Module 8 mathematics

Texts

BT1. Mathematics

ST 1. The language of mathematics

ST 2. A quotation from Alfred North Whitehead

I. Робота в парах. Прочитайте та перекладіть подані інтернаціональні слова:

algebra	['ælʤibrə]
arithmetic	[ə'riθmətik]
axiom	['æksiəm]
form	[fo:m]
fundamental	[,fʌndə'mentl]
geometry	[d3i'omitri]
mathematics	[,mæθi 'mætiks]
number	['nʌmbə]
operation	[,ope'rei∫ən]
position	[pə'zi∫n]
postulate	['postjulit]
real	[`riəl] _
stereometry	[stiəri'omitri]
theorem	[`θiərəm]
trigonometry	[,trigə`nomitri]

II. Робота в парах. Прочитайте та перекладіть наступні англійські слова. Порівняйте їх з українськими словами, які мають той же корінь.

algebra arithmetic axiom form fundamental geometry mathematics number operation position postulate real stereometry theorem trigonometry

алгебра арифметика аксіома форма, утворювати фундаментальний геометрія математика номер, число операція, дія позиція, положення постулат реальний, дійсний стереометрія теорема тригонометрія

III. Прочитайте подані слова та запам'ятайте їх значення.

1. admit	[əd'mit]	допускати
2. concern	[kən'sə:n]	стосуватись, відноситись
3. deal	[di:l]	мати справу
4. dimension	[di'men∫n]	вимір, величина
5. equation	[ik'wei∫n]	рівняння
6. magnitude	['mægnitju:d]	величина
7. mean	[mi:n]	означати
8. originate	[ə'riʤineit]	давати початок
9. prove	[pru:v]	доводити
10. proof	[pru:f]	доказ
11.solve	[solv]	вирішувати, розв'язувати
12. solution	[sə'lu:∫n]	розв'язок
13. statement	['steitmənt]	твердження
14. take	[teik]	брати
15. treat	[tri:t]	трактувати

IV. Прочитайте та перекладіть базовий текст №1 (BТ 1).

MATHEMATICS

1. Mathematics is the science of numbers, their combinations and the forms of bodies.

2. The word «mathematics» takes its origin from the Greek word «matema» meaning «science».

3. Mathematics is divided into several branches, such as arithmetic, algebra, geometry, and so on.

4. Arithmetic is the science of real numbers and fundamental operations with them.

5. The fundamental operations are: addition, subtraction, multiplication and division.

6. Algebra deals mainly with the solution of equations.

7. Geometry is the science of position, form and magnitude.

8. Plane geometry (planimetry) is the science which treats plane figures.

9. It is geometry of two dimensions.

10. Solid geometry (stereometry) is the science which treats figures in space.

11. It is geometry of three dimensions.

12. Mathematical theories are based on the simplest notions and principals which are called axioms, theorems and postulates.

13. An axiom is a statement admitted to be true without proof.

14. A theorem is a statement to be proved.

15. A postulate is a statement admitted to be possible.

V. Назвіть дієслова, які закінчуються на:

• глухий приголосний, крім [t]

• голосний, або дзвінкий приголосний, крім [d]

• звуки [t], або [d]

divide	research	prove
treat	base	form
found	develop	concern
admit	force	solve

VI. Прочитайте англійські дієприкметники (А,В,С). Зверніть увагу на читання закінчення. Виберіть відповідний український еквівалент.

А.

researched	вимушений
based	досліджений
forced	розвинутий
developed	заснований

В.

proved	віднесений
concerned	утворений
formed	доведений
solved	вирішений

С.

admitted	заснований
treated	трактований
founded	поділений
divided	допущений

VІI.Знайдіть у тексті наступну інформацію

1. Mathematics, its branches.

2. What does mathematics (arithmetic, algebra, geometry, planimetry, stereometry) study?

3. Simplest mathematical notions: theorem, axiom, postulate.

4. Word «mathematics», its origin.

5. Arithmetic (algebra, geometry) as a branch of mathematics.

VІІI.Дайте відповідь на запитання.

1. What is mathematics?

2. What is the origin of the word "mathematics"?

3. Into what branches is mathematics divided?

4. What is arithmetic?

5. What are the fundamental arithmetical operations?

6. What does algebra deal with?

7. What is geometry?

8. What is plane geometry?

9. What is solid geometry?

10. What are mathematical theories based on ?

11. What is an axiom?

12. What is a theorem?

13. What is a postulate?

IX. Перекладіть наступні інтернаціональні слова:

basic	character	abbreviation
characteristic	civilized	concept
communication	compactness	aspect
confuse	construct	definition
determine	economy	express
familiar	formal	idea
general	letter	numeral
ordinal	phrase	reason
person	political	presentation
product	royal	specialize
style	supernational	base
symbol	student	term
transition	verbally

Х. Прочитайте та перекладіть додатковий текст №1(SТ1).

THE LANGUAGE OF MATHEMATICS

What distinguishes the language of science from language we ordinarily understand the word? How is that scientific language international? The supernational character of scientific concepts and scientific language is due to the fact they are set up by the best brains of all countries and all times

A. Einstein.

One of the foremost reasons given for the study of mathematics is, to use a common phrase that "mathematics is the language of science". This is not meant to imply that mathematics is useful only to those who specialize in science. No, it means that even a layman must know something about the foundations, the scope and the basic role played by mathematics in our scientific age.

The language of mathematics consists mostly of signs and symbols and, in a sense, is an unspoken language. There can be no more universal or more simple language, it is the same throughout the civilized world, though the people of each country translate it into their own particular spoken language. For instance, the symbol 5 means the same to a person in England, Italy or any other country; but in each country it may be called by a different spoken word. Some of the best known symbols of mathematics are the arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and the signs of addition (+), subtraction (-), multiplication (x), division (:), equality (=) and the letters of the alphabets: Greek, Gothic and Hebrew (rather rarely).

Symbolic language is one of the basic characteristic of modern mathematics for it determines its true aspect. By the aid of symbolism mathematicians can make transitions in reasoning almost mechanically by the eye and leave their mind free to grasp the fundamental ideas of the subject matter. Just as music uses symbolism for the representation and communication of sounds so mathematics expresses quantitative relations and special forms symbolically. Unlike the common language, which is the product of custom, as well as a social and political movements, the language of mathematics is carefully and often ingeniously designed. By virtue of compactness, it permits a mathematician to work with ideas which expressed in terms of common language are unmanageable. This compactness makes for efficiency of thought.

Mathematical language is precise and concise, so that it is often confusing to people unaccustomed to its forms. The symbolism used in mathematical language is essential to distinguish meanings often confused in common speech. Mathematical style aims at brevity and formal perfection.

Let us suppose we wish to express in general terms the Pythagorean theorem, well-familiar to every student through his high-school studies. We may say: "We have a right triangle. If we construct two squares each having an arm of the triangle as a side and if we construct a square having the hypotenuse of the triangle for its side, then the area of the third square is equal to the sum of the areas of the first two".

But no mathematician expresses himself that way. He prefers: "The sum of theе squares on the sides of a right triangle equals the square on the hypotenuse. In symbols this may be stated as follows: C² = A² + B² .This economy of words makes for conciseness of presentation, and mathematical writing is remarkable because it encompasses much in few words. In the study of mathematics much time must be devoted 1) to the expressing of verbally stated facts in mathematical language, that is, in the signs and symbols of mathematics;2) to the translation of mathematical expressions into common language. We use signs and symbols for convenience. In some cases the symbols are abbreviations of words, but often they have no such relation to the thing they stand for. We cannot say why they stand for what they do, they mean what they do by common agreement or by definition.

A student must always remember that the understanding of any subject in mathematics presupposes a clear and definite knowledge of what precedes. This is the reason why "there is no royal road to mathematics and why the study of mathematics is discouraging to weak minds, those who are not able and willing to master the subject".

XІ. Дайте відповідь на наступні запитання.

1. Why is it so important to know mathematics?

2. What is the distinction between common language and the language of mathematics?

3. What are the best known signs and symbols in mathematics?

4. What do we use signs and symbols for?

5. What are the main characteristics of the language of science?

6. How do mathematicians prefer to express themselves in their mathematical writing?

7. What relation may the symbols have to the things they stand for?

8. What must students of mathematics always remember?

9. Why is the study of mathematics sometimes discouraging people?

10. What are the basic operations of arithmetic?

11. How is the language of mathematics designed?

12. How can mathematicians make transitions in reasoning?

13. What does mathematical style aim at?

XII. Прочитайте та перекладіть додатковий текст 2 (ST 2), звертаючи увагу на спеціальну лексику уроку.

A QUOTATION FROM ALFRED NORTH WHITEHEAD

“Mathematics is often considered a difficult and mysterious science because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than a symbolism which we do not understand. Also, a symbolism, which we only partially understand and are unaccustomed to use, is difficult to follow. In exactly the same way the technical terms of any profession or trade are incomprehensible to those who have not been trained to use them. But this is not because thеу are difficult in themselves. On the contrary they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification. It is not only of practical use, but of great interest. For it represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other. If anyone doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of the following equation which represents one of the fundamental laws of Algebra: X+V=V+X. Without symbols, this becomes: "If a second number be added to any given number the result is the same as if the first given number had been added to the second number (commutative law for addition)".

"This example shows that, by the aid of symbolism, we can, make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain".

"It is a profoundly erroneous truism, repeated by all copybooks and еminent people when they are making speeches that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments".

XIII. Дайте відповіді на наступні запитання.

1. Why mathematics is considered a difficult and mysterious science?

2. What is the language of mathematics?

3. What was the symbolism introduced for?

4. What kinds of transitions can we make by the aid of symbolism?

XIV. Анотуйте англійською мовою основний зміст додаткових текстів. Користуйтесь українським варіантом як ключем.

Supplementary texts (STI, ST2) provide information on.........................................

В додаткових текстах (STI, ST2) йдеться про математику як мову науки. Її часто вважають важкою і загадковою наукою. Причиною цього є те, що деякі математичні терміни і символи є незрозумілими для тих, хто непідготовлений до їх використання. Насправді вони не є важкими самі по собі. Навпаки, вони впроваджені для спрощення речей. Математична мова є точною і стислою. Немає більш універсальної або більш npocтої мови. Вона однакова в усьому цивілізованому світі. За допомогою математичних символів і термінів можливою є передача великого обсягу інформації засобом небагатьох слів.