english for mathematicians

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.

А В1. fraction

2. whole numbers

3. irrational numbers

4. differential equations

5. concept

6. point

7. line

8. triangle

9. equality

10. axiom

a) аксиома

b) иррациональные числа

c) дробь

d) целые числа

e) концепция

f) точка

g) дифференциальные уравнения

h) равенство

i) линия

j) треугольник

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 mathe-matics 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

VI. Укажите правильный перевод подчеркнутой части предложения.

1. The incidental members were called “auditors”.

a) называют b) назывались

c) названный

2. This part of mathematics is termed the mathematics of number.

a) будет определена b) была определена

c) определяется

3. From the concepts and axioms theorems are deduced.

a) выводят b) выводятся

c) были выведены

4. Mathematics as a science, viewed as a whole, is a collection of branches.

a) рассматривается b) рассмотрела

c) рассматриваемая

5. A second branch is geometry consisting of several geometries.

a) состоящая b) состояла

c) состоит

6. These concepts must verify explicitly stated axioms.

a) установив b) установленные

c) устанавливающие

UNIT II



A B1.abstraction

2. concept

3. irrational number

4. negative number

5. notion

6. addition

7. multiplication

8. variable

9. function

10. quantitative values

a. умножение

b. сложение

c. количественные величины

d. концепция

e. иррациональное число

f. отрицательное число

g. понятие

h абстракция

i. переменная

j. функция


1. The basic concepts of the main branches of mathematics are … from experience.

a) abstractions b) notions

c) concepts c) functions

2. Irrational numbers, ... , and so forth are not wholly abstracted from the physical experience.

a) functions b) negative numbers

c) quantitative values d) abstractions

3. The notion of ... 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.

a) a function b) an abstraction

c) a variable d) an addition

4. The concepts of a... , or relationship between variable, is almost totally a mental creation.

a) irrational number b) negative number

c) multiplication d) function

5. The gradual introduction of new ... in any field enables mathematics to grow rapidly.

a) notions b) concepts

c) functions d) quantitative values

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.


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.


a. The basic and new concepts c. Modern mathematicians

b. The basic concepts d. Irrational numbers


1. But it is noteworthy, that many more concepts arc introduced which are, in essence, creations of human mind.

a) представлены b) были представлены

c) представляли

2. Irrational numbers, negative numbers, and so forth are not wholly abstracted from the physical practice.

a) не отделяются b) не отделяли

c) не были отделены

3. Later concepts are built on earlier notions.

a) строили b) строятся

c) были построены

4. The basic concepts of the main branches of mathematics are abstractions from experience, implied by their obvious physical counterparts.

a) подразумеваемые b) подразумевают

c) подразумевали

5. 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.

a) можно будет применить b) могли применить

c) могут применяться

6. 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.

a) играла b) играемая

c) играющая

UNIT III



A B1.axiom

2. concept

3. theorem

4. deduce

5. calculus

6. function

7. differential equation

8. prove

9. geometry

10. pure mathematics

а) концепция

b) чистая математика

с) геометрия

d) аксиома

e)дифференциальное уравнение

f) доказывать

g) выводить

h) функция

i) вычисления

j) теорема


1. ... constitute the second major component of any branch of mathematics.

a) geometries b) theorems

c) differential equations d) axioms

2. The objective of mathematical activity consists of the ... deduced from a set of axioms.

a) functions b) theorems

c) concepts d) notions

3. The … of number give rise to the results of algebra.

a) axioms b) functions

c) calculus d) geometries

4. Mathematical ... must be deductively established and proved.

a) axioms b) concepts

c) theorems d) numbers

5. Some mathematicians claim that ... is the most original creation of human mind.

a) pure mathematics b) geometry

c) theorems d) concepts

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.


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


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


1. Up to the XIX -century axioms were considered as basic self-evident truth about the concepts involved.

a) считались b) считаются с) считают

2. Mathematical theorems must be deductively established and proved.

a) должны будут основываться и доказываться

b) должны быть основаны и доказаны

c) должны были быть основаны и доказаны

3. Much of the scientific knowledge is proved by deductive reasoning.

a) обоснованием b) были обоснованы

c) будут обоснованы

4. New theorems are proved constantly, even in such old subjects as algebra and geometry and the current developments are as important as the older results.

a) доказываются b) доказали

c) были доказаны

5. In ancient times the axioms and theorems were regarded as necessary.

a) рассматриваются b) рассматривались

c) были рассмотрены

6. The objective of mathematical activity consists of the theorems deduced from some sets of axioms.

a) выведенные b) выводили

c) выводят

UNSOLVED MATHEMATICAL PROBLEMS UNIT I



А Б1. solution

2. replace

3. significance

4. methods

a) заменить

b) значение

c) решение

d) методы


1. Every age has its own problems that it solves or … by new ones.

a) replaces b) seeks

c) requires d) tests

2. It is by the … of problems that the researcher tests the temper of his steel.

a) question b) solution

c) investigation d) generation

3. The deep … of certain problems for the advance of mathematical science are not to be denied.

a) glance b) advance

c) significance d) importance

4. What new … in the wide and rich field of math – cal thought can the new centuries disclose?

a) methods b) facts

c) solutions d) ideas

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 of science. 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.


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.


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


1. Mathematical minds of coming generations will solve they problems.

a) наступать b) наступающих

c) будет наступать

2. The following conclusion either solves mathematical problems or casts aside by new ones.

a) следующий b) следовал

c) будет следовать

3. Mathematics is the science dealing with many subjects.

a) имеющая дело b) иметь дело

c) будет иметь дело

4. Mathematicians working in any given branch discover new concepts.

a) работающие b) работать

c) работая

UNIT II



A B1. descent a). решение

2. solution

3. evidence

4. theorem

5. axiom

6. assertion

7. calculation

b). понижение

c). теорема

d). аксиома

e). доказательство

f). вычисление

g). утверждение


1. The mathematicians of past centuries knew the … of difficult problems.

a) value b) solution

c) importance d) idea

2. “Problem of quickest …” was proposed by J. Bernoulli.

a) solution b) descent

c) integers d) evidence

3. We study Fermat’s … xn + yn = zn (x, y, z integers).

a) theorem b) axiom

c) assertion d) descent

4. They attempt to prove the impossibility of Fermat’s … .

a) problem b) theory

c) theorem d) example

5. The … of variations owes its origin to the problem of Bernoulli and to similar problems.

a) calculus b) axiom

c) theorem d) assertion

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.


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.


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


1. The students know the clearness and ease of understanding a mathematical theory.

a) понимания b) понимая

c) поняв

2. The mathematicians of past were accustomed to devoting themselves to the solution of mathematical problems.

a) посвящению b) посвящая

c) посвятив

3. “Problem of quickest descent”, was proposed by Bernoulli.

a) будет предложена b) была предложена

c) была бы предложена

4. The attempt to prove the impossibility of Fermat's theorem offers a striking example.

a) докажет b) доказав

c) доказать d) доказал бы

UNIT III



A

1. application

2. equation

3. curved lines

B

a) окружность

b) цельный

c) возводить в куб

4. surface

5. polyhedron

6. icosahedron

7. linear

8. differential

9. cube

10. circle

11. integral

d) линейный

f) многогранник

e) дифференциал

g) двадцатигранник

h) поверхность

i) кривая линия

j) уравнение

k) применение


1. The same special problem finds… in the most diverse and unrelated branches of mathematics.

a) duplication b) calculation

c) equations d) application

2. The students know the theory of … and the theory of equation.

a) curved lines surfaces b) polyhedra, curved lines

c) icosahedron, surfaces d) linear differential equations

3. The rules of … with natural numbers were discovered in this fashion.

a) cube b) squaring of the circle

c) calculation d) duplication

4. The oldest problems in the theory of curves and the differential and ... calculus belong to mechanics astronomy and physics.

a) integral b) linear

c) squaring d) cube

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.


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.


a. The sources of mathematical problems

d. The rules of calculation

c. The application of mathematical problems

d. Galois’ theory of equations

VI. Выберите правильный перевод подчеркнутой части предложения.

1. After referring to the general importance of problem, we were able to find its solution.

a) ссылаясь b) при ссылке на

c) сославшись

2. We study the problems properly belonging to mechanics.

a) принадлежащие b) принадлежали бы

c) принадлежали

3. Mathematical thought involves special kind of thinking.

a) мышление b) мыслит

c) будет мыслить

4. The human mind encouraged by the success of its solutions became convinced of its independence.

a) поддерживать b) поддержанный

c) поддерживал

UNIT IV


II. Выберите в колонке В эквиваленты к словам в колонке А.

A B1. division

2. curve

3. hypotheses

4. deduction

5. requirement

6. reason

a) вычитание

b) деление

c) гипотезы

d) требование

e) кривая

f) причина


1. New questions open up new . . . of mathematics.

a) solutions b) divisions

c) problems d) experiences

2. 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 … .

a) hypotheses b) solutions

c) deductions d) requirements

3. The demand for logical … by means of finite number of processes is simply the requirement of rigour in reasoning.

a) reason b) deduction

c) division d) solution

4. The theory of algebraic … experienced a considerable simplification.

a) curves b) deduction

c) methods d) solution

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 while we 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 proof is 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.


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.


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


1. Modern methods of carrying out mathematic operations become sophisticated through modern computers.

a) выполнения b) выполнить

c) выполняя

2. There is a universal philosophical necessity of our understanding.

a) понимания b) понимающий

c) понимавший

3. Only by satisfying this claim the problems attain their full effect.

a) удовлетворяя b) удовлетворив

c) удовлетворяющий

4. The rigorous methods are easier to understand.

a) понять b) понял бы

c) понятный

UNIT V



A

1. inequality

2. triangle

3. rectangle

4. axis

5. isosceles

6. figure

7. segment

B

a) неравенство

b) отрезок

c) треугольник

d) цифра, фигура

e) прямоугольник

f) равнобедренный

g) ось

II. Заполните пропуски подходящими по смыслу словами.

1.Here is the double … a>b>c.

a) equality b) inequality

c) triangle d)rectangle

2. Who can do without the figure of the … , the circle with its centre, or with the cross of three perpendicular axes?

a) triangle b) rectangle

c) axes d) circle

3. On the diagram there is the cross of three perpendicular … .

a) axes b) rectangles

c) triangles d) inequalities

4. Geometric … are graphic formulas and no mathematician can do without them.

a) figures b) segments

c) circles d) axes

5. Students draw segments and … closed in one another.

a) points b) rectangles

c) formulas d) symbols

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 and are 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 most cases 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.


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.


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


1. Draw three points following one another on a straight line.

a) следующие b) следование

c) следует

2. This way for finding general methods is certainly the most fruitful one.

a) нахождения b) находить

c) находящий

3. We seek for methods without having a definite problem.

a) не имея b) не иметь

c) не имевший

4. After finding this standpoint, the problem becomes more accessible to our investigations.

a) нашел b) находивший

c) находя

UNIT VI



A B

1. hypotenuse

2. parallelepiped

3. to square the circle

4. perpetual

a) бесконечный

b) гипотенуза

c) параллелепипед

d) искать квадратуру круга


1. The ratio of the … to the side of an isosceles right triangle is irrational.

a) hypotheses b) hypotenuse

c) parallel d) ratio

2. Our task is to … .

a) to square the circle b) seek

c) investigate d) show

3. The proof of the axiom of … is very important.

a) hypotenuses b) triangles

c) rectangles d) parallelepiped

4. The problem of “… motion” is the most important one to science.

a) inexhaustible b) hypotenuse

c) perpetual d) triangle

Text VI

Occasionally it happens that we seek the solution under insufficient hypotheses 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 the impossibility of the perpetual motion in the sense originally presupposed.

This conviction of the solvability of every mathematical problem is a 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.


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.


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


1. As soon as one problem is solved the other comes on it’s place.

a) решена b) решили

c) будет решена d) решили бы

2. There are linear differential equations, having a prescribed monogramic group.

a) имеющие b) имевшие

c) иметь d) имели

3. The question is raised whether mathematics can like other scinces.

a) поднять b) подняв

c) будет поднят d) поднят

4. The efforts to construct a perpetual motion machine were not futile.

a) создав b) создающие

c) создал d) создавать

GREEK SCHOOLS OF MATHEMATICS UNIT I



А В1. diminish

2. number

3. numeration system

4. symbol

5. content

6. property

7. doctrine

a) содержание

b) учение

c) уменьшать

d) свойство

e) число

f) числовая система

g) символ


1. Pythagoras founded a community which embraced both mystical and rational.

a) members b) axioms

c) doctrines d) theorems

2. They used their own non-positional … system.

a) coordinate b) arithmetic

c) numeration d) geometric

3. Standard Greek alphabet was supplemented by special … .

a) quantities b) theorems

c) symbols d) words

4. The Pythagorean school was the most influential in determining both the nature and … of Greak mathematics.

a) content b) system

c) power d) diminish

5. The symbols represented a … instead of a word.

a) property b) number

c) axiom d) theorem

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 they had 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 Babylonia on Greek 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 Pythagorean school was the most influential in determining both the nature and content Greek mathematics. Its leader Pythagoras founded a community 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.


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


a. Great minds of Greece

b. Numeration system

c. The Pythagorean school


1.They were taught entirely by word of mouth.

a) обучаются b) обучили бы

c) обучались

2. The sources of help were controlled by the brotherhood.

a) контролируются b) контролировали бы

c) контролировались

3. The symbols represented a number instead of a word.

а) представляли b) представили бы

с) были представлены

4. Written documentation was not permitted.

a) не разрешала бы b) не была разрешена

c) не будет разрешена

UNIT II



А В1. basic element

2. properties of similar figures

3. properties of parallel lines

4. the sum of the angles of a

triangle

5. practical computational aspect

6. theory of proportions

7. incommensurable quantities

8. polyhedra

9. geometrical magnitude

10. real number

а) свойства подобных фигур

b) сумма углов треугольника

c) базисный элемент

d) теория пропорции

e) свойства параллельных линий

f) практический и расчетный

аспект

g) геометрическая величина

h) действительное число

i) несоизмеримые количества

j) многогранники


1. In geometry they (the Pythagoreans) developed the properties of … .

a) number b) parallel line

c) similar figures d) equation

2. The Pythagoreans used to prove that “the sum of the angles of any triangle is equal to … .

a) 180 degrees b) right angles

c) two right angles d) three right angles

3. The Pythagoreans were aware of the existence of at least three of the regular … .

a) triangles b) squares

c) polyhedral solids d) circles

4. The Pythagoreans discovered the … of a side and a diagonal of a square.

a) angle of a square b) sides of a triangle

c) angle of a triangle d) incommensurability

5. The Pythagoreans developed a fairly complete theory of … .

a) fractions b) proportions

c)real numbers d) numbers

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 the mystical 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.


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


a. Non-positional numeration system

b. Number as the basic element of all studies

c. Incommensurable quantities

d. The properties of geometry


1. In geometry were developed the properties of parallel lines.

a) были развиты b) развили бы

c) будут развиваться d) развиватья

2. The theory was limited to commensurable magnitudes.

a) была ограничена b) ограничила

c) ограничивала d) ограничить

3. To discover the incommensurability of a side and a diagonal of a square is our UNIT.

a) были открыты b) открылись

c) открыли d) открыть

4. To contribute in a noteworthy manner to Greek geometrical algebra was very important.

a) внесли вклад b) вносили бы вклад

c) был внесен вклад d) вносить

UNIT III



А

1. rational approximation

2. fraction

3. rational number

В

a)отношениедвух переменных b) действительное число

c) геометрическая величина

4. ratio of two integers

5. real number

6. geometrical magnitude

7. regular pentagon

8. numeration system

9. incommensurable quantities

10. diagonal of a square

d) рациональное число

e) правильный пятиугольник

f) дробь

g) рациональное приближение

h) диагональ квадрата

i) числовая система

j) несоизмеримые количества


1. The same geometric procedure can be adapted to the side and … .

a) diagonal of a square b) diagonal numbers

c) regular pentagon d) star pentagon

2. A number can’t be expressed as the … .

a) rational approximation b) irrational approximation

c) ratio of associated pairs d) ratio of two integers

3. The first pair of segments to be incommensurable is the side and diagonal of a … .

a) square b) right triangle

c) number d) regular pentagon

4. The ration of associated pairs of the numbers give closer and closer … .

a) rational number b) fraction

c) rational approximation d) irrational number

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 indepen-dent of commensubarility. This was accomplished by Eudoxus (c. 370 B. C.). The details of the gradual transition from a theory of proportions which included incommensurable 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.


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


a. The concept of irrational number

b. Geometrical magnitudes

c. The discovery of incommensurables


1. The ratio of associated pairs of side and diagonal numbers give rational approximations to y2.

a) соединять b) соединенные

c) соединение d) соединены

2. The discovery of incommensurables and its significance can hardly be overappreciated.

a) могут быть переоценены b) могли бы переоценить

c) смогут быть переоценены d) могли бы быть переоценены

3. The discovery was made by some Pythagoreans.

a) сделало b) будет сделано

c) сделало бы d) было сделано

4. This number can’t be expressed as the ratio of two integers.

a) не может быть выражена b) не могло быть выражено

c) не сможет быть выражена d)не смогло бы быть выражено

UNIT IV



А В1. property

2. natural number

3. proof

4. concept

5. right triangle

6. abstraction

7. solid geometry

8. deductive aspect of geometry

9. rational doctrines

10. pure mathematics

a) концепция

b) правильный треугольник

d) свойство

c) натуральное число

e) пространственная геометрия

f) доказательство

g) дедуктивный аспект геометрии

h) чистая математика

i) абстракция

j) рациональные доктрины


1. Under Plato’s influence they emphasized … to the extent of ignoring all practical applications.

a) geometry b) axioms

c) pure mathematics d) new theory

2. They proved the fundamental theorems of a plane and … .

a) curve b) plane geometry

c) symbol d) solid geometry

3. The Egyptians were able to prove the … property.

a) right angle b) theorem

c) information d) number

4. The ancient’s knew the … of some theorems.

a) subjects b) proofs

c) groups d) columns

Text IV

The "Pythagorean" theorem is one 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 prior to the 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.


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


1. Much of this information was known to the ancients of earlier times.

a) известна b) была известна

c) была бы известна d) будет известна

2. The deductive aspect of geometry was exploited in the works of the Pythagoreans.

a) разработал b) разработался бы

c) был разработан d) будет разработан

3. They proved that fundamental theorems of plane and solid geometry were known to them.

a) доказали b) были доказаны

c) доказывались d) доказанные

4. The mysticism of the celebrated school aroused suspicion of the people.

a) вызывает b) вызвало

c) вызовет d) вызвало бы

DESCARTES'S AND P. FERMAT'S COORDINATE GEOMETRY UNIT I



A B1.plane

2.curve

3. equation

4. independent variable

5. real number

6. property

7. relation

8. conic section

9. n-dimensional space

10. dependent variable

11.function

12.surface

13.calculus

14.dimension

15.bulk

a) действительное число

b) отношение

c) плоскость

d) н-мерное пространство

e) исчисление

f) уравнение

g) измерение

h)функция

i) коническое сечение

j) зависимая переменная

k) кривая

l) поверхность

m) свойство

n) объем

o) независимая переменная


1. Appolonius derived the bulk of his geometry of the … from the geometrical equivalents.

a) conic section b) polyhedra

c) sequence d) surface

2. Oresme represented certain laws by graphing the … against the independent one.

a) mental notion b) dependent variable

c) set of theorems d) relation

3. A correspondence between curve in the plane and the … in two variables is used in analytic geometry.

a) equation b) inequality

c) equality d) sequence

4. In the history of mathematics much will be said about of … .

a) fact and concept b) property and relation

c) function and dimension d) surface and plane

5. In Oresme’s work there are the notions of the subject from … to three.

a) one- dimensional space b) five- dimensional spaces

c) six – dimensional spaces d) two dimensional spaces

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".


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


a. Who invented analytic geometry?

b. Analytic geometry

c. Algebraic symbolism

d. R. Descartes and P. Ferma


1. Every student of mathematics meets the remarkable subject called Analytic Geomenry.

a) называемый b) назывался

c) называется d) будет называться

2. This is the task of proving a theorem in geometry.

a) доказывать b) доказательства

c) доказывающая d) докажут

3. There is no unanimity of opinion among historians of mathematics concerning who invented analytic geometry.

a) интересующихся b) интересуются

c) для интереса d) интерес

4. In the history of mathematics much will be said about far-reaching concepts as of “function’ and “dimension”.

a) сказать b) сказали

c) скажут d) сказали бы

UNIT II



A

1.applied mathematics

2.axiomatic deductive

construction

3.conclusion

4.coordinate geometry

5.mathematical theory of

probability

6. theory of number

B

a)аксиоматическая дедуктивная конструкция

b) вывод

c) прикладная математика

d) теория чисел

e) геометрия координат

f) математическая теория

вероятности


1. The conclusions of theorems will be derived from … .

a) axioms b) definitions

c) laws d) proofs

2. P. Fermat and R. Descartes founded the … .

a) general method b) applied mathematics

c) theory of numbers d) coordinate geometry

3. The method of Descartes will be … for all thoughts.

a) conclusion b) rule

c) axiomatic deductive construction d) code

4. P. Fermat shared with Pascal the honour of creating the … .

a) applied mathematics b) pure mathematics

c) solid mathematics d) mathematical theory of

probability

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.


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


1. R. Descartes devoted himself to critical and profound thinking about the nature of truth.

a) мыслив b) мышление

c) мыслящий d) мыслить

2. Fermat sought a general method of working with curves.

a) работа b) работающий

c) работая d) работал

3. Descartes succeeded in creating a new way of representing and analyzing curve.

a) создание b) создает

c) создал d) создаст

4. Fermat did seek a general method of working with curves.

a) работал b) будет работать

c) работа d) работать

5. P. Fermat was busy creating and offering to the world his famous theorems.

a) представил b) представлением

c) представит d) представлять

UNIT III



A B1.figures

2.straight line

3.circle

a) эллипс

b) круг, окружность

c) количество

4.ellipse

5. quantity

d) прямая линия

e) фигура, цифра, диаграмма


1. The simplest figure in geometry is a … .

a) rectangle b) straight line

c) cube d) segment

2. Euclidean geometry confines itself to figures formed by straight lines. and … .

a) forms b) figures

c) circles d) curves

3. The scientists noticed that algebra could be employed to reason about abstract and unknown … .

a) quantites b) curves

c) lines d) truths

4. …. became important in the seventeenth century because they described the paths of the planets and projectiles such as cannon balls.

a) ellipses, parabolas, and hyperbolas b) surfaces

c) conic sections d) straight lines

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.


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


a. Euclidean geometry as the basis of mathematics

b. Descartes’s general method for solving problems

c. A need to work with a curve


1. Euclidean geometry confines itself to figures formed by straight lines and circle.

a) образованные b) образуют

c) образование d) образовать

2. They proposed to borrow all that was the help of the other.

a) заимствовали b) заимствуют

c) заимствовать d) заимствовали бы

3. Descartes decided to solve all the necessary problems.

a)решая b) решить

c) будет решать d) решил

4. Algebra could be employed to reason about abstract and unknown quantities.

a) могла быть использована b) может использовать

c) сможет использовать d) могла использовать

5. Proceeding from the simplex to the complex was Descartes’s principle.

a) переход b) переходящий

c) перешедший d) переходить

UNIT IV



A B1.ordinate

2. perpendicular

3. abscissa

4. smooth curve

5. summarize

6. calculate

a) непрерывная кривая

b) абсцисса

c) вычислять

d) ордината

e) суммировать

f) перпендикуляр


1. Each equation involving x and y can be pictured as a curve by interpreting x and y as … .

a) relation of points b) coordinates of points

c) mental notions d) symbols

2. Since each of these pairs of coordinates represents a point on the curve we can plot these points and join them by a … .

a) ordinary line b) smooth curve

c) vertical line d) perpendicular

3. The horizontal line is called the … .

a) X-axis b) Y-value

c) Y-axis d) X-value

4. The distance from P to the Y-axis, is called … .

a) origin b) ordinate of P

c) abscissa of P d) X-axis

5. The vertical line is called the … .

a) Y-value b) X-value

c) X-axis d) Y-axis

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 use of coordinates that made the work of Descartes and Fermat so important; coordinates were used effectively in antiquity, especially in the geometry of Appolonius, and again in the fourteenth century in a more primitive form in the latitude of forms of Oresme. Descartes saw as the objective of his work the cooperation of algebra and geometry to the end that mathematics might have the best aspects of both branches. In the end, however, it turned out that geometry lost popularity in the partnership. Pure geometry was so overshadowed that it made little progress during the next century and a half, during which time infinitesimal analysis went through a progress of arithmetization that amounted almost to a revolution.


1. Who introduced the conic sections in astronomy?

a) Kepler b) Descartes

c) Fermat d) Appolonius

2. What makes it possible the scientific applications of mathematics?

a) the association of equation and curve.

b) the combination of algebra and mathematical theory of probability.

c) the association of line and circle.

d) the combination of physics and mechanics.

3. How is the number of the distance from O to the foot of the perpendicular from P to the X-axis called?

a) ordinate of P b) Y-axis

c) Origin d) Abscissa of P (X- value)

4. What is the heart of Descartes’s and Fermat’s idea?

a) to each curve there belongs an equality that uniquely describes many points on the line.

b) to each curve there belongs an equality that uniquely describes the points on that smooth curve.

5. What shall we receive, if we plot points and join them?

a) smooth curve b) cube

c) rectangle d) triangle


a. Coordinate system of Descartes

b. Kepler’s astronomical calculations

c. Horizontal and vertical lines

d. Abscissa and ordinate


1. The curve itself is then described algebraically by stating some equation.

a) путем установления b) будет путем установления

c) была бы путем установления d) устанавливать

2. Each equation involving x and y can be pictured as a curve.

a) включавшие b) включающееся

c) включающее d) включать

3. Each equation involving x and y can be pictured as a curve by interpreting x and y as coordinates of points.

a) интерпретацией b) интерпретировалась

c) будет интерпретироваться d) интерпретировать

4. The conic sections in astronomy became basic in astronomical calculations including eclipses and paths of comets.

a) включавшие b) включали

c) включат d) включать

5. The conic sections possess the properties that make these curves employed in lenses, telescopes, microscopes.

a) применяемые b) применявшие

c) применявшиеся d) применять

ANALYSIS INCARNATE - LEONARD EULER UNIT I



А

1. research

2. range

3. probability

4. condition

5. measure

6. approximate

7. decimal

8. conclusion

9. entire

10. variation

В

a) изменение, переменный

b) приближенное значение

c) целый

d) измерение

e) ряд

f) вероятность

g) заключение

h) условие

i) десятичный

j) исследование


1. The lead in scientific … was taken by the various royal academies.

a) calculus b) method

c) research d) axiom

2. Euler became famous for the wide … subjects he covered.

a) range b) research

c) stage d) matter

3. He contributed new ideas to calculus geometry, algebra, number theory, calculus of variations … and topology.

a) conditions b) curvatures

c) approximates d) probability

4. Euler became famous for his great output of original mathematics and for the … of subjects he covered.

a) equation b) wide range

c) measure d) variation

Text I

Though P. Fermat and R. Descartes founded analytic geometry they did not advance the subject far enough and did not elaborate it purely analytically either. A century later L. Euler (1707-1783) a Swiss mathematician who lived the greater part of his life in Russia, engaged in scientific research, lecturing and textbook writing in St. Petersburg Academy, developed the subject matter of both Plane and Solid Analytic Geometry far beyond its elementary stages. Euler's mathematical career opened when Analytic Geometry (made public in 1637) was ninety years old, the calculus about fifty. In each of these fields a vast number of isolated problems were solved, but no systematic unification of the whole of the then mathematics, pure and applied, was made. In particular, the powerful analytic methods of Fermat, Descartes, Newton and Leibnitz were not exploited to the limit of what they were capable, especially in Calculus, Geometry and Mechanics, where Euler proved himself the master.

In the XVIII c. the Universities were not the principal centres of science in Europe. The lead in scientific research was taken by the various royal academies. In Euler's case St. Petersburg and Berlin furnished the sinews of mathematical creation. Both of these foci of creativity owed their inspiration to the restless ambition of Leibnitz. These academies were like some of these today: they were research organizations which paid their leading members to produce scientific research. Euler became famous for his great output of original mathematics and for the wide range of subjects he covered. He contributed new ideas to calculus, geometry, Algerba, Number Theory, Calculus of variations, probability and Topology. He also worked in many areas of applied mathematics, such as Acoustics, Optics, Meachanics, Astronomy, Ballistics, Navigation, Statistics and Finance. His industry was as remarkable as his genius. Euler was the most prolific mathematician in histoty; his scientific heritage is vast, a list of some 850, works of which 550 were published in the lifetime. Euler wrote his great memoirs quite easily and total blindness during the last seventeen years of his life did not regard his unparalleled productivity. He overcame the difficulty of blindness chiefly by means of his remarkable memory. Indeed, if anything, the loss of his eyesight sharpened Euler's perception of the inner world of his imagination.


1. Who founded analytic geometry?

a) Lagrange and Fermat b) P. Fermat and R. Descartes

c) Euler and Fermat d) Lagrange and Euler

2. Where did Euler live the greater part of his life?

a) in Russia b) Alexandria

c) Switzerland d) Greece

3. What did Euler become famous for … .

a) his great out put of original mathematics.

b) his great out put of analitic geometry.

c) his great out put of calculas.

d) his great out put of mechanics.

4. In what parts of mathematics did Euler especially prove himself as a master?

a) in differential equations.

b) in analytic geometry and caleulus.

c) in calculus, geometry and mechanics.

d) in arithmetics and physics.


a. New discovery in analytic geometry

b. Euler is innovator of mathematics

c. Euler’s genius and remarkablity of his industry

d. Euler is a swiss scientist


1. Euler engaged in scientific research lecturing many subjects.

a) читая лекция b) читающий лекции

c) читать лекции d) читал бы лекции

2. In each of the fields a vast number of isolated problems were solved.

a) решались b) решая

c) будут решаться d) решали бы

3. The powerful analytic methods were not exploited to the limit.

a) не будут разработаны b) не разрабатываются

c) не были разработаны d) не разрабатывались бы

4. These academies were research organizations which paid their leading members to produce scientific research..

a) чтобы проводить научно-исследовательскую работу.

b) проводя научно-исследовательскую работу.

c) проводящим научно-исследовательскую работу.

UNIT II



A

1. curvarture

2. general

3. equation

4. differential

B

a) целый

b) переменный

c) искривление

d) десятичный

5. variation

6. entire

7. decimal

8. probability

9. research

e) вероятность


g) дифференциальный

h) исследование

i) общий


1. Algebraic equation of Analytic geometry furnishes answers to as many … places as individual cases require.

a) decimal b) equation

c) theory d) integral

2. Algebraic … of Analytic Geometry is a much simpler tool.

a) doctrine b) equation

c) range d) curve

3. Differential geometry got its real start in Euler’s study of lines of … .

a) condition b) curvature

c) result d) analysis

4. Euler gave his … expressing a necessary condition for a minimizing curve.

a) differential equation b) analytic procedure

c) condition of the problem d) algebraic symbol

Text II

Euler first gave the examples of those long analytic procedures in which conditions of the problem are first expressed by algebraic symbols and then pure calculation resolves the difficulties. He skillfully applied his analytic method to geometry and мechanics. Where the synthetic methods of Euclidean geometry required elaborate and complicated constructions and furnished lenghts that could be measured only approximately, algebraic equation of Analytic geometry is a much simpler tool and furnishes answers to as many decimal places as individual cases require. Euler improved the basic concepts of mathematical analysis, promoted differential and integral calculus, fathered the theory of linear differential equations and devised methods for their approximate solution. His treatises "Introduction to the Analysis of Infinities", "Differential Calculus" and "Integral Calculus" which for the most part present Euler's own results served as an encyclopedia in mathematical analysis of the period. Euler's contemporaries called him "Analysis Incarnate". One of the most remarkable features of Euler's universal genius was its equal strenght in both of the main currents of mathematics, the continuous and the discrete. "Read Euler, he is teacher of us all . . .", Laplace so amptly assessed his worth. But Euler was far more than a textbook writer. He enriched mathematics with beautiful new results. Differential Geometry got its first real start in Euler's study of lines of curvature (1760) and the Calculus of Variations took an independent status, when Euler (1736) gave his diffrential equation expressing a necessary condition for a minimizing curve.


1. How did Euler express conditions of problems.

a) by algebraic symbols b) by calculus

c) by algebraic equations d) logically

2. What resolved the difficulties of long analytic procedures?

a) scientific knowledge b) pure calculation

c) approximate solution d) study of lines

3. How did Euler’s contemproraries call him?

a) “Founder of Mathematics” b)“Greator of Analytic

geometry”

c) “Analysis Incarnate” d) “Founder of pure calculus”

4. What was the first real start in Euler’s study of lines of curvature?

a) analytic geometry b) mathematics

c) integral calculus d) differential geometry


a. Euler is as the author of an encyclopedia of mathematical analysis of that period

b. Euler’s study of lines on curvature

c. Euler is “Analysis Incarnate”

d. New beautiful results of Euler


1. The synthetic methods of Euclidean geometry furnished lengths that could be measured only approximately.

a) могли бы быть измеренными b) может быть измеренным

c) можно будет измерить d) измерить

2. Euler gave his differential equation expressing a necessary condition.

a) выражать b) выразил

c) выражение d) выражающий

3. He enriched mathematics with beautiful new results.

a) обогащает b) обогатил

c) обогатить d) обогащая

4. The calculus of Variations took an independent status.

a) получает b) получит

c) почило d) получая

5. Differential Geometry got its first real start in Euler’s study of lines of curvature.

a) получает b) получила

c) будет получать d) получил

UNIT III



A

1.solution

2.curvarture

3.conclusion

4.limit

5.differential

6.condition

7.analysis

8.equation

9. rule

B

а) заключение, вывод

b) дифференциал

c) условие

d) анализ

e) правило


g) решение

h) искривление

i) ограничить

III. Заполните пропуски подходящими по смыслу словами

1. Curiously enough in arriving at his … theoretical, Euler sought to “rid” mathematical analysis.

a) method b) results

c) conclusions d) problems

2. The entire exposition is … to pure analysis.

a) limited b) replaced

c) needed d) dealt

3. Euler wrote about the … of the calculus.

a) methods b) solutions

c) rules d) problems

4. Euler sought to “rid” mathematical … of geometrical, mechanical and physical interpretation.

a) measure b) analysis

c) curvature d) probability

5. Euler used algorithmic devices for the … of problems.

a) solution b) measure

c) equate d) variation

Text III

Curiously enough, in arriving at his theoretical conclusions, by working on practical tasks in different fields, Euler sought to "rid" mathematical analysis of geometrical mechanical and physical interpretation and couch it in purely analytic form. Thus he wrote, here the entire exposition is limited to pure analysis and, hence, not a single drawing was needed to set out the rules of this calculus". In an effort to replace synthetic methods by analytic Euler was succeeded by Lagrange, who dealt not with special concrete cases and tasks, but sought for abstract generality. Nevertheless, Euler was excelled either in productivity or in the skillful and imaginative use of algorithmic devices for the solution of problems.

The contribution that this illustrious scientist made to mathematics is truly enormous. We have every right to entitle him the 18th century Mathematician Number One whose works left their imprint on almost all branches of mathematics. L. Euler was buried in 1783 near Lomonosov's grave in the old cemetery of the Alexandro-Nevsky Monastery in Leningrad's (then St. Petersburg) necropolis. Even when he was compelled to emigrate to Germany, Russia ever remained in Euler's heart and mind and he never ruptured ties with the St. Petersburg Academy. To this day the great mathematician's descendants live in the this country, whose people will always revere his name. In 1983 scientists around the world extensively commemorated the 275th birthday and death bicentennial of this great scientist.


1. How did Euler couch mathematical analysis of geometrical, mechanical and physical interpretation.

a) in purely analytic from b) in form of differential equations

c) in logical from d) in Physical from

2. How did Lagrange reach the replace of synthetic methods by analytic one?

a) he sought for special forms.

b) he dealt not with special concrete cases and facts.

c) he did not deal with special concrete cases and tasks but sought for abstract generality.

d) he founded special concrete cases and tasks.

3. Who was the Mathematician Number one of the eighteenth century.

a) Lagrange b) Descartes

c) Fermat d) Euler

4. Where was Euler compelled to emigrate to?

a) Russia b) Alexandria

c) Germany d) Switzerland


a. The enormous contribution of Euler in mathematics

b. Euler sought to couch mathematical analysis in purely analytic from

c. Euler was succeeded by Lagrange

d. Euler is the eighteenth – century Mathematician Number One


1. Euler sought to “rid” mathematical analysis of geometrical and physical interpretation.

a) пытается освободить b) пытался освободить

c) попытается освободить d) пытался бы

2. In an effort to replace synthetic methods by analytic Euler was succeded by Langrange.

a) заменить b) замена

c) заменили d) заменят

3. We have every right to entitle him the eighteenth century Mathematician Number One.

a) будем называть b) назвали

c) называть d) назовут

4. Here the entire exposition is limited to pure analysis.

a) ограничивало b) ограничено

c) ограничит d) ограничивающий

5. Even when he was compelled to emigrate to Germany, Russia ever remained in Euler’s heat.

a) эмигрировать b) эмигрировал бы

c) эмигрировал d) будет эмигрировать

PART II Linear programming

2.1 Linear programming is a mathematical programming technique most closely associated with operations research and management science. In business linear programming is used for finding the optimal uses of the firm’s limited resources. A linear programming problem is often referred to as an allocation problem because it deals with allocation of resources to alternative uses.

Linear programming involves the formulation and solution of a class of business problems by the optimization of a linear mathematical function subject to linear inequalities. Here the term linear has a specific meaning. It will be treated in detail in the forthcoming section, together with the properties of linear programming.

Several sample problems will be presented subsequently to facilitate the understanding of linear programming properties and formulation. The graphic method is treated in this chapter to provide a conceptual understanding of the optimization procedures. The algebraic method is not contained in this chapter and will not be treated separately, since the solution process by the simplex method, treated in detail.

2-2. Properties of linear programming

Let us consider the following problem to examine the properties of linear programming.

The Riggs Manufacturing Company specializes in the manufacture of two types of televisions sets, regular and color. Production takes place on two assembly lines. Regular televisions sets are assembled on Assembly Line I and color television sets are assembled on Assembly Line II. Because of the limitation of the assembly line capacities, the daily production is limited to no more than 80 regular televisions sets on Assembly Line I and 60 color televisions sets on Assembly Line II. The production of both types of television sets requires electronic components. The production of each of these television sets requires five units and six units of electronic components, respectively. The electronic components are supplied by another manufacturer, and the supply is limited to 600 units per day. The company has 160 employees; i.e. the labor supply amounts to 160 man-days. The production of one unit of regular television requires 1 man-day of labor, whereas 2 man-days of labor are required for a color television set. Each unit of these televisions is sold in the market at a profit of $50 and $80 respectively. How many units of regular and color television sets should the company produce in order to obtain a maximum profit?

In this problem, the major objective of the manufacturer is to maximize dollar profits. To achieve this objective, the manufacturer must determine the optimal number of regular and color television sets to be produced daily. The production of regular and color television sets is variable, depending upon the available resources of the company. In linear programming, the number of these television sets to be produced is called the decision variable (or choice variable). Let x1 and x2 be the number of regular and color televisions to be produced daily. Since a regular television set is sold at a profit of $50 and a color television set at a profit of $80, the daily profit is expressed in the form

F=50x1 + 80x2

The total profit is a linear function of two decision variables x1 and x2. This profit function is called the objective function. The term linear is used to describe a directly and precisely proportional relationship between variables; i.e. the exponents of all variables must be one. All units of regular and color television sets produced for sale are sold, and the unit profit contribution of each of the two products remains the same at the various sales levels.

All linear programming problems contain functional restrictions depicted by linear inequalities. Linearity in the restrictions implies that the amount of available resources to produce each product is uniform, regardless of the number of units produced within the relevant range. In the case of the television manufacturer, production is influenced by the available resources. The two assembly lines have limited capacity to produce regular and color television sets. Since no more than 80 regular television sets can be assembled on Assembly Line I and 60 color television sets on Assembly Line II per day, we have two functional restrictions on the assembly lines x1≤80, x2≤60.

These restrictions are expressed in terms of linear inequalities called side constraints.

There is another side constraint in the daily requirement of the electronic components, so that we have 5x1 + 6x2≤600

The number of available employees is limited to 160 man-days. If this were not the case, the manufacturer would no doubt attempt to produce up to full production capacities of the two assembly lines. Therefore, the limited labor supply also introduces a similar side constraint

x1 + 2x2≤160

Finally, the values of decision variables must be either positive or zero; i.e. they cannot be negative. If the manufacturer does not produce any regular television sets, the value of the decision variable x1 is zero. If he does produce them, the value of the decision variable x1 becomes positive. In other words, the solution to the linear programming problem cannot be negative. The decision variables then can be expressed in the form x1≥0 and x2≥0

This restriction is called the non-negativity condition.

The problem in linear programming is that of determining the values of the decision variables which maximize (or minimize) the value of the objective function, subject to the linear side constraints. Hence, linear programming problems always contain the three major element, objective function, non-negative decision variables, and side constraints.

2-3. Assumptions of linear programming

From the description of the properties of linear programming, it is evident that a linear programming problem is based on a number of assumptions. Linear programming may not provide a desired optimal solution when the underlying assumptions are removed from the problem. An application of linear programming business decision making is limited by the following assumptions.

2-3.1. As indicated in Sec. 2-2, the primary assumption of linear programming is the linearity in the objective function and in the side constraints. This implies that the measure of effectiveness and utilization of each resource must be directly and precisely proportional to the level of each individual activity. Also the activities must be additive. The total measure of effectiveness must be the sum of the measure of effectiveness of each individual activity. The total amount of resources utilized for all activities must be exactly equal to the sum of the resources utilized for each individual activity. Joint interactions are impossible for amount of resources used in some of the activities. In the process of petroleum refining, for example, gasoline is produced as a primary product and asphalt as a by-product. If asphalt is to be produced alone, crude oil will still have to be consumed. The quantity of crude oil consumed if gasoline and asphalt are produced at the same time will be less than the sum of the quantities of crude oil consumed separately for each for each product. These two activities are not additive in their resource utilization.

2-3.2 Divisibility

Linear programming presupposes the complete divisibility of the resources utilized and the units of output produced. That is, it is assumed that the decision variables can take on fractional values. Resources and activities are considered to be continuous within a relevant range. Therefore, linear programming allows a production program which uses 600 units of electronic components and 86 man-hours of labor time to produce 80 units of regular television sets and 33 units of color television sets per day. It is entirely appropriate and practically feasible to have fractional values in the resource utilization and production activities in many business situations. However, there are also occasions in which fractional values are neither permissible nor practical. Integer programming is a special technique which can be used for finding non-fractional values of resource usage and decision variables.

2-3.3 Finiteness

The need for optimal decision making in business arises from a relative scarcity of productive resources. The problem in linear programming is the optimal allocation of limited resources to alternative activities to achieve a specific business objective. There must be a finite number of resource restrictions and available alternative activities. Goods are produced with a limited number of resources; therefore, an infinite number of resource restrictions are impossible. If the decision maker is faced with an unlimited number of alternative activities, linear programming can no longer be used for finding an optimal solution.

2-3.4 Certainty and Static Time Period

It is assumed that the coefficients of the decision variables in linear programming are known with certainty; all the coefficients, such as unit profit contribution, prices, and the amount of resources required per unit of output, are known constants. The available resources are also known with accuracy. This assumption is reasonable if the variance of the input-output coefficients is not significant; i.e. unit profit contribution and resource usage per unit of output do not fluctuate widely. Thus, linear programming implicitly assumes static time period. In reality, the input-output coefficients are neither known with certainty nor constants. Therefore, a number of special techniques have been used, such as parametric programming and sensitivity analysis.

2-4. Mathematical formulation of linear programming problem

If, instead of two decision variables and four side constraints, we had n decision variables and m side constraints in the problem of Sec. 2-2, we would have the following type of mathematical formulations:

Maximize

F = c1x1 + c2x2 + …+ cnxn

Subject to

a11x1 + a12x2 + … + a1nxn ≤ b1

a21x1 + a22x2 + … + a2nxn ≤ b2

……………………………..

am1x1 + am2x2 +… + amnxn ≤ bm

and x1≥ 0, x2 ≥ 0, …, xn ≥ 0

where aij, bi, cj = given constants

xj = decision (or choice) variables

m = number of side constraints

n = number of decision variables

The above problem is formulated in a more general form:

Maximize

F =

subject to

(for i =1, 2,…, m)

and xj ≥ 0 (for j = 1, 2, …, n)

where Σ= Greek letter capital sigma, which means “the sum of”.

The above problem is formulated in a more general form:

Minimize Z= Subject to

(for i = 1, 2, …, m)

and xj ≥ 0 (for j = 1, 2, …, n)

The basic difference between the maximization and the minimization problems in linear programming is found in the signs of the inequalities of the side constraints. The side constraints are expressed by the “less than or equal to” signs (i.e., ≤) in the maximization problem; whereas those of the minimization problem are expressed by the “greater than or equal to” signs (i.e. ≥)1[1].

2-5. Formulation of linear programming problems

The usefulness of linear programming as a tool for optimal decision making and resource allocation is based on its applicability to many business problems. The effective use and application of linear programming require, as a first step, the formulation of the model when the problem is presented. Several examples are given in this section to illustrate the formulation of linear programming problems.

Example 2-1. A furniture manufacturer.

A small furniture manufacturer produces thee different kinds of furniture: desks, chairs, and bookcases. The wooden materials have to be cut properly by machines. In total, 100 machine-hours are available for cutting. Each unit of desks, chairs, and bookcases requires 0.8 machine-hour, 0.4 machine-hour, and 0.5 machine-hour, respectively.

This manufacturer also has 650 man-hours available for painting and polishing. Each unit of desks, chairs, and bookcases requires 5 man-hours, 3 man-hours, and 3 man-hours for painting and polishing, respectively. These products are to be stored in a warehouse which has a total capacity of 1.260 sq. ft. The floor space required by these three products are 9 sq ft, 6 sq ft, and 9 sq ft, respectively, per unit of each product. In the market, each product is sold at a profit of $30, $16, and $25 per unit, respectively. How many units of each product should be made, to realize a maximum profit?

Formulation of Example 2-1. Let x1, x2, x3 be the number of units of desks, chairs, and bookcases to be produced, respectively. Since 100 total machine-hours are available for cutting, the production of x1, x2, and x3 should utilize no more than the available machine-hours. Therefore, the mathematical statement of the first side constraint is in the form:

0.8x1 + 0.4x2 + 0.5x3 ≤ 100

Also, no more than 650 man-hours and 1.260 sq ft are available for painting and polishing and storing, respectively. Therefore, these two side constraints are in the form:

5x1 + 3x2 + 3x3 ≤ 650

9x1 + 6x2 + 9x3 ≤ 1.260

Finally, the decision variable must be nonnegative.

The problem them can be formulated as follows:

Maximize

F = 30x1 + 16x2 + 25x3

subject to

0.8x1 + 0.4x2 + 0.5x3 ≤ 100

5x1 + 3x2 + 3x3 ≤ 650

9x1 + 6x2 + 9x3 ≤ 1.260

1

and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Example 2-2. The riverside rubber company

The Riverside Rubber Company specializes in the production of three different kinds of tires: premium tires, deluxe tires, and regular tires. These three different tires are produced at the company’s two different plants, with different production capacities. In a normal 8-hours working day, Plant A produces 50 premium tires, 80 deluxe tires, and 100 regular tires. Plant B produced 60 premium tires, 60 deluxe tires, and 200 regular tires. The monthly demand for each of these is know to be 2,500 units, 3,000 units, and 7,000 units, respectively. The company finds that the daily cost of operation is $2,500 in Plant A and $3,500 in Plant B.

Find the optimum number of days of operations per month at the two different plants to minimize the total cost while meeting the demand.

Formulation of Example 2-2. Let the decision variables x1 and x2 represent the number of days of operation in each of these plants. The objective function of this problem is the sum of the daily operational costs in the two different plants; that is,

Z = 2,500x1 + 3,500x2

Given the total revenue, the profit is increased by reducing the total cost. The objective, then, is to determine the value of the decision variables x1 and x2, which yield the minimum of total cost subject to side constraints. The production of each of three different tires must be at least equal to or greater than the specific quantity in order to meet the demand requirement. In no event should the production be less than the quantities of each product demanded.

Together with the side constraints, the problem then can be formulated as:

Minimize

Z = 2,500x1 + 3,500x2

Subject to

50x1 + 60x2 ≥ 2,500

80x1 + 60x2 ≥ 3,000

100x1 + 200x2 ≥ 7,000

and x1 ≥ 0, x2 ≥ 0.

Example 2-3 A local auto shop. A local auto shop needs 400 cans of paints per month, the supply consisting of three different colors. The cost per can of each of these is given as follows:

Blue $4.00 per can

Brown $4.50 per can

Black $4.25 per can

The auto-repair work requires at least 80 cans of brown paint, not more than 160 cans of blue paint, and at least 40 cans of black paint. How many cans of each of these paints should be purchased to minimize the total cost?

Formulation of Example 2-3 To set up this situation as a linear programming problem, an objective function stated in terms of cost is required. Let the decision variables x1, x2 and x3 represent the number of cans of blue paint, and black paint required, respectively. Then, the objective function is expressed as

Z = 4.000x1 + 4.50x2 + 4.25x3

Naturally, the value of this function must be minimized, subject to the monthly requirement constraints.

Since the auto-repair work does not require more than 160 cans of blue paint, x1 must be less than or equal to 160 cans,

x1 ≤ 160

The direction of inequality sings of the second and the third constraints is different from that of the first, because the auto-repair work requires at least 80 cans of brown paint and 40 cans of black paint. Hence

x2 ≥ 80

x3 ≥ 40

The final constraint is specified by the fact that the monthly requirement of these paints is 400 cans.

x1 + x2 + x3 = 400

The problem is then summarized in the general linear programming form as

Minimize

Z = 4.00x1 + 4.50x2 + 4.25x3

Subject to

x1 ≤ 160

x2 ≥ 80

x3 ≥ 40

x1 + x2 + x3 = 400

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

(In this particular problem, the two nonnegative constraints x2 ≥ 0 and x3 ≥ 0 may be redundant)

Example 2-4

THE TIFFANY INVESTMENT COMPANY The Tiffany Investment Company possesses a large amount of cash to invest – say, $1,000,000. There are five different investment choices, each with different growth and income potentialities: common stocks, mutual funds, municipal bonds, saving certificates, and real estate investments. The current returns from the investment of each of these five choices are also different. The current returns on investment are known for each of the investment opportunities as follows:

Current annual yield percent

Common stocks 10

Mutual funds 8

Municipal bonds 6

Savings certificates 5

Real estate investment 9

In this case, management believes that the current yield will persist in the future and wishes to diversify the investment portfolio of the company to obtain maximum returns. Because of the risk element involved, management restricts the investment in common stocks to not more than the combined total investment in municipal bonds, savings certificates, and real estate investment. Total investment in mutual funds and real estate combined must be at least as large as that in municipal bonds. Also, management wishes to restrict its investment in mutual funds to a level not exceeding that of savings certificates. Determine the optimum allocation of investment funds among these five choices and the actual amount of returns from investment under the above conditions.

Formulation of Example 2-4 In this investment problem, the decision that needs to be made concerns how to allocate the available investment funds into the various alternative choices. In this instance, there are five decision variables. Let the decision variables x1, x2, x3, x4, and x5 represent the percentage of the total investment fund to be allocated into common stocks, mutual funds, municipal bonds, savings certificates, and real estate investment, respectively. Then the objectives is to maximize the total returns from the different investment choices. The objective function is

F = 0.10x1 + 0.08x2 + 0.06x3 + 0.05x4 + 0.09x5

As mentioned earlier, the side constraints must be linear inequalities of the decision variables. The first side constraint comes from the restriction of investment in common stocks to not more than the combined total investment in municipal bonds, savings certificates, and real estate investment. This restriction can be expressed in a linear inequality form

x1 ≤ x3 +x4 + x5

The second side constraint comes from the restriction that total investment in mutual funds and real estate combined must be at leats as large as that in municipal bonds.

Formulation of Example 2-4 In this investment problem, the decision that needs to be made concerns how to allocate the available investment funds into the various alternative choices.

Formulation of Example 2-4

In this investment problem, the decision that needs to be made concerns how to allocate the available investment funds into the various alternative choices. In this instance, there are five decision variables. Let the decision variables x1, x2, x3, x4, and x5 represent the percentage of the total investment fund to be allocated into common stocks, mutual funds, municipal bonds, savings certificates, and real estate investment, respectively. Then the objective is to maximize the total returns from the different investment choices. The objective function is

F=0,01x1+0,08x2+0,06x3+0,05x4+0,09x5

As mentioned earlier, the side constraints must be linear inequalities of the decision variables. The first side constraint comes from the restriction of investment in common stocks to not more than the combined total investment in municipal bonds, savings certificates, and real estate investment. This restriction can de expressed in a linear inequality form

x1 ≤ x3+x4+x5

The second side constraint comes from the restriction that total investment in mutual funds and real estate combined must be at least as large as that in municipal bonds.

x2+x3≥x3

The third side constraint is the fact that the restriction of investment in savings certificates.

x2≤x4

Finally, the aggregate of total investment in different choices must be equal to 1; that is 100 (or $1 million).

x1+x2+x3+x4+x5=1

Here is the problem, summarized in linear programming form, together with the conditions of nonnegativity:

Maximize

F=0,01x1+0,08x2+0,06x3+0,05x4+0,09x5

subject to

x1≤x3+x4+x5

x2+x3≥x3

x2≤x4

x1+x2+x3+x4+x5=1

and

x1≥0,x2≥0,x3≥0,x4≥0,x5≥0

2-6 Graphic representation linear programming problem can easily be solved by a graphic method when it involves two decision variables. It is also possible to solve a linear programming problem with three decision variables by a graphic method. But its presentation is not as easy as in the problem with two decision variables, because it requires three dimensions to illustrate. A graphic method is not practical when there are more than three decision variables, because there is no way of presenting more than three dimensions in space.

2-6.1 A Maximization Problem

To illustrate the graphic method, let us again consider the Riggs Manufacturing Company’s production problem as given in Sec. 2-2. The problem is summarized in a linear programming form:

Maximize

F = 50x1 + 80x2

subject to

x1 ≤ 80 (Assembly Line I)

x2 ≤ 60 (Assembly Line II)

5x1 + 6x2 ≤ 600 (electronic components)

x1 + 2x2 ≤ 160 (labor supply)

and x1 ≥ 0, x2 ≥ 0

where x1= number of regular television sets to be produced

x2= number of color television sets to be produced

Assembly-line constraints

Let us draw a two-dimensional graph, with the production of regular television sets shown on the horizontal axis and the production of color television sets shown on the vertical axis.

Because of the limited capacity in Assembly Line I, where regular television sets are produced, no more than 80 units of x1 can be produced per day; that is, x1≤80. The side constraint for Assembly Line I consists of two parts: an equality part and an inequality part. Thus, the number of the regular television sets assembled will be either x1=80 or x1<80. Together with the nonnegativity condition x1≥0, the inequality side constraint for Assembly Line I can be represented by the shaded area shown in Fig. 2-1.

A daily production of color television sets is limited to 50 units when Assembly Line II is fully utilized; that is, x2≤50. As in the case of Assembly Line I, together with the nonnegativity condition, the inequality side constraint for Assembly Line II can be represented by a shaded area as shown in Fig. 2-2.

Figures 2-1 and 2-2 are combined to obtain an area which simultaneously satisfies the two assembly-line constraints. Then the production of x1 and x2 can be represented by the shaded rectangle area shown in Fig. 2-3.

Electronic-components constraint. The production of x1 and x2 is also limited by the available electronic components. The inequality

5x1+6x2≤600

may be drawn by the line MM in Fig. 2-4. If the entire electronic components are used for the production of x1, 120 units of x1 can be produced. Similarly, if all the available components are used for the production of x2, 100 units of x2 can be produced. The line MM is drawn by connecting these two extreme points. Any point on the line MM indicates a different production program with full utilization of available electronic components. For example, at the point M1, the production combination of 30 units of x1 and 75 units of x2 is possible; at the point M2, 60 units of x1 and units of x2; at the point M3, 84 units of x1 and 30 units of x2. All these points lie on the straight line MM, because they indicate different production programs of x1 and x2 with full utilization of available electronic components. A different product combination is also possible on any points to the left of the line MM, but not to the right, because of the limited availability of electronic components.

Labor-supply constraint.

A similar procedure applies to the drawing of the labor-supply inequality constraint x1+2x2≤160. If all the available workers are fully employed for the production of regular television sets alone, 160 units of x1 will be produced per day. Similarly, 80 units of x2 will be produced per day when all the available workers produce only color television sets. The line LL in Fig. 2-5 is drawn by connecting these two extreme points.

Any point on the line LL represents all combinations of x1 and x2 with full utilization of available labor supply. The line LL divides the plane into two parts. Different combinations of x1 and x2 are possible in the lower-left part of the line LL, because all points in the left part satisfy the inequality constraint x1 + 2x2 < 160. All points in the upper-right part of the line LL cannot be considered, because they would be represented by the inequality x1 + 2x2 < 160. That is, all combinations of x1 and x2 in the upper-right part of the plane require more than 160 workers.

Since all these inequality constraints must be satisfied simultaneously, they are combined in Fig. 2-6. The shaded area, denoted by a hexagon OABCDE, is the area which satisfies all the side inequality constraints at the same time, as well as the nonnegative conditions. It is called the feasible region, or admissible region. All points in the shaded area, including the boundaries of the hexagon, are the only ones that satisfy all the restrictions and therefore are feasible. Consequently, all points outside the shaded area are not feasible.

Let us choose any two points Q and R to illustrate the production feasibility. Obviously, these two points lie outside the feasible region. At Q the production is not feasible because it violates the capacity restriction of Assembly Line I. Similarly. Point R is also not feasible, as it violates the electronic components’ supply restriction.

The next step is to find a point from this feasible region that maximize the value of the objective function, i.e. profits. In order to find this point let us consider an arbitrary case in which the daily profit is $2000. Then the objective function is 2.000 = 50x1 + 80x2. This equation is represented by a straight line P1 by connecting two extreme points, as shown in Fig. 2-7. Two extreme points are found by setting x2=0 and then x1=0.

If x2 = 0

2.000 = 50x1 + 80(0)

x1 = 40

And if x1 = 0

2.000 = 50(0) + 80x2

x2 = 25

And so all points on the line P1 yield a daily profit of $2,000 with different combinations of x1 and x2.

Let us consider another arbitrary case, in which the daily profit is $4,000. The objective function 4,000 = 50x1 + 80x2 is represented by a straight line P2 by connecting two extreme points x1 = 80 and x2 = 50. All points on the line P2 yield a daily profit of $4,000 with different combinations of x1 and x2. The line P2 is farther right from the origin and is parallel to the line P1, and so on. If we draw a new straight line P4, which is parallel to the line P1, and move as far as possible from the origin to the right, then the line P4 is tangent to one of the corner boundary points C. The lines P1, P2, P3, and P4 are known as profit contour lines, or isoprofit lines. Clearly, the line P4 is preferable to other profit contour lines P1, P2, and P3, because the largest profit is found on any point on this line. There is only one point C that lies on the line P4 and is in common with the feasible region. At this corner point C, where x1 = 60 and x2 = 50, the maximum daily profit of $7,000 is obtainable. The optimal solution, then, is to produce 60 units of regular television sets and 50 units of color television sets per day. The maximum profit obtainable is $7,000 per day.

To verify the optimal solution at Point C, we test the objective function F = 50x1 + 80x2 at each of the five corner points of the feasible region A, B, C, D, and E.

A: 50 (80) + 80 (0) = 4,000

B: 50 (80) + 80 ( ) = 6,666

C: 50 (60) + 80 (50) = 7,000

D: 50 (40) + 80 (60) = 6,800

E: 50 (0) + 80 (60) = 4,800

Again, we see that the maximum daily profit of $7,000 is obtained at Point C.

The daily production of 60 regular television sets and 80 color television sets must satisfy the side constraints.

Assembly Line I: 60<80

Assembly Line II: 50<60

Electronic components:

5(60) + 6(50) = 600

Labor supply: 60 + 2(50) = 160

Thus, the available electronic components and labor force are fully utilized are point C. However, there is still the unutilized capacity to produce 20 regular television sets in Assembly Line I and 10 color television sets in Assembly Line II.

2-6.2 A Minimization Problem

A minimization problem of linear programming can also be solved by the graphic method. Let consider the problem of the Riverside Rubber Company, as described in Example 2-2, to illustrate the minimization problem.

The problem is stated in the linear programming form:

Minimize

Z = 2,500x1 + 3,500x2

Subject to

50x1 + 60x2 ≥ 2,500 (premium tires)

80x1 + 60x2 ≥ 3,000 (deluxe tires)

100x1 + 200x2 ≥ 7,000 (regular tires)

and x1 ≥ 0, x2 ≥ 0

where x1 = number of days of operation in Plant A

x2 = number of days of operation in Plant B

To simplify the computation, the three side constraints can be expressed in the reduced form. In doing this, both sides of the inequalities are divided by 10 in the first constraint, 20 in the second, and 100 in the third. Thus, we get

5x1 + 6x2 ≥ 250 (premium tires)

4x1 + 3x2 ≥ 150 (deluxe tires)

x1 + 2x2 ≥ 70 (regular tires)

The three constraints are represented in Fig. 2-8, following the same procedures discussed in the preceding section. It is to be noted that the side constraints in a minimization problem are by “greater than or equal to” (≥). They must be represented by all points on and to the right of the three lines (ABCD) drawn. Therefore, all combinations of x1 and x2 that simultaneously satisfy the three side constraints must fall on or to the right of the convex polyhedral set ABCD. This is the shaded area in Fig. 2-8, and represents the feasible region.

The four corner points of the feasible region are

A (70,0) B (20,25)

C ( ,27 ) D (0,50)

To find the optimal solution, these four corner points are tested out with the objective function Z = 2,500x1 + 3,500x2.

A: 2,500 (70) + 3,500 (0) = 175,000

B: 2,500 (20) + 3,500 (25) = 127,500

C: 2500 (16 ) + 3,500 (27 ) = 138,888

D: 2,500 (0) + 3,500 (50) = 175,000

Test results indicate that the minimum monthly operating cost of $127,000 is found at Point B. To achive this, the company must operate 20 days in Plant A and 25 days in Plant B per month.2[2]

The monthly supply requirements are tested at Point B, substituting x1 = 20 and x2 = 25.

Premium tires: 50 (20) + 60 (25) = 2,500

Deluxe tires: 80 (20) + 60 (25) = 3,100

Regular tires: 100 (20) + 200 (25) = 7,000

We see that when two plants are operated as indicated, the company exactly meets the monthly supply requirements of 2,500 premium tires and 7,000 regular tires. There will be a surplus of 100 deluxe tires after meeting the supply requirements of 3,000 units, because the combined production of deluxe tires is 3,100 units at Point B.

Part III GRAMMAR REFERENCE (The Infinitive) Инфинитив

Инфинитив (the Infinitive) – неопределенная форма глагола, отвечает на вопрос что делать? или что сделать? Формальным признаком инфинитива является приинфинитивная частица to, не имеющая самостоятельного значения. Однако в некоторых случаях инфинитив употребляется без этой частицы (после модальных, вспомогательных глаголов и др.)

Являясь неличной формой глагола, инфинитив не выражает лица, числа и наклонения.

Формы инфинитива.

Indefinite Continuous Perfect

Active to write to be writing to have written

Passive to be written - to have been written

Обратите внимание на перевод инфинитива в зависимости от его формы:

I like to study mathematics. Я люблю изучать математику.

I am glad to be helping them. Я рад, что сейчас помогаю им.

He is glad to have devoted himself to the solution of difficult problems.

Он рад, что посвятил себя решению трудных задач.

We are glad to have been learning the theory of algebraic curves.

Мы рады, что учили (на протяжение некоторого времени) теорию алгебраических кривых.

She is glad to be invited to the conference.

Она рада, когда ее приглашают на конференцию.

The students are glad to have been helped to find the answers to old unsolved problems.

Студенты рады, что им помогают найти ответы на старые нерешенные проблемы.

2

Функции инфинитива в предложении.

Подлежащее.

To give a concise definition of what is mathematics is impossible.

Дать краткое определение, что такое математика - невозможно.

Составного сказуемого.

His task is to study difference- equation.

Его задача – изучить разностное уравнение.

Часть составного глагольного сказуемого.

She must repeat the theme “Methods of indirect measurement”.

Она должна повторить тему «Методы косвенного измерения».

He began to do the laboratory work. Он начал выполнять лабораторную работу.

Дополнение.

She had promised me to work with many new configurations and new curves.

Она обещала мне работать с многими новыми конфигурациями и новыми кривыми.

Определение.

The task to be done has been explained by the teacher.

Задание, которое надо выполнять было объяснено учителем.

Euler was the first to give the examples in which conditions of the problem are expressed by a algebraic symbols.

Эйлер первый дал примеры, в которых условия задачи выражены алгебраическими символами.

Обстоятельство цели.

To discuss the equation of a curve Descartes infroduced X-axis and Y-axis.

Чтобы исследовать уравнение кривой Декарт ввел оси X и Y.

You must work much in order to master this subject.

Вы должны много работать, чтобы овладеть этим предметом.

Вводный член предложения.

To tell you the truth we shan’t be able to finish this work today.

По правде говоря, мы не сможем закончить эту работу сегодня.

The Objective Infinitive Construction (Объектный инфинитивный оборот)

В английском языке после некоторых переходных глаголов употребляется сложное дополнение, представляющее собой сочетание существительного в общем падеже или личного местоимения в объектном падеже с инфинитивом. Такое сочетание обычно называют Objective Infinitive Construction.

Объектный инфинитивный оборот употребляется:

1) после глаголов to know знать; to want хотеть; to wish желать; to find находить, узнавать; to expect предполагать; to like любить, нравиться; to think думать; to believe полагать, считать; to consider считать; to suppose полагать; to assume допускать, предполагать в действительном залоге.

We know Alexandria to be the mathematical centre of the ancient world.

Мы знали, что Александрия была математическим центром древнего мира.

He wanted me to use the graphic for the solution of two variables

Он хотел, чтобы я использовал график для решения двух переменных.

We would like him to take this alignment chart for his calculations.

Мы хотели бы, чтобы он взял эту горизонтальную проекцию для своих вычислений.

We found her to be a very capable mathematician.

Мы нашли, что она очень способный математик.

He consideres these graphic solutions of spherical triangles to be of great importance.

Он считает, что эти графические решения сферических треугольников очень важными.

We expect this geometric figure to be a regular plolyhedron.

Мы предполагаем, что эта геометрическая фигура – правильный многогранник.

We see the mathematical science of today be manifold and extensive.

Мы видим, что математическая наука сегодня разнообразна и обширна.

She students heard the professor speak about the question of the impossibility of certain solutions.

Студенты слышали, как профессор говорил о проблеме невозможности некоторых решений.

Oборот for + Object + Infinitive (for – Phrase)

Этот оборот представляет собой сочетание предлога for с существительным в общем падеже (или личным местоимением в объектном падеже), за которым следует инфинитив. Инфинитив показывает, какое действие должно быть совершено лицом, обозначенным существительным или местоимением.

Этот оборот может входить в состав любого члена предложения и переводиться на русский язык при помощи инфинитива или придаточного предложения:

This is for you to decide how to act. (в составе сказуемого)

Вы должны решить как действовать.

The first thing for me to do is to start Первое, что я должен сделать, это

the experiment. (в составе подлежащего)

начать опыт.

I am waiting for you to begin the research. (в составе дополнения)

Я жду, чтобы вы начали исследование.

He opened the door for us to enter. (в составе обстоятельства)

Он открыл дверь, чтобы мы вошли.

Predicative Infinitive Construction (Предикативный инфинитивный оборот)

Predicative Infinitive Construction, называется также Subjective Infinitive Construction, состоит из существительного в общем падеже или личного местоимения в именительном падеже (подлежащее) и глагола в личной форме (обычно в страдательном залоге) + инфинитив (сказуемое) и употребляется:

1) с теми же глаголами, что и объектный инфинитивный оборот, но в страдательном залоге. На русский язык предложения, содержащие предикативный инфинитивный оборот, переводятся сложноподчиненным предложением, состоящим из неопределенно- личного главного предложения и дополнительного придаточного предложения, вводимого союзами что и как:

The Greeks is said to form mathematics as a scientific discipline.

Говорят, что греки формировали математику как отрасль науки.

The Greeks is believed to have regarded a straight line as infinity.

Полагают, что греки уже рассматривали прямую линию как бесконечность.

They seem to distinguish mathematical objects and the mathematical method.

Они, кажется, отличают математические объекты и математический метод.

She appears to be a very good specialist in this subject.

Кажется, она хороший специалист в этой области.

This axiom is likely to be the solution of every problem.

Эта аксиома, вероятно, является решением каждой проблемы.

The Gerund (Герундий)

Герундий – это неличная форма глагола, обладающая признаками как существительного, так и глагола.

Герундий выражает действие, представляя его как название процесса. Герундий образуется путем прибавления окончания –ing к основе глагола.

В русском языке соответствующей формы нет, поэтому герундий переводится отглагольным существительным, инфинитивом, придаточным предложением или деепричастием:

I. Формы герундия.

Tense Active Passive

Indefinite sending being sent

Perfect having sent having been sent

Обратите внимание на перевод герундия в зависимости от формы:

He is fond of studying the mathematical theory of probability.

Он любит изучать математическую теорию вероятности.

I am not fond of being read to. Я не люблю, когда мне читают.

We remember having read very much about two French mathematicians R. Descartes and P. Fermat.

Мы помним, что читали очень много о двух французских математиках: Рене Декарте и Пьере Ферма.

The students remember having been read a lot of the importance of Analytic Geometry.

Студенты помнят, что им много читали о важности аналитической геометрии.

II. Функции герундия в предложении.

1. Подлежащее.

Our being invited to take part in such conferences is very important for us.

То что нас приглашают принимать участие в таких конференциях, очень важно для нас.

Your studying much now will help you in your future work.

То, что вы сейчас много занимаетесь, поможет вам в вашей будущей работе.

Proving theorems is his hobby. Доказательство теорем - его любимое занятие.

2. Часть сказуемого.

My favourite occupation is analyzing curves.

Мое любимое занятие анализ кривых линий.

3. Дополнение.

He doesn’t like studying geometric figures and curves.

Он не любит изучать геометрические фигуры и кривые линии.

She insisted on adopting the new decision of the task.

Она настаивала на принятии нового решения задачи.

The students must avoid making such mistakes.

Студенты должны избегать таких ошибок или: студенты должны стараться не делать таких ошибок.

I remember having told you about Euclidean geometry.

Я помню, что говорил вам о Евклидовой геометрии.

4. Определение.

He never missed an opportunity of listening to this lecturer.

Он никогда не упускал возможности послушать этого лектора.

5. Обстоятельство.

After proving mathematical theorems he made a short summary of it.

Доказав математические теоремы, он кратко изложил их содержание.

In spite of being tired the mathematicians continued their discussion.

Несмотря на усталость, математики продолжали свои прения.

You will never be able to draw up a graph of an equation without knowing rule well.

Вы никогда не сможете правильно составить график уравнения, не зная хорошо правило.

(The Participle) Причастие

Причастие (the Participle) – это неличная форма глагола, совмещающая в себе свойства глагола, прилагательного и наречия.

Participle I выражает действие, одновременное с действием сказуемого;

Participle II выражает действие, одновременное с действием сказуемого или предшествующее ему;

Perfect Participle выражает действие, предшествующее действию сказуемого.

I. Формы причастия

Participle I Participle II Perfect Participle

Active writing - having written

Passive being written written having been written

The Participle I.

II. Функции причастия.

A second branch of Mathematics is geometry consisting of several geometries.

Второй отраслью математики является геометрия, состоящая из нескольких геометрий.

They watched a drawing up a schedule.

Они наблюдали за составляющимся графиком.

The described modern terminology and symbolism are a relatively new development.

Описанная современная терминология и символизм – это относительно новое развитие.

The method used depends on the chosen material.

Используемый метод зависит от выбранного материала.

1. Обстоятельство.

They spent the whole day learning modern methods of arithmetic operations.

Они провели весь день, изучая современные методы арифметических действий (операций).

Using only a straightedge and a compass the Greeks performed many constructions.

Используя лишь линейку и компас, греки строили много сооружений.

While solving this problem I came across many difficulties.

Решая (когда я решал) эту задачу я встретился со многими трудностями.

When invented this method was used for finding the equations of loci.

После того, как метод изобрели, его использовали для нахождения уравнений траекторий.

Being invited the celebrated mathematician said he would not be able to deliver a lecture.

Когда известного математика пригласили, он сказал, не сможет прочитать лекцию.

Having carried out most arithmetic operations they decided to show the results to the professor.

Выполнив большинство арифметических действий, они решили показать результаты профессору.

Having been given all the instructions we began calculations.

После того как были даны все инструкции мы начали вычисление.

2. Часть сказуемого.

Participle входит в состав времен группа continuous.

The students of our group are studying Klein’s work on the icosahedra.

Студенты нашей группы сейчас изучают работу Ф. Клейна о двадцатигранниках.

She was explaining the theory of differential equations during half an hour yesterday

Она объяснена теорию дифференциальных уравнений в течение получаса вчера.

Participle II входит в состав:

Времен группы Perfect.

I have solved the problem with the regular polyhedra.

Я уже решил задачу с правильным многогранником.

The teacher will have given examples with the duplication of the cube by the end of the lesson.

Учитель даст примеры с удвоением куба к концу урока.

Форм страдательного залога:

The laboratory work on the theme “The squaring of the circle” will be carried out in this classroom.

Лабораторная работа по теме: «квадратура круга» будет проводиться в этом классе.

Lectures on “Integrals” are attended by many students.

Многие студенты посещают

лекции по «интегралам».

Если подлежащее в главном придаточном обстоятельственном предложениях различны, то в английском языке возможна (в русском – невозможна) замена придаточного предложения причастным оборотом, сохраняющим свое подлежащее. Такой оборот придаточного предложения, называется самостоятельным, или абсолютным, причастным оборотом.

When our students had written the test we went to show it to the professor = Our students having written the test. we went to show it to the professor.

Когда наши студенты написали тест, мы пошли показать его профессору.

All preparations being made (= when all preparation were made), we started the experiment.

Когда все приготовления были сделаны, мы начали эксперимент.

The conditions permitting (= if the conditions permit), we shall start our work.

Если условия позволят, мы начнем нашу работу.

There are a lot of unsolved problems in mathematics, some of them having been solved.

В математике есть много нерешенных задач, причем некоторые из них решены.

VOCABULARY

A

add [Фd] прибавлять

addend [у'dend] слагаемое

addition [у'dшыуn] сложение

additional [у'dшыуnl] добавочный, дополнительный

algebraic equality [,Фldпш'breшшk ш:'kwщlшtш]

алгебраическое равенство

amount, v [у'maunt] составлять сумму,

равняться

applied mathematics [у'plaшd ,mФЕш'mФtøks]

прикладная математика

area ['Ууrшу] область, сфера

axiom ['Фksшуm] аксиома

axiomatic [,Фksшуu'mФtшc] неопровержимый

axes ['Фksш:z] осьaxiomatic deductive

construction[,Фksшуu'mФtшk dш'dфktшve kуn'strфkыуn]

самоочевидная

дедуктивная

конструкция

B

bulk [bфlk] объем

C

calculate ['kФlkjuleшt]вычислять,

рассчитывать

calculation [,kФlkju'leшыуn] вычисление

calculator ['kФlkjuleшtу] вычислитель

calculus ['kФlkjulуs] исчисление

concise [kуn'saшs] сжатый

conclude [kуn'klu:d]заключать, делать

вывод

conclusion [kуn'klu:пуn] вывод

conclusive [kуn'klu:sшv] заключительный

circle ['sу:kl] окружность

circulant ['sу:kjulуnt] циркулянт

circular ['sу:kjulу]круговой,

циклический

circularly ['sу:kjulуlш] кругообразно

circulate ['sу:kjuleшt] циркулировать

cube, n [kju:b] куб

cube, v [kju:b] возводить в куб

cubage ['kju:bшdп]нахождение объема;

кубатура

cubic ['kju:bшk] кубический

cuboid ['kju:bщшd] прямоугольный параллелепипед

commensurable

magnitudes[kу'menыуrуbl 'mФgnшtju:dz]

соизмеримые

величины

conic section ['kщnшk 'sekыуn] коническое сечение

coordinate geometry [kуu'щ:dшneшt dпш'щmшtrш]

геометрия координат

coordinate system [kуu'щ:dnшt 'sшstшm]

система координат

curve, n [kу:v] кривая

curve, v [kу:v]

искривляться

D

define [di'faшn] определять

definite ['definшt] определенный

definition [,defi'nшыуn] определениеdependent

variable

[dш'pendуnt

vУуrшуbl]зависимая переменная

differential [,dшfу'renыуl] дифференциальный

differentiable [,dшfу'renыуbl] дифференцируемый

differentiability [dшfу,renыiу'bшlшtш]

дифференцируемость

dimension [dш'menыуn] измерение

dimensional [dш'menыуnl] пространственный

domain [dуu'meшn] область, сфера

divide [dш'vaшd] делить

divisible [dш'vшzуbl] делимый

division [dш'vшпn] деление

divisor [dш'vaшzу] делитель

duplicate ['dju:plшkeшt] удваивать

duplicated ['dju:plшkeшtшd] дублирующий

duplication [,dju:plш'keшыуn] удвоение

duplicator ['dju:plшkeшtу]копировальный

аппарат

E

ellipse [i'lшps] эллипс

elliptic [i'lшptшk] эллиптический

equate [i'kweшt] равняться

equation [i'kweшыn] уравнение

equality [ш"kwщlшtш] равенство

equalize ['ш:kwуlaшz] равнять, уравнивать

F

figure ['figу]фигура, цифра,

диаграмма

fraction ['frФkыn] дробь, доля, частица

fractional ['frФkыуnl] дробный

H

hyperbola [haш'pу:bуlу] гипербола

I

icosahedra [,aшkуusу'hш:drу] двадцатигранники

icosahedral [,aшkуusу'hш:drуl] двадцатигранный

icosahedron [,aшkуusу'hш:drуn]

двадцатигранник

integral ['шntшgrуl] цельный

integrate ['шntшgreшt] интегрировать

integrity [in'tegrшtш] целостность

internal [in'tу:nl] внутренний

internally [in'tу:nуlш] внутренне

incommensurable [,шnkу'menыуrуbl]несоизмеримый,

иррациональный

incommensurable magnitudes

[шnkу'menыуrуbl 'mФgnшtju:dz]

несоизмеримые

величины

incommensurate [,шnkу'menыуrшt]несоизмеримый,

несоизмеримый

independent variable [,шndш'pendуnt 'vУуrшуbl]

независимая

переменная

indirect measurement [,шndш'rekt 'meпуmуnt]

косвенное измерение

invariable [in'vУуrшуbl]неизменный,

неизменяемый,

постоянный

invariance [in'vУуrшуns] инвариативность

invariant [in'vУуrшуnt] инвариантный

involve [in'vщlv] возводить в степень

involving [in'vщlvшт] возведение в степень

L

line, n [laшn] линия

line, v [laшn] проводить линию

linear ['lшnшу] линейный

linear equation ['lшnшуш'kweшыn]

линейное уравнение

limitless ['lшmшtlшs] безграничный

M

measure ['meпу] мера

measurability [,meпуrу'bшlшtш] измеримость

measurable ['meпуrуbl] измеримый

measurement ['meпуmуnt] измерение

measured ['meпуd] измеренный

multiple, a ['mфltшpl]многократный,

сплошной, составной

multiple, n ['mфltшpl] кратное число

multiplication [,mфltшplш'keшыn]

умножение,

увеличение

multiply ['mфltшplaш] умножать

N

notional ['nуuыуnl]смысловой,

воображаемый

notion ['nуuыуn] понятие

numeration system ['nju:mу'reшыуn 'sшstшm]

система чисел

n-dimensional space [,endш'menыуnl 'speшs]

н-мерное пространство

n-dimension [уn dш'menыуn] н-мерное измерение

O

ordinate ['щ:dшneшt] ордината

P

parabola [pу'rФbуlу] парабола

parabolic [,pФrу'bщlшk] параболический

pentagon ['pentуgуn] пятиугольник

pentahedral [,pentу'hш:drуl] пятигранный

pentahedron [,pentу'hш:drуn] пятигранник

perpendicular, n [,pу:pуn'dikjulу] перпендикуляр

perpendicular, adj [,pу:pуn'dikjulу]перпендикулярный

вертикальный

perpendicular axes [,pу:pуn'dшkju:lу 'Фksш:s]

перпендикулярная ось

plane, n [pleшn] плоскость

plane, adj [pleшn] плоскоский, ровный

plane geometry ['pleшn dпш'щmшtшi]

планиметрия

polyhedra ['pщlш'hedrу] многогранники

polyhedron ['pщlш'hedrуn] многогранник

precise [prш'saшs] точный

precision [prш'sшпn] точность

proof [pru:f] доказательство

projective geometry [prу'dпektшv dпш'щmшtrш]

планиметрия

proper ['prщpу]присущий,

свойственный

property ['prщpуtш] свойство

prove [pru:v] доказывать

Q

quantitative ['kwщntшteшtшv] количественный

quadratic surface ['kwщdrшtшk 'sу:fis]

квадратичная

поверхность

quadratic equation ['kwщdrшtшk ш'kweшыуn]

квадратичное

уравнение

quantitative ['kwщntшtуtшv] количественный

quantity ['kwщntшtш] величина, количество

R

realm [relm] сфера

rectangular [rek'tФтgjulу] прямоугольный

rectangle ['rek,tФтgуl] прямоугольник

rational approximation

['rФыnуl у'prщksшmeшыn]

пропорциональное

приближение

regular pentagon ['regjulу 'pentуgуn]

правильный

пятиугольник

regular polyhedral solid

['regjulу'pщlш'hedrуl 'sщlшd]

правильное

многогранное

геометрическое тело

relation [rш'leшыуn]отношение,

соотношение

relativity [,rшlу'tшvшtш]относительность,

теория относительности

relate [rш'leшt]устанавливать связь

или отношение между

чем-либо

rational ['reшыуnl] рациональный

ration ['reшыуn]отношение,

соотношение,

коэффициент,

пропорция

real number ['rшуl 'nфmbу] действительное число

S

segment ['segmуnt] сегмент, отрезок

signify ['signifai] значить

solution [s'lju:ыуn] решение

solve [sщlv] решать

square, v [skwУу] возводить в квадрат

square, adv [sшkwУу]в квадрате,

перпендикулярно

squaring ['skwУуrшт] квадратура

subtract [sуb'trФkt] вычитать

subtraction [sуb'trФkыn] вычитание

subtrahend ['sфbtrehуnd] вычитаемое

surface ['sу:fis] поверхность

summarize ['sфmуraшz] суммировать

sum [sфm] сумма

summation [sф'meiыn] суммирование

summary ['sфmуri] суммарный

smooth curve ['smu:е 'kу:v] непрерывная кривая

T

tetrahedron ['tetrу'hedrуn] четырехгранник

tetrahedral ['tetrу'hedrуl] четырехгранный

triangle [traш'Фтgl] треугольник

triangular [traш'Фтglу] треугольный

U

unity [ 'ju:nıtı] единица, число

unsolved [ `ôn`sùlvd ] нерешенный

V

value, n ['vælju:] величина

value, v ['vælju:] ценить

valuation [,vælju'eıû n] оценка

volume ['vщljum] объем

voluminous [vу'lju:mnуs] объемистый

W

within [ within ] внутри, в пределах

X

xi [ `eksøz ] греч. кси (буква)

Y

yield [ ji:ld ] производить

Z

zero [ `zøórou ]нуль,

нулевая точка

3[1] A linear programming problem with mixed constraints will be presented and discussed in detail in Sec. 3-5

4[2] The optimal condition can be also treated by drawing a number of parallel isocost lines. If we move an isocost line as far as

possible to the left from the feasible region, the line will be tangent to one of be corner boundary points B.

34

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