The Natural Numbers

1. So far we have encountered several different systems that are capable of representing the natural numbers. One system consists of the common Arabic numerals, with their addition and multiplication tables. A second one consists of the Roman numerals, with their tables. The third system consists of points on a line and the appropriate constructions for adding or multiplying them. This variety of representations raises the question "What is the natural number system?"

2. We may try to answer this question by listing some characteristics that all these number systems have in common. All three, for example, obey the five laws. But this is not an adequate answer, because all number systems as we have defined the term, will obey these five laws. And we intend to produce some number systems that are not interchangeable with the natural number system at all. To define the natural number system, we must list not merely characteristics that all of its representations have in common. We must note particularly its distinguishing characteristics. This is done by choosing the defining characteristics in such a way that all systems that have these characteristics must be isomorphic to each other. Such a section of charac­teristics that effectively defines one and only one structure is called a system of axioms for the structure. Here is a system of axioms for the natural number system (not including 0), first formulated by the Italian mathematician G. Peano.

3. A set of elements is called a natural number system if it has the following characteristics:

(1) It contains an element called 1.

(2) For every member in the system, there is another member (and only one) called its successor.

(3) Two distinct members do not have the same successor.

(4) There is no member of the system that has 1 as its successor.

(5) If a set of elements belonging to the system contains 1, and, for each member that it contains, also contains its successor, than this set contains the whole system.

4. Notice that addition and multiplication are not mentioned in these axioms at all. G. Peano defined these operations in terms of his axioms as follows: For any natural numbers x and y,

let x+l = the successor of x;

let x+ (the successor of y)= the successor of (x+y);

let x 1 = x

let x (the successor of y) = x y + x

With these definitions it is possible to prove that the natural number system obeys the five laws.

5. What G. Peano did for the natural number system is typical of the way in which mathematical structures are studied today. In modern mathematics, a mathematical structure is often defined as a set of objects that satisfies a specified set of axioms.

Questions:

1. What does the first system consist of? 2. How many laws do all three systems obey? 3. What must we do to define the natural number system? 4. What are the characteristics of a natural number system? 5. How did G. Peano define addition and multiplication? 6. How is a mathematical structure defined in modern mathematics?

II. Письменно переведите 2 и 5 абзацы текста.

III. Поставьте специальные вопросы к следующим предложениям.

1. Ordinary people communicate by means of the formalized language of maths (How…?). 2. The students of our group have got two exams today (Who …?). 3. Algebraists learnt to think in terms of equations (In what way …?). 4. There are seven faculties for sciences at our university (How many …?). 5. We will be considering complex numbers next term (When …?). 6. We are substituting unknowns for irrationals to get a right result (Why …?).

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

1. The teacher (to explain) how to mark off the line with numbers right now.

2. The graph (to approach) the axis of x, but never (to reach) it. 3. They (to start) a new series of experiments next week. 4. We (to go) down in the lift when it suddenly stopped. 5. Marie and Pierre Curie (to discover) radium and in 1903 (to win) the Nobel Prize. 6. When he (to have) a problem to solve, he will work at it until he (to find) an answer.

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

1. This problem is (difficult) than the first problem. 2. The (long) he refuses to recognize the impossibility of the solution, the (bad) for him. 3. The contribution of the ancient Greeks to geometry is much (great) than the formulas of the Egyptians. 4. This year’s exam was (difficult) than last year’s. 5. If you need any (far) information, call the office. 6. Their house is (old) in the village.

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

1. Do you know how far (there is/it is/it was) from here to the university? 2. (There will be/It will be/There is) a supermarket opposite the park in two years. 3. Why not take a taxi? (There is/It is/Is it) a long way from your house to the airport. 4. Last Monday (it was/it wasn’t/there was) a party next door. (There was/It wasn’t/It is) easy for me to get to sleep. 5. (There wasn’t/It wasn’t/There weren’t) anything interesting in yesterday’s news programmes. 6. (It is/There is/There was) ten miles to the nearest petrol station.

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

1. Let us consider this sequence. 2. Let the radius of the circle be R. 3. Let C denote a subfield of the set F of complex numbers. 4. Let me see what you are doing. 5. Let the above condition hold in this case. 5. Let there be no doubt in your minds about our intentions.

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

1. Her new car is really nice, but I don’t like (her/it’s/its) colour. 2. Do you think that most people are happy in (they/their/theirs) jobs? 3. Where did you spend (your/yours/his) holiday? 4. Last night I went out for a meal with a friend of (mine/me/my). 5. This is their computer, that computer is (them/their/theirs) too. 6. We are going to the cinema. Why don’t you come with (our/us/we)?

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

1. If a curve is symmetric with respect to both (axis/axes/axeses), is it symmetric with respect to the origin? 2. All these facts may serve as reference (datum/data/datas). 3. Analytic methods give us a means of finding the equations of (locus/locuses/loci). 4. There (was/were/will be) three symposia held on the problems of pollution last year. 5. If you go to bed late, (there is/it is/there will be) difficult to get up early in the morning. 6. The economic crisis of the 1990s (were/are/was) the heaviest one for our country.