches of mathematics.
2. What problem plays a chief and historically important part in the foundations of Geometry?
a) the problem of the shortest line.
b) the problem of the longest line.
c) the problem of Three Bodies.
d) “the problem of quickest descent”.
3. Where do the first and oldest problems in every field of mathematics spring from?
a) the nature b) experience
c) the human mind d) the external phenomena
4. What happens with the human mind, encouraged by the success of its solution?
a) it becomes unconvinced of its independence.
b) It becomes convinced of its independence.
c) It becomes sure in itself.
d) It becomes clear.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The sources of mathematical problems
d. The rules of calculation
c. The application of mathematical problems
d. Galois’ theory of equations
Text IV
In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new divisions of mathematics and whilewe seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus simultaneously advance most successfully the old theories, thanks' to this ever-recurring interplay between pure thought and experience.
It remains to discuss briefly what general requirements may be proposed and laid down for the solution of a mathematical problem. I want first of all say this: that it shall be possible to establish the correctness of the solution by means of .a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This demand for logical deduction by means of a finite number of processes is simply the requirement of rigour in reasoning. Indeed, this requirement of rigour, which became proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and on the other hand, only by satisfying this claim do the problems attain their full effect.
Besides it is an error to believe that rigour in the proofis the enemy of simplicity. On the contrary, we find it proved by numerous examples that the rigorous method is at the same time the simpler and worthy in the long run and easier to understand. The very effort for rigour helps us come across a simpler method of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigour. Thus, the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of a more rigorous function-theoretical methods and the introduction of transcendental curves.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What play does the outer world come into?
a) It forces upon us new questions from actual experience.
b) It opens up new divisions of mathematics.
c) It gives us answers to the old questions.
d) It does not come into play.
2. What does the outer world open up?
a) new questions from actual experience.
b) new divisions of mathematics.
c) new field of knowledge.
d) old unsolved problems.
3. What does the very effort for rigour help us to do?
a) to solve mathematical problems.
b) to come across a simpler method of proof.
c) to find new problems.
d) to make an exact formulation of the problem.
4. What is the main demand for the solution of a mathematical problem?
a) it must be exactly formulated.
b) it must be solved.
c) it must not be understandable.
d) it must be based on experience.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The outer world again comes into play
b. The solution of mathematical problem
c. The very effort for rigour helps us come across a simpler method of proof
d. The theory of algebraic curves
Text V
To the new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena of the external world. Likewise the geometric figures are signs or symbols of space intrusion and are used as such by all mathematicians. Who does not always use along with the double inequality a>b>c the picture or drawing of three points following one another on a straight-line as the geometrical idea of "betweenness"? Who does not make use of drawings of segments and rectangles closed in one another, when it is required to prove with perfect rigour a difficult theorem on the continuity of functions or the existence of points of condensation? Who can do without the figure of the triangle, the circle with its centre, or with the cross of three perpendicular axes? The arithmetical symbols are written diagrams and the geometric figures are graphic formulas and no mathematician can do without them or avoid them.
Some remarks upon the difficulties which mathematical problems may offer and the means of overcoming and coping with them may be worth discussing. If we do not manage andare not able to solve a mathematical problem the reason often consists in our failure to recoenize the more general standpoint from which the problem under study appears only as a single link in a chain of related problems. After finding this standpoint, the problem becomes more accessible to our investigations and we possess then a method which is applicable also to related problems. This way for finding general methods is certainly the most fruitful and the most certain; for who seeks for methods without having: a definite problem in mind seeks for the most part in vain.
In dealing with mathematical problems, specialization plays, to my mind a still more important part than. generalization. Perhaps in mostcases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one at issue were either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and I think, that it is used wherever it is possible, though sometimes unconsciously.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What do the new concepts correspond?
a) new signs b) new symbols
c) new solutions d) new problems
2. What does the formula a>b>c mean?
a) It is a double inequality b) It is a double equality
c) It is an equation d) It is triangle
3. What are the arithmetical symbols?
a) graphic formulas b) written diagrams
c) double inequality d) algebraic figures
4. When does the problem become more accessible to our investigations?
a) after finding proper standpoint.
b) after finding the solution of concepts.
c) after making drawing.
5. What does the solution of the problem depend on?
a) on it’s formulation.
b) on it’s source.
c) on finding out easier problems and on their solving.
d) on it’s difficulties.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The phenomena of the external world
b. The double inequality a>b>c
c. Mathematical symbols
d. Mathematical symbols and the rule for overcoming mathematical difficulties
Text VI
Occasionally it happensthat we seek the solution under insufficienthypotheses or in an incorrect sense and for that reason do not surmount the difficulty. The problem then arises to show the impossibility of the solution under the conditions specified. Such proofs of impossibility were effected by the ancients, for instance, when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question of the. impossibility of certain solutions plays a great part and we realize in this way that old and difficult problems, such as the proof of tile axiom of parallels, the squaring the circle, of the solution of equations of the fifth degree by radicals found fully satisfactory and rigorous solutions, although in a different sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares but which as yet no one supported by a proof or refuted) that every definite mathematical problem must necessarily be settled, either in the form of a direct answer to the question posed, or by the proof of the impossibility of its solution and hence the necessary failure of all attempts.
Is this axiom of the solvability of every problem a peculiar characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences there exist also old problems which were handled in a manner most satisfactory and most useful to science by the proof of their impossibility. For example, the problem of perpetual motion. The efforts to construct a perpetual motion machine were not futile as the investigations led to the discovery of the law of the conservation of energy, which, in turn, explained theimpossibility of the perpetual motion in the sense originally presupposed.
This conviction of the solvability of every mathematical problemisa powerful stimulus and impetus to the researcher. We hear within us the perpetual call. There is the problem. Seek its solution. You can find it for in mathematics there is no futile search even if the problem defies solution. The number of problems in mathematics is inexhaustible and as soon as one problem is solved others come forth in its place. Permit me in the following to dwell on particular and definite problems, drawn from various departments of mathematics, whose discussion and possible solution may result in the advancement and progress of science.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Where do we seek the solution of mathematical problems?
a) In the experience.
b) In the outer world.
c) In the hypotheses.
2. What mathematical figure has hypotenuse?
a) circle.
b) right triangle.
c) rectangle.
3. What is the number of mathematical problems?
a) many.
b) exhaustible.
c) inexhaustible.
d) definite.
4. What is the conviction of the solvability of ever mathematical problem for the researcher?
a) powerful stimulus and impetus.
b) problem.
c) difficulty.
d) stimulus.
5. What happens as soon as one problem is solved?
a) others come forth in its place.
b) all other problems are also solved.
c) we can solve the next problems.
d) all other problems disappear.
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
1. The hypotenuse of an isosceles right triangle
2. The problem of perpetual motion
3. The number of problems in mathematics is inexhaustible
4. Unsolved problems
Text I
Great minds of Greece such as Thales, Pythagoras, Euclid, Archimede, Appolonius, Eudoxus, etc. produced an amazing amount of first class mathematics. The fame of these mathematicians spread to all corners of the Mediterranean world and attracted numerous pupils. Masters and pupils gathered in schools which though theyhad few, buildings and no campus were truly centres of learning. The teaching of these schools dominated the entire life of the Greeks.
Despite the unquestioned influence of Egypt and BabyloniaonGreek mathematicians, the mathematics produced by the Greeks differed fundamentally from that which preceded it. It were the Greeks who founded mathematics as a scientific discipline. The Pythagoreanschool was the most influential in determining both the nature and content Greek mathematics. Its leader Pythagoras foundedacommunity which embraced both mystical and rational doctrines.
The original Pythagorean brotherhood (c. 550—300 B. C.) was a secret aristocratic society whose members preferred to operate from behind the scenes and, from there, to rule social and intellectual affairs with an iron hand. Their noble born initiates were taught entirely by word of mouth. Written documentation was not permitted, since anything written might give away the secrets largely responsible for their power. Among these early Pythagoreans were men who knew more about mathematics then available than most other people of their time. They recognized that vastly superior in design and manageability Babylonian base-ten positional numeration system might make computational skills available to people in all walks of life and rapidly democratize mathematics and diminish their power over the masses. They used their own non possitional numeration system (standard Greek alphabet supplemented by special symbols). Although there was no difficulty in determining when the symbols represented a number instead of a word, for computation the people of the lower classes had to consult an exclusive group of experts or to use complicated tables and both of these sources of help were controlled by the brotherhood.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Who produced an amazing amount of first-class mathematicians?
a) Great minds of Greece.
b) Great minds of Mediterranian world.
c) Great minds of Babylonia.
2. Who influenced Greek mathematics?
a) Egypt and Fromce b) Egypt
c) Egypt and Babylonia d) Babylonia
3. What did the Pythagoreans represent instead of a word?
a) number b) hieroglyph
c) symbol d) figure
4. What did the people of the lower classes use for their communication?
a) coordinate system b) complicated tables
c) book d)"Pythagorean" theorem
5. What did Pythagoras community embrace?
a) property b) numeration system
c) mystical and rational doctrines d) real number
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Great minds of Greece
b. Numeration system
c. The Pythagorean school
Text II
For Pythagoras and his followers the fundamental studies were geometry, arithmetic, music, and astronomy. The basic element of all these studies was number not in its practical computational aspect, but as the very essence of their being; they meant that the nature of numbers should be conceived with the mind only. In spite of themystical nature of much of the Pythagorean study the members of community contributed during the two hundred or so years following the founding of their organization, a good deal of sound mathematics. Thus, in geometry they developed the properties of parallel lines and used them to prove that “the sum of the angles of any triangle is equal to two right angles”. They contributed in a noteworthy manner to Greek geometrical algebra, and they developed a fairly complete theory of proportional though it was limited to commensurable magnitudes, and used it to deduce properties of similar figures. They were aware of the existence of at least three of the regular polyhedral solids, and they discovered the incommensurability of a side and a diagonal of a square.
Details concerning the discovery of the existence of incommensurable quantities is lacking, but it is apparent that the Pythagoreans found it as difficult to accept incommensurable quantities as to discover them. Two segments are commensurable if there is a segment that “measures” each of them – that is, it contains exactly a whole number of times in each of the segments.
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1. What study was the fundamental for Pythagoras?
a) mathematics b) philosophy
c) arithmetic, geometry, music and astronomy
d) algebra
2. What was the basic element of all studies?
a) word b) number
c) symbol d) point
3. What property did the Pythagoreans develop in geometry of?
a) angles b) diagonal
c) parallel lines d) sides
4. What did the Pythagoreans discover?
a) a diagonal of a square b) a right angle
c) a computational aspect d) a parallel line
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Non-positional numeration system
b. Number as the basic element of all studies
c. Incommensurable quantities
d. The properties of geometry
Text III
The fact that there revealed pairs of segments for which such a measure does not exist provides the incommensurable case. It is possible that the first pair of segments found to be incommensurable is the side and diagonal of a regular pentagon, the favourite-figure of the Pythagoreans because its diagonals form the star pentagon, the distinctive, mark of their society. This same geometric procedure can also be adapted to the side and diagonal of a square. Here there exists an association with the so-called Pythagoreans side and diagonal numbers. The ratio of associated pairs of these numbers gives successively closer and closer rational approximations to √2; in fact, they are the approximations obtained by computing successive convergents of the continued fraction form of √2. This is reflected in modern mathematics in the concept of irrational number, a number that cannot be expressed as the ratio of two integers, e. g., п, e, 1/2. This devastating discovery was ascribed to Pythagoras himself, but more probably it was made by some Pythagorean. Since the philosophy of the Pythagorean school was that whole numbers or whole numbers in ratio are the essence of all existing things, the followers of that school regarded the emergence of irrationals as a "logical scandal". As the revelation of geometrical magnitudes whose ratio cannot be represented by pairs of integers led to the "crisis" in the foundations of their mathematics, the Pythagoreans tryed to conceal their greatest discovery. A Pythagorean Hippasus (c. 400 B. C.) who first brought out the irrationals from concealment into the open supposedly perished in a shipwreck at sea. But great discoveries could not be suppressed. The discovery of incommensurables was a turning-point, a landmark in the history of mathematics, and its significance can hardly be over appreciated. It resulted in a need to establish a new theory of proportions independent of commensubarility. This was accomplished byEudoxus (c. 370 B. C.). The details of the gradual transition from a theory of proportions which includedincommensurable quantities to a clear realization of .the concept of an irrational number covered a wide range of sophisticated mathematical topics and this concept was fully clarified only in the nineteenth century by R. Dedekind and G. Cantor. In mathematics of today the irrationals form an important subset of real numbers the basis of both algebra and analysis.
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1. What was the distinctive mark of the Pythagoreans society?
a) segment b) side and diagonal
c) polyhedral d) star pentagon
2. What gives the ratio of pairs of numbers?
a) real number b) approximation to 0
c) rational approximation to y2 d) irrational number
3. Why cannot a number be expressed as the ratio of two integers? Because it was … .
a) irrational number b) rational number
c) real number d) fractional number
4. Who first brought out the irrationals from concealment?
a) Cantor b) Eudoxus
c) Hippasus d) Descart
5. What new theory did Eudoxus establish?
a) a theory of numbers b) a theory of proportions
c) a theory of equations d) the“Pythagorean” theory
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The concept of irrational number
b. Geometrical magnitudes
c. The discovery of incommensurables
Text IV
The "Pythagorean" theorem isone of the most important propositions in the entire realm of geometry. There is no doubt, however, that the "Pythagorean property": c2 =a2 + b2 was known priortothe time of Pythagoras; there existed clay tablet texts which contained columns of figures related to Pythagorean triples. The frequent textbook reference to Egyptian "rope-stretchers" and their knotted surveying ropes as proof that these ancients knew the theorem was erroneous. While it isknown that the Egyptians realized as early as 2000 B. C. that 42+ 32=52, there is no evidence that the Egyptians knew or were able to prove the right angle property of the figure involved. Pythagoras is credited with the proof of this property which is true for all right trianges, and for all natural numbers. Although much of this information was known to the ancients of earlier times, the deductive aspect of geometry was exploited and advanced considerably in the work of the Pythagoreans.
The mysticism of this celebrated school aroused the suspicion and dislike of the people who finally drove the Pythagoreans out Crotona. A Greek seaport in Southern Italy and burnt their buildings. Pythagoras was murdered but his followers were scattened to other Greek centres and continued his teachings. The Pythagoreans were credited with giving the subject of mathematics special and independent status. They were the first group to treat mathematical concepts as abstractions and they distinguished mathematical theory from practices or calculations. They proved the fundamental theorems of plane and solid geometry and of “arithmetica”- the theory of numbers.
More widely known than the Pythagoreans was the Academy of Plato which had Aristotle as its most distinguished students. The latter then founded his own school at the time of Plato’s death pupils were the most famous philosophers, mathematicians and astronomers of their age. Under Plato’s influence they emphasized pure mathematics to the extent of ignoring all practical applications and they added immensely to the range of mathematics.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What is one of the most important proportions in the entire realm of geometry?
a) theory of proportions b) theory of numbers
c) the “Pythagorean” theorem d) numeration system
2. What was the “Pythagorean property?”
a) rational approximation b) ratio of two integers
c) sum of the angles is equal to two right angles d) c2 =a2 + b2
3. What was the “Pythagorean property” true for?
a) diagonals b) right angles
c) right triangle and all natural numbers d) irrational numbers
4. What was considerably advanced in the work of the Pythagoreans?
a) written documentation b) numeration system
c) incommensurable quantities d) deductive aspect of
geometry
5. What did they distinguish mathematical theory from?
a) deductive aspect b) theory of number
c) practice or calculation d)geometrical
magnitudes
6. Who was the most distinguished student in the Academy of Plato?
a) Plato b) Aristotle
c) Pythagoras d) Cantor
V. Выберите заголовок для данного текста в соответствии с его содержанием.
a. The “Pythagorean property”
b. The Pythagorean theorem
c. The academy of Plato
d. Aristotle – the founder of his own school
Text I
A correspondence is similarly established between the algebraic and analytic properties of the equation f (x, y) = 0, and the geometric properties of the associated curve. The task of proving a theorem in geometry will cleverly be shifted to that of proving a corresponding theorem in algebra and analysis.
There is no unanimity of opinion among historians of mathematics concerning who invented Analytic Geometry, nor even concerning what age should be credited with the invention. Much of this difference of opinion is caused by a lack of agreement regarding just what constitutes Analytic Geometry. There are those who, favouring Antiquity as the era of the invention, point out the well-known fact that the concept of fixing the position of a point by means of suitable coordinates was employed in the ancient world by the Egyptians and the Romans in surveying, and by the Greeks in map-making. And, if Analytic Geometry implies not only the use of coordinates but also the geometric interpretation of relations among coordinates then a particularly strong argument in favour of crediting the Greeks is the fact that Appolonius (c. 225 B. C.) derived the bulk of his geometry of the conic sections from the geometrical equaivalents of certain Cartesian equations of these curves, the idea which originated with Menaechmus about 350 B. C.
Others claim that the invention of Analytic Geometry should be credited to Nicole Oresme, who was born in Normandy about 1323 and died in 1382 after a career that carried him from a mathematics professorship to a bishopric. N. Oresme in one of his mathematical tracts, anticipated another aspect of Analytic Geometry, when he represented certain laws by graphing the dependent variable against the independent one, as the latter variable was permitted to take on small increments. Advocates for N. Oresme as the inventor of Analytic Geometry see in his work such accomplishments as the first, explicit introduction of the equation of a straight line and the extension of some of the notions of the subject from two-dimensional space to three, and even four-dimensional spaces. A century after N. Oresme's tract was written, it enjoyed several printings and in this way it may possibly exert some influence on the succeeding mathematicians.
However, before Analytic geometry could assume its present highly practical form, it had to wait the development of algebraic symbolism, and accordingly it may be more correct to agree with the majority of historians, who regard the decisive contributions made in the seventeenth century by the two French mathematicians, R. Descartes (1596-1650) and P. Fermat (1601-1663), as the essential origin of at least the modern spirit of the subject. After the great impetus given to the subject by these two men, we find Analytic Geometry in a form with which we are familiar today. In the history of mathematics a good deal of space will be devoted to R. Descartes and P. Fermat, for these men left very deep imprints on many subjects. Also, in the history of mathematics, much will be said about the importance of Analytic geometry, not only for the development of Geometry and for the theory of curves and surfaces in particular, but as an indespensable force in the development of the calculus, as the influential power in molding our ideas of such farreaching concepts as those of "function" and "dimension".
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Why was not there unanimity of opinion among scientists about the invention of analytic geometry? Because of … .
a) the position of a point b) a lack of agreement
c) suitable coordinates d) a map-making of the Greeks
2. Who derived the bulk of geometry?
a) Appolonius b) Fermat
c) Menaechmus d) Descartes
3. Who left deep imprints on many subjects?
a) R. Descartes and P. Fermat b) Appolonius
c) Menaechmus c) N. Oresme
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Who invented analytic geometry?
b. Analytic geometry
c. Algebraic symbolism
d. R. Descartes and P. Ferma
Text II
Applied mathematics in the modern sense of the term was not the creation of the engineer or the engineering-minded mathematician. The two great thinkers who founded this subject one was a profound philosopher, the other was a scientist in the realm of ideas. The former Rene Descartes devoted himself to critical and profound thinking about the nature of truth, and the physical structure of the universe. The latter Pierre Fermat, lived an ordinary life as a lawyer and civil servant, but in his spare time he was busy creating and offering to the world his famous theorems. The work of both men in many fields will be immortal. R. Descartes proposed to generalize and extend the methods used by mathematicians in order to make them applicable to all investigations. In essence, the method will be an axiomatic deductive construction for all thoughts. The conclusions will be theorems derived from axioms. Guided by the methods of the geometers Descartes carefully formulated the rules that would direct him in his search for truth. His story of the search for method and the application of the method to problems of philosophy was presented in his famous "Discourse on Method". The method Descartes abstracted from mathematics and generalized he then reapplied to mathematics; with it he succeeded in creating a new of representing and analyzing curves. This creation, now know as coordinate geometry, is the basis of all modern applied mathematics.
P. Fermat, despite the brief amount of time he was able to spend on mathematics and the pleasure-seeking attitude with which he approached it, established himself as one of the truly great mathematicians of all times. His contributions to the calculus were first rate though some what overshadowed by those of Newton and Leibnits. He shared with Pascal the honour of creating the mathematical theory of probability and shared with Descartes the creation of coordinate geometry, and founded the theory of numbers. In all these fields this "amteur" produced brilliant results. Though not concerned with a universal method in philosophy, Fermat did seek a general method of working with curves and here his thoughts joined company with those of Descartes’s.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What creations did Fermat share with Pascal?
a) coordinate geometry b) theory of numbers
c) mathematical theory of probability d) calculus system
2. What method in philosophy did Fermat seek?
a) method of approximate calculation.
b) general method of working with curves.
c) simplex method.
d) Euclid’s synthetic method.
3. What was R. Descartes?
a) а profound philosopher.
b) аn engineer.
c) a astronomer.
d) a physicist in the field of mechanics.
4. What is the basis of all modern applied mathematics?
a) methods of geometry b) the coordinate geometry
c) applied mathematics d) famous theorems
5. What method of Descartes is for all thoughts?
a) axiomatic deductive construction b) rule
c) code d) conclusion
V. Выберите заголовок для текста, в соответствии с его содержанием.
a. R. Descartes and P. Fermat as great founders of coordinate geometry
b. The famous work of R. Descartes “Discourse of method”
c. Immortal works of R. Descartes and P. Fermat
Text III
One must understand why it was that the great mathematicians of the time were so much concerned with the study of curves. In the early part of the seventeenth century mathematics was still essentially a body of geometry and the heart of this body was Euclid's contribution. Euclidean geometry confines itself to figures formed by straight lines and circles, but in the seventeenth century the advances of science and technology produced a need to work with many new configurations. Ellipses, parabolas and hyperbolas became important because they described the paths of the planets and projectiles such as cannon balls.
Both Descartes and Fermat recognized that geometry supplied information and truth about the real world. They also appreciated the fact that algebra could be employed to reason about abstract and unknown quantities; and it could be used to mechanize the reasoning process and minimize the effort needed to solve problems. Therefore they proposed to borrow all that was best in geometry and algebra and correct the defects of one with the help of the other. In Descartes's general study of method he decided to solve all problems by proceeding from the simple to the complex. Now, the simplest figure in geometry is the straight line. He therefore sought to approach the study of curves through straight lines and he found the way to do this.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What is the simplest figure in geometry?
a) circle b) surface
c) straight line d) triangle
2. How did Descartes decide to solve all problems in the general study of method?
a) by proceeding from the simple to the complex.
b) by proceeding from the complex to the simplex.
c) by studying the planes.
d) by studying the curves.
3. What was mathematics in the early part of the seventeenth century?
a) a body of geometry b) a body of curves
c) a body of information d) a body of lines
4. What does geometry confine itself to figures formed by straight lines and circles?
a) pure geometry b) analytic geometry
c) Euclidean geometry d)Pythagorean geometry
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Euclidean geometry as the basis of mathematics
b. Descartes’s general method for solving problems
c. A need to work with a curve
Text IV
To discuss the equation of a curve Descartes introduced a horizontal line called the X-axis, a point O on the line called the origin, and a vertical line through O called Y-axis. If P is any point on a curve, there are two numbers that describe its position. The first is the distance from O to the foot of the perpendicuar, from P to the X-axis. This number, called X-value, is the abscissa of P. The second number is the distance from P to the Y-axis, called Y-value or ordinate of P. These two numbers are called the coordinates of P and are generally written as P (x, y). The curve itself is then described algebraically by stating some equation which holds for x and y values of points on that curve and only for those points.
The heart of Descartes's and Fermat's idea is the following. To each curve there belongs an equation that uniquely describes the points of that curve and no other points. Conversely, each equation involving x and y can be pictured as a curve by interpreting x and y as coordinates of points.
Thus formally stated: the equation of any curve is an algebraic equality which is satisfied by the coordinates of all points on the curve but not the coordinates of any other point.
Since each of these pairs of coordinates represents a point on the curve we can plot these points and join them by a smooth curve. The more coordinates we calculate, the more points can, be plotted and the more accurately the curve can be drawn.
Beyond the analysis of properties of individual curves, the association of equation and curve makes possible a host of scientific applications of mathematics. Among the practical applications of mathematics we shall merely mention that all the conic sections possess the properties that make these curves effectively employed in lenses, telescopes, microscopes, X-ray machines, radio antennas, searchlights and hundreds of other major devices. When Kepler introduced the conic sections in astronomy they became basic in all astronomical calculations including those of eclipses and paths of comets.
To summarize, it was not so much the