Unit 3. Texts for extracurricular work

Date: 2015-10-07; view: 349.

Task 6. Translate into English

Task 5. Determine the fallacy in the given statements

1. A right angle is equal to an angle which is greater than a right angle.

2. A part of a line is equal to the whole line.

3. The sum of the lengths of two sides of any triangle is equal to the length of the third side.

4. Every triangle is isosceles.

5. Every ellipse is a circle.

6. If two opposite sides of a quadrilateral are equal, the other two sides must be parallel.

1. Геометрия занимается построениями и изучением свойств и отношений геометрических фигур и тел.

2. Геометрические фигуры определяются через понятие множества точек.

3. Геометрические построения должны выполняться только с помощью циркуля и линейки.

4. Начальное понятие геометрии – точка. Точка в геометрии не имеет измерения. Точка – это определенное положение в пространстве.

5. Отрезки прямой, лучи, углы, круги, и треугольники – это геометрические фигуры на плоскости, т.е. множество точек, лежащих в одной плоскости.

6. Существуют различные типы углов, треугольников и многоугольников.

7. Мы находим площади геометрических фигур умножая длину на ширину.

8. Многогранники -- это геометрические тела, каждая грань которых многоугольник.

Text 1. Mathematics is the Queen of natural knowledge (3350 characters)

Mathematics grew up with civilization as man's quantitative needs increased. It arose out of practical and man's needs. As soon as man began to count even on his fingers mathematics began. It was the first of sciences to develop formally. It is growing faster today than in its early beginnings. New questions are always arising partly from practical problems and partly from pure theoretical problems. In each generation men have developed new methods and ideas to solve these problems.

Why has mathematics become so important in recent years? Can the new electronic brains solve mathematical problems faster and more accurately than a person and eliminate the needs for mathematicians?

To answer these questions we need to know what mathematics is and how it is used. Mathematics is much more than the arithmetic, which is the science of number and computation. It is algebra, which is the language of symbols, operations and relations. It is much more than the geometry, which is the study of shape, size and space. It is much more than the statistics, which is the science of interpreting data and graphs. It is much more than the calculus, which is the study of change, limits, and infinity. Mathematics is all these and more.

Mathematics is a way of thinking and a way of reasoning, where new ideas are being discovered every day. It is way of thinking that is used to solve all kinds of problems in the government and industry.

Mathematics method is reasoning of the highest level known to man and every field of investigation – be it law, politics, psychology, medicine or anthropology – has felt its influence.

There are various ways in which mathematics serves scientific investigation:

1. Mathematics supplies a language for the treatment of the quantitative problems of the physical and social sciences.

2. Mathematics supplies science with numerous methods and conclusions.

3. Mathematics enables the science to make predictions.

4. Mathematics supplies science with ideas to describe phenomena.

The language of mathematics is precise and concise; it is a language of symbols that is understood in all civilized nations of the world. Mathematics style aims at brevity and formal perfections.

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase «crisis of foundations» describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Some disagreement about the foundations of mathematics continues to present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gцdel's incompleteness theorems which (informally) imply that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gцdel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.

Task 1. Fill in the table with the English equivalents of mathematical concepts from the text:

Text 2.Fields of mathematics(4820 characters)

Pure mathematics. Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Quantity.Thestudy of quantity starts with numbers, first the familiar natural numbers and integers («whole numbers») and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.

As the number system is further developed, the integers are recognized as a subset of the rational numbers. These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of «infinity». Another area of study is size, which leads to the cardinal numbers, and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Structure.Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Space. The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincarй conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.

Change. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.