I. Найдите в тексте интернациональные слова, переведите их.
II. Выберите в колонке В эквиваленты к словам колонки А.
| А
1. rational approximation
2. fraction
3. rational number
4. ratio of two integers
5. real number
6. geometrical magnitude
7. regular pentagon
8. numeration system
9. incommensurable quantities
10. diagonal of a square
| В
a)отношениедвух переменных b) действительное число
c) геометрическая величина
d) рациональное число
e) правильный пятиугольник
f) дробь
g) рациональное приближение
h) диагональ квадрата
i) числовая система
j) несоизмеримые количества
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III. Заполните пропуски подходящими по смыслу словами.
1. The same geometric procedure can be adapted to the side and ….
a) diagonal of a square b) diagonal numbers
c) regular pentagon d) star pentagon
2. A number can’t be expressed as the ….
a) rational approximation b) irrational approximation
c) ratio of associated pairs d) ratio of two integers
3. The first pair of segments to be incommensurable is the side and diagonal of a ….
a) square b) right triangle
c) number d) regular pentagon
4. The ration of associated pairs of the numbers give closer and closer ….
a) rational number b) fraction
c) rational approximation d) irrational number
Text III
The fact that there revealed pairs of segments for which such a measure does not exist provides the incommensurable case. It is possible that the first pair of segments found to be incommensurable is the side and diagonal of a regular pentagon, the favourite-figure of the Pythagoreans because its diagonals form the star pentagon, the distinctive, mark of their society. This same geometric procedure can also be adapted to the side and diagonal of a square. Here there exists an association with the so-called Pythagoreans side and diagonal numbers. The ratio of associated pairs of these numbers gives successively closer and closer rational approximations to √2; in fact, they are the approximations obtained by computing successive convergents of the continued fraction form of √2. This is reflected in modern mathematics in the concept of irrational number, a number that cannot be expressed as the ratio of two integers, e. g., п, e, 1/2. This devastating discovery was ascribed to Pythagoras himself, but more probably it was made by some Pythagorean. Since the philosophy of the Pythagorean school was that whole numbers or whole numbers in ratio are the essence of all existing things, the followers of that school regarded the emergence of irrationals as a "logical scandal". As the revelation of geometrical magnitudes whose ratio cannot be represented by pairs of integers led to the "crisis" in the foundations of their mathematics, the Pythagoreans tryed to conceal their greatest discovery. A Pythagorean Hippasus (c. 400 B. C.) who first brought out the irrationals from concealment into the open supposedly perished in a shipwreck at sea. But great discoveries could not be suppressed. The discovery of incommensurables was a turning-point, a landmark in the history of mathematics, and its significance can hardly be over appreciated. It resulted in a need to establish a new theory of proportions independent of commensubarility. This was accomplished byEudoxus (c. 370 B. C.). The details of the gradual transition from a theory of proportions which includedincommensurable quantities to a clear realization of.the concept of an irrational number covered a wide range of sophisticated mathematical topics and this concept was fully clarified only in the nineteenth century by R. Dedekind and G. Cantor. In mathematics of today the irrationals form an important subset of real numbers the basis of both algebra and analysis.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. What was the distinctive mark of the Pythagoreans society?
a) segment b) side and diagonal
c) polyhedral d) star pentagon
2. What gives the ratio of pairs of numbers?
a) real number b) approximation to 0
c) rational approximation to y2 d) irrational number
3. Why cannot a number be expressed as the ratio of two integers? Because it was ….
a) irrational number b) rational number
c) real number d) fractional number
4. Who first brought out the irrationals from concealment?
a) Cantor b) Eudoxus
c) Hippasus d) Descart
5. What new theory did Eudoxus establish?
a) a theory of numbers b) a theory of proportions
c) a theory of equations d) the“Pythagorean” theory
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. The concept of irrational number
b. Geometrical magnitudes
c. The discovery of incommensurables
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