Numbers systems of Mathematics

1. Mathematicians study numbers and develop new number systems in a specific field of maths — number theory — which is the oldest and purest branch of maths. The generators of classical number theory - the ancient Greek mathematicians — studied numbers with no immediate applications in mind. The main value of numbers for them was that “numbers are amusing and challenging to the human imagination” and they assigned all kinds of mysterious meanings and interpretations to numbers: the number 2 for them stood for female, 3 stood for male, 4 — for justice, 5 - for marriage because it is the union of the first odd and the first even number, etc.

2. Although applica­tions were not the main objective (aim) of the classical number theory, Greek investigators discovered many curious and fascinating number properties and gave birth to theoretical pure maths. They were the first to formulate the abstract notion of “number” that constituted a grand advance of the human intellect. The positive integers or natural numbers were the foundation of all classical maths and the major ancient Greeks’ thesis was that “number is the essence of reality, that scientists should study nature quantitatively and express the results in terms of math laws (rules) and theories”.

3. In maths there exist various ways to study numbers — one way of further extension, generalization and synthesis when mathematicians build up number concepts of great complexity and generality. Another method is analysis when mathematicians arrive at the essence of numbers; when they break down the complexities and study the original primitive positive integers and their properties. Both ways are of great importance.

4. Nowadays mathematicians separate the number systems of maths into five principal stages. Each stage has got a long history of its development and recognition. They are: 1. The system of natural numbers or positive integers only; 2. The next stage comprises positive as well as negative integers and zero; 3. The rational numbers which combine integers and fractions; 4. The real numbers that include irrational numbers such as π. 5. The complex numbers that contain the so-called “imaginary” number . In modern maths there are several new number systems. Of these modern systems three occupy an exceptionally significant place within maths, viz. (namely), quaternions (triplets), matrices and transfinite numbers.

5. Some comments are necessary, indeed. The word “rational” does not mean “reasonable” — it comes from the word ratio or quotient of two integers. Don't think that the word “imaginary” means that these numbers are mystical or unreal in the everyday sense of the word, or that “complex” means “complicated’. Imaginary numbers and complex numbers have had very “real” applications in many branches of maths and science. It is interesting to mention that the number 0 (zero) originally had signified an empty place only. Modern mathematicians recognize zero as any other number and not just a symbol for an empty space. Zero is a meaningful math object with the properties defined by a set of rules. Zero is neither “more real” or “less real” than any other number.

Questions:

1. What field of maths deals with the number theory? 2. Who first formulated the abstract notion of “number”? 3. What methods of studying numbers do there exist? 4. How many principal stages do mathematicians separate the number system into? Enumerate them. 5. What new number systems do you know? 6. Can we say that the number 0 (zero) signifies an empty place only? Why?

II. Письменно переведите 2 и 5 абзацы текста.

III. Поставьте специальные вопросы к следующим предложениям.

1. Other writers define a point as a pure position (Who…?). 2. Euclid constructed a whole system of geometry from these few evident axioms (What…?). 3. There were three important statements in the last article (How many…?). 4. Distance education has got a lot of implications (What…?). 5. She will represent this mathematical relation by a formula (How…?). 6. Now we are adding two numbers to check the result (Why…?).

IV. Раскройте скобки, поставив глаголы в нужном по смыслу времени. Переведите письменно предложения на русский язык.

1. He (to make) a report on analytical methods next month. 2. She (to wear) a beautiful black and white silk blouse now. 3. It all happened while we (to live) in Bristol. 4. I will join you as soon as I (to get) a note from you. 5. My friend (to move) to a new flat next week. 6. This time next month the students of our group (to take) an exam in geometry.

V. Раскройте скобки, употребив нужную степень сравнения прилагательног o. Переведите письменно предложения на русский язык.

1. Air travel is (expensive) than any other form of modern transport. 2. The (soon) you take your medicine, the (good) you will feel. 3. This definition of an angle is (precise) than that one. 4. The weather in this part of the country is much (cold) than anywhere else. 5. What we need is a (good) job. 6. What is the (quick) way of getting from here to the station?

VI. Раскройте скобки, выбрав правильный вариант. Переведите письменно предложения на русский язык, обращая внимание на разницу в переводе предложений, содержащих it или there.

1. (There will be/There are/It is) about 80 different types of plastics in the world at present. 2. (It wasn’t/There wasn’t/There weren’t) easy to find your house yesterday. 3. When you come tomorrow, (there is/there will be/it will be) your friend at the station meeting you. 4. (There are/There is/It is) too much sugar in the tea, I can’t drink it. 5. Economic crises (is/are/were) the most characteristic features of modern civilization. 6. (It is/There is/It will be) a photograph of the village where my parents were born.

VII. Переведите письменно на русский язык следующие предложения, обращая внимание на побудительную форму повелительного наклонения с глаголом let.

1. Let us determine the algebraic properties of these numbers. 2. Let them consider three types of motion. 3. Let me help you with your bag. 4. Let this assumption be false. 5. Let the children go to bed early during the week. 6. Let the same results hold for negative numbers.

VIII. Раскройте скобки, выбрав подходящее местоимение. Переведите письменно предложения на русский язык.

1. The company has offices in many places but (it/its/her) head office is in Paris. 2. Thank you for (yours/your/you) letter, I was glad to hear from (him/hers/you).

3. Some people talk about (them/their/theirs) jobs all the time. 4. This watch is a gift from my uncle, he gave it to (my/mine/me) last year. 5. We were staying in a nice hotel, (its/our/his) room was very comfortable. 6. She told me of a friend of (she/her/hers) who was not reliable.

IX. Раскройте скобки, выбрав правильный вариант. Вспомните правила образования множественного числа существительных латинского и греческого происхождения. Переведите письменно предложения на русский язык.

1. There (is/are/was) two foci in a regular oval, called an ellipse. 2. The conic sections will be defined with reference to a (foci/focus/focuses) and a (directrix/directrices/derectrici). 3. The notion of a four dimensional geometry is a very helpful one in studying physical (phenomenon/phenomena/phenomenons). 4. The common point of two rays (are/is/were) a vertex of the angle. 5. The length of a diameter is equal to twice the length of (a radius/radii/radiis). 6. A cylinder is a circular prism, the bases of which (is/are/was) equal circles that are parallel to each other.