2015-10-22
1282
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 maths a good deal of space will be devoted to R. Descartes and R Fermat, for these men left very deep imprints on many subjects. Also, in the history of maths, 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 indispensable force in the development of the calculus, as the influential power in molding our ideas of such far-reaching concepts as those of "function" and "dimension".
Thus, applied maths in the modern sense of the term was not the creation of the engineer or the engineering-minded mathematician. Of 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, JI Fermat, lived an ordinary life as a lawyer and civil servant, but in his spare time he was busy creating arm offering to the world his famous theorems. The work of both men in many fields will be immortal. 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.
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Vocabulary
| majority | большинство |
| decisive | решающий |
| essential | существенный |
| spirit | дух |
| impetus | толчок |
| familiar | знакомый |
| a good deal of = much | |
| to leave deep imprints | оставить глубокий отпечаток |
| indispensable | необходимый, обязательный |
| influential | влиятельный |
| to mold | формировать, создавать |
| profound | глубокий |
| to offer | предлагать |
| in order to | для того, чтобы |
1. Answer the following questions:
1) What is the two great mathematicians’ contribution to math science?
2) Why is analytic geometry so important?
3) What do you know about each of these great thinkers?
4) Are their methods applicable to all investigations?
5) Are you familiar with their ideas and what do you think they are for mathematics?
6) What did Rene Descartes devote himself to?
2. Find Russian equivalents for the English word combinations:
Profound thinking; the nature of truth; curves and surfaces; indispensable force; decisive contribution; a good deal of space; influential power; the realm of ideas; to offer to the world; immortal work; to generalize and extend the methods; in order to make them applicable; deductive construction; to derive from axioms.
3. Translate from Russian into English.
1) Он посвятил себя критическому и глубокому мышлению о природе истины.
2) Труды Ферма и Декарта бессмертны.
3) Декарт предложил обобщить методы, используемые математиками, что применять их во всех исследованиях.
4) Эти великие люди дали большой толчок развитию аналитической геометрии.
5) Прикладная математика в современном смысле этого термина не была творением инженера или инженерно-мыслящего математика.
4. Find suitable adjectives for the following nouns:
Construction, thinking, ideas, concepts, importance, methods, term, spirit, historian, space, force, contributions, nature, universe.
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Text. 15.
Algebra
Algebra originated in the Middle East. Earlier than 1000 BC, the Babylonians developed an arithmetical system for solving problems that could be written algebraically. This was in advance of other systems, notably that of the Ancient Egyptians, who were able to solve the same problems, but did so by using geometry. The word algebra comes from Arabic and translates into English as reunion. It describes a system of mathematics which performs calculations by firstly rewriting, that is, transposing them, and then reducing them to their simplest form.
Algebra is the branch of mathematics which studies the structure of things, the relationship between things and quantity. It looks different from arithmetic when it is written. Arithmetic uses numbers and the four operators (plus, minus, multiply and divide). Algebra uses symbols, usually letters, and the operators. Actually, it is not very different from arithmetic; what can be done in algebra can be done in arithmetic. There, are good mathematical reasons, however, why algebra is used instead of arithmetic.
Firstly, by not using numbers, mathematicians are able to set out arithmetical laws. In this way they are able to understand the system of numbers more clearly. Secondly, by using algebra, mathematicians are able to perform calculations where unknown quantities are involved. This unknown is usually represented by x. Solutions can then be applied not just to the immediate problem, but to all problems of the same nature by the use of a formula. A common algebraic problem to solve in school exams would be, for example: find x where 3x + 8 = 14. A third reason for the use of algebra rather than arithmetic is that it allows calculations which involve change in the relationship between what goes into the problem and what comes out of it, that is, between input and output. It is an algebraic formula which allows a business to calculate its potential profit (or loss) over any period of time.
It is possible to classify algebra by dividing it into four areas. Firstly, there is elementary algebra in which symbols (such as x and y, or a and b) are used to denote numbers. In this area, the rules that control the mathematical expressions and equations using these symbols are studied. Then, there is abstract or modem algebra in which mathematical systems consisting of a set of elements and several rules (axioms) for the interaction of the elements and the operations are defined and researched. Thirdly, there is linear algebra (linear equations) in which linear transformations and vector spaces, including matrices, are studied. Finally, there is universal algebra in which the ideas common to all algebraic structures are studied.
Like all branches of mathematics, algebra has developed because we need it to solve our problems. By avoiding the use of numbers we are able to generalise both the problem and the solution.
Vocabulary
| to originate | происходить, брать начало |
| to be in advance | опережать |
| notably | исключительно, особенно, весьма |
| reunion | воссоединение |
| to transpose | переносить в другую часть уравнения с обратным знаком |
| to reduce | сводить к ч.-л. (зд.) |
| to look different from | отличаться от |
| instead of | вместо |
| to set out laws | излагать законы (правила) |
| in this way | таким образом |
| rather than | а не; скорее … чем |
| to allow | разрешать, позволять |
| to denote | обозначать |
| to consist of | состоять из |
| to avoid | избегать |
1. Answer the following questions:
1) Where and when did algebra originate?
2) In what way does it differ from geometry?
3) What does algebra describe?
4) How can we classify algebra?
5) When did you begin learning algebra and how did you do in algebra at school?
6) What does linear algebra deal with?
7) What is modern algebra concerned with?
8) Is algebra or geometry more difficult for you?
2. Find the English equivalents for the Russia ones in the right column.
| состоять из | to apply to the immediate problem |
| решать проблему | common to all structures |
| несколько правил | in advance of other systems |
| общие для всех структур | over any period of time |
| применять к непосредственной проблеме | to consist of |
| отношения между вещами | relationships between things |
| раньше других систем | to calculate profits and losses |
| за какой-нибудь период времени | to perform calculations |
| высчитать прибыль и потери | to solve a problem |
| сделать вычисления | several rules |
3. Try to remember the verbs:
to solve, to include, to consist of, to come out of, to perform, to avoid, to denote, to define, to describe, to apply, to represent, to allow, to interact, to use, to transpose, to involve.
4. Pair work. Discuss the text. Tell each other what interesting and unknown to you things you have found in the text.
Text. 16.
Algebra
Algebra is a branch of mathematics that uses variables to solve equations. When solving an algebraic problem, at least one variable will be unknown; Using the numbers and expressions that are given, the unknown variable(s) can be determined. Early Algebra.The history of algebra began in ancient Egypt and Babylon, the Rhind Papyrus, which dates to 1650 В. С. E..provides insight into the types of problems being solved at that time. The Babylonians are credited with solving the first quadratic equation, the Babylonians were also the first people to solve indeterminate equations, in which more than one variable is unknown. The Greek mathematician Diophantus continued the tradition of the ancient Egyptians and Babylonians into the common era. Diophantus is considered the "father of algebra," and he eventually furthered the discipline with his book Arithmetka. In the book he gives many solutions to very difficult indeterminate equations.
By the ninth century, an Egyptian mathematician, Abu Kamil, had stated and proved the basic laws and identities of algebra. In addition, he had solved many problems that were very complicated for his time Medieval Algebra. During medieval times. Islamic mathematicians made great strides in algebra. They were able to discuss high powers of an unknown variable and could work out basic algebraic polynomials. All of this was done without using modern symbolism. In addition, Islamic mathematicians also demonstrated knowledge of the binomial theorem.
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The study of algebra went on to become more interdisciplinary as people realized that the fundamental principles could be applied to many different disciplines. Today, algebra continues to be a branch of mathematics that people apply to a wide range of topics. Current Status of Algebra. Today, algebra is an important day-to-day toolkit is not something that is only used in a math course. Algebra can be applied to all types of real-world situations. For example, algebra can be used to figure out how many right answers a person must get on a test to achieve a certain grade.
Vocabulary
| to determine | определять |
| expression | выражения |
| to provide insight into | дает понимание |
| to be credited with | приписывать к-л. ч-л. |
| indeterminate | неопределимый, переменный |
| to further | продвигать |
| to state | формулировать, излагать |
| identity | тождество |
| complicated | сложный |
| medieval | средневековый |
| to make strides | делать успехи |
| power | степень |
| range | область, сфера, круг |
| variable | переменная величина |
1. Answer the following questions:
1) How can the unknown variables be determined?
2) What types of problems could ancient mathematicians solve?
3) What is the contribution of Diophantus to maths?
4) What principle of math could b applied to many different disciplines?
5) What other professions use algebra and in what way?
6) What is the future of algebra?
2. Find in the text English equivalents for the Russian word combinations:
кроме того, применять к разным дисциплинам, неизвестная переменная величина, нерешенные уравнения, излагать законы, современная алгебра, приписывать решение уравнений, продолжать традицию, вычислить сколько правильных ответов, достичь успехов, заказывать продукты, бесчисленные примеры, отвечать (подходить к) нуждам людей, помогать в изучении.
3. Fill in the necessary word.
1) In the book he gives many ___ to very different indeterminate ___.
2) Today, algebra ___ to be a branch of mathematics.
3) When companies ___ budgets, algebra is used.
4) It will continue ___ to better suit peoples’s needs.
5) The Babylonians ___ with solving the first ___ equation.
6) An Egyptian mathematician had stated and ___ the basic ___ and ___ of algebra.
4. Humor to Fit any occasion. Can you understand it?
He who devotes sixteen hours a day to hard study may become as wise at sixty as he thought himself at twenty.
Mary Wilson Little.
Text. 17.
LINEAR ALGEBRA
Linear algebra like several other math disciplines may be considered from two different points of view: 1) as a branch of maths of independent interest with a development and with problems of its own; 2) as a tool for other math disciplines and for math physics, A large class of math problems is generally called "linear", the simplest problem is the following: Let a and b be two given (real or complex) numbers - to find a number x that satisfies the equation ax=b. The problem has a unique solution x, if, and only if, a ^0; no solution at all if a=0, b^0; an infinity of solutions, viz., all real (or complex) x if a=0 and b=0. This statement comprises the whole theory of the problem.
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One of the main technical devices of linear algebra is the theory of determinants and for a long time it has been the only part of linear algebra which was studied systematically. This is the more surprising as the notion of matrix — the main object of modem linear algebra — is evidently more fundamental than that of a determinant because a determinant is only a certain number associated with a given square matrix. Whilst the notion of determinant was discovered by Leibnitz (c. 1690), the notion of matrix appeared only much later in 1854 in a paper by Cayley and independently in 1867 in a paper by Laguerre. Since then linear algebra and matrix calculus have developed into a vast domain of maths closely connected with a good many other math branches, such as the theory of groups, the theory of invariants, tensor calculus, the theory of systems of differential equations, etc. Linear algebra provides the methods of proof as well as the adequate algebraic formalism for a considerable part of analytic geometry. It has served as a model in recent developments of analysis (theory of integral equations and of linear transformation in infinite- dimensional spaces) which have considerably advanced this branch of maths and proved important in modern physics.
Vocabulary
| to consider | рассматривать |
| point of view | точка зрения |
| own | собственный |
| to satisfy | удовлетворять, отвечать ч-л. |
| unique | уникальный |
| statement | утверждение |
| to comprise | включать в себя, охватывать |
| main | главный, основной |
| certain | определенный |
| square | квадрат |
| independently | независимо от |
| since then | с тех пор |
| a vast domain | огромная область |
| to be closely connected | быть тесно связанным |
1. Answer the following questions:
1) What do we call “linear” in mathematics?
2) What is linear algebra connected with?
3) In what way are analytic geometry and algebra connected?
4) What definition can be applied to linear algebra?
5) What statement comprises the whole theory of the problem?
2. Translate from Russian into English.
1) Линейная алгебра превратилась в огромную область математики.
2) Она очень тесно связана со многими другими областями математики.
3) Один из главных технических механизмов линейной алгебры является теория детерминантов.
4) Эта теория долгое время была единственной частью линейной алгебры, которую систематически изучали.
5) Главная цель современной линейной алгебры очевидно более основательная, чем теория детерминантов.
3.Work in pairs. Discuss the problem how linear algebra developed into present form.
PART III
TESTS
The verb “to be”
1.My brother … in good health.
A) am B) is C) are
2. He … at school yesterday.
A) are B) were C) was
3. … she at home two hours ago?
A) was B) were C) shall be
4. Her parents … poor, she must earn her living herself.
A) was B) am C) are
5. … she … a student in a year?
A) shall be B) will be C) were
6. Nick … very busy today.
A) is B) am C) will be
7. … the weather fine?
A) was B) were C) am
8. I … in Moscow in two weeks.
A) will be B) am C) shall be.
9. My address … not difficult to remember.
A) am B) is C) are
10. This book … of no interest to us.
A) are B) am C) is
11. The paper … on the shelf yesterday.
A) were B) was C) are
12.We … students of the medical Institute soon.
A) will be B) shall be C) were
13.Helen … not in the street yesterday.
A) were B) was C) are
14.This little boy … five years old.
A) is B) are C) am
15. … Baikal the deepest lake in the world?
A) am B) are C) is
16. She …a old sick woman.
A) am B) is C) are
17. They … at the station last night.
A) are B) was C) were
18. The mountains … not very high in Great Britain.
A) is B) are C) was
19. … this flat comfortable?
A) is B) are C) am
20. Olga and Nick … at the station in 2 hours.
A) shall be B) will be C) were
21. They … good doctors in 6 years.
A) shall be B) will be C) are
22. She … ill and could not go there.
A) is B) was C)were
23.It … spring soon.
A) will be B) shall be C) is
24. Olga … very busy tomorrow.
A) is B) was C) will be
The verb “to have”
(have, has, had, will have, shall have)
1. What flat… your friend…?
A) do … have B) does have C) have
2. These students … English books.
A) has B) have C) do have
3. He … … breakfast in the morning.
A) don’t have B) have C) doesn’t have
4. Now his parents … a comfortable flat.
A) have B) has C) had
5. They … English yesterday.
A) has B) will have C) had
6. These students … … five examinations next year.
A) will have B) had C) does … have
7. … you … more or less free time now then you … last year?
A) does … have B) had C) do … have
8. We … no lectures on Sunday.
A) has B) don’t have C) have
9. They usually … their entrance examinations in June.
A) have B) has C) will have
10. This toy dog … a very interesting story.
A) have B) has C) don’t have
11. He … … any news from his friend.
A) didn’t have B) shan’t have C) had
12. They … … plenty of time to relax in summer.
A) had B) will have C) doesn’t have
