The Advantages and disadvantages of certain numeration systems

1. The concept of a number did not appear all of a sudden. Scientists do not have enough evidence to fix the period in history of the invention or dis­covery of cardinal numbers. The origin of number and counting is hidden behind countless prehistoric ages. The earliest documents available show that the number concept is equally present in many ancient civilizations. Counting represents a very important milestone in the progress of civilization. The first requirement in computation is a system of numerals, i.e., a way to write numbers.

2. Numeration first evolves through the use of spoken and later on written languages. Some ancient tribes used a base of 2 and 3 to count by (1-2, 2-1,2-2) (1-2-3, 3-1, 3-3). Historical records give evidence of the astronomical and arithmetical achievements of the ancient Babylonians, Sumerians and Chinese. Sometime before 2000 B.C. the Babylonians developed a base-sixty or sexagesimal system of numeration with the positional principle which is still useful in astronomical calculations. The Babylonians of 2000 B.C. were well-trained and skillful calculators.

3. The early Egyptian numeration system used a base often with no more than three symbols to express any number less than 100 — one for units, one for tens, and one for hundreds. The zero symbol is unnecessary. The ancient Greeks’ nonpositional numeration system employs twenty-four letters of their alphabet to produce letter-numerals and special symbols (M = myriad) for large numbers. To tell a number from a word the ancient Greeks used an accent (stress) at the end of a number sign or a stroke over it. The traditional Chinese-Japanese numeration system is a base-ten system with nine numerals and symbols for the place value. Numbers go from the top downward or from left to right. The Mayan numeration system (400 A.D.) uses the base twenty with positional notation and a special symbol for zero.

4. The present-day number-symbols are Hindu characters, but the details of the exact formation of the Hindu-Arabic symbolic system are missing. Our number system uses only the symbols 0, 1, 2... 9; it has base ten and positional notation. Thus any integer can be expressed with these symbols in various combinations and arrangements. It is not known when or by whom zero (nought) was invented. Historians think that zero was introduced by the Hindus or the Babylonians not later than in the ninth century A. D. and probably as early as the second century В. С. The invention of zero and our number system is one of the greatest achievements of the human race, without which the progress of science, industry, and commerce could be impossible. This new system was introduced in Europe by the Arabs, or Moslems, at about the beginning of the tenth century. These new numbers were used, and finally, after about five centuries, the decimal system won the battle.

5. Binary system is of recent origin and extremely important in cybernetics. It needs only a sequence of two digits, 0 and 1, to represent numbers of any size.

Questions:

1. When did the cardinal numbers appear? 2. What is the first requirement in computation? 3. Was the Greeks’ numeration system positional or nonpositional? 4. In what way did the ancient Greeks tell a number from a word? 5. How many digits does the binary system need to represent numbers of any size? 6. By whom was zero invented?

II. Письменно переведите 1 и 4 абзацы текста.

III. Поставьте специальные вопросы к следующим предложениям.

1. New geometries find invaluable application in the modern development of analysis (Where …?). 2. There are two meanings of this term (How many …?). 3. The math language has got some design and rules (What …?) 4. The algebraists learnt to think in terms of equations (How …?). 5. We were sitting in a café at 2 o’clock yesterday (When …?). 6. Scientists will know the results of the experiment in a meek (What …?).

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

1. The professor (to present) his viewpoint at the next conference. 2. We (to discuss) that important law while they were away. 3. I (to have) no time to help you yesterday. 4. I (to stay) with my parents at the moment, though I have my own flat. 5. If father (to have) some time in the evening, he will upgrade your computer. 6. This time next week I (to enjoy) the extraordinary underwater world on the Hawaii islands.

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

1. The (fundamental) are the laws, the (universal) is their use. 2. We choose the magnitude of an angle (large) we please. 3. Engineering develops so quickly that (late) model of a machine today becomes out-of-date in a few years. 4. Gauss was interested in (deep) ideas than the simple numerical computations. 5. The exam was quite easy – much (easy) than we expected. 6. Why does she come to see me at the (bad) possible moment?

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

1. How far (is it/is there/it is) from Minsk to Moscow? 2. I wanted to visit the museum but (it wasn’t/it isn’t/there wasn‘t) enough time. 3. (It is/It was/There was) an accident in King Street but (there wasn’t/it wasn’t/it was) very serious. 4. I don’t think (there are/there will be/it will be) any problems with the exam in algebra. 5. (Will it be/there will be/will there be) many people at the party? 6. A lot of cities are not safe, (there is/it is/there will be) dangerous to go out alone.

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

1. Let point P be the point of intersection of two circles. 2. Let us leave the case at the station. 3. Let the authorities do something about the problem. 4. Let them not say anything about it till they hear all the arguments. 5. Let there be various ways of evaluating formulae. 6. Let the students keep in mind that algebraic fractions are essentially the same as arithmetic ones.

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

1. That is a good idea, but (he/his/him) is better. 2. I am going to a wedding on Saturday. A friend of (me/mine/my) is getting married. 3. The room is large but (its/her/their) windows are not large. 4. Their university is in Regent Street, (our/ours/us) is in the centre of the city. 5. Did your sister pass (her/hers/its) exams? 6. Where are my keys? Where did I put (they/them/their)?

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

1. The graph of an equation is the (loci/locuses/locus) of the points whose coordinates satisfy the equation. 2. To coordinate numbers to points in space we’ll employ a coordinate system consisting of three mutually perpendicular (axes/axis/axises). 3. The area of an ellipse equals times the product of the long and the short (radius/radii/radices). 4. Matrices of single row or single column type (is/are/was) of very common occurrence. 5. The hypothesis given (were/is/are) contrary to our statement. 6. There (are/is/was) crises in our life which lead to great instability in economic affairs.