III. Number these events in the order they happened. Look at the text for reading and check your answers.



a. Drinfel’d was elected a corresponding member of the National Academy of Sciences of Ukraine

b. He was born February 14, 1954 in Kharkov

c. He announced a proof of the Langland’s conjectures

d. Drinfel’d is the Harry Pratt Judson Distinguished Service Professor at the University of Chicago

e. Drinfel’d was awarded the Fields Medal

f. His new work appeared in a book form

g. He represented the Soviet Union at the International Mathematics Olympiad in Bucharest, Romania

h. He was awarded the Candidate of Sciences degree and Doctor of Sciences degree from the Steklov Mathematical Institute

i. The same year he entered Moscow State University

j. He graduated from the university in 1974

 

III. Interview your partner about this great mathematician (Work in pairs).

IV. Ask and answer the following questions in pairs.

1. What kind of school establishment (secondary school, lyceum) did you finish last summer?

2. Which subjects were you good at?

3. Which was your best subject?

4. Have you ever solved the problem in Mathematics that only few people from your class could solve? What problem was it?

5. Have you ever participated in the Olympiad in Mathematics (English)? When and where was it held?

6. Have you ever won? Were you awarded or not? When?

7. Would you like to be a representative of your group in the Olympiad in Mathematics (English) held in Donetsk National University? Why?

Text 9

Reading and Speaking

CULTURAL ASSOCIATIONS OF SOME NUMBERS

I. Choose the number and read the text about it.

A. (book closed) Tell your partner what you have read about this number.

B. (book open) Listen to your partner and help him/her to tell about the chosen number.

Not surprisingly, the number 1 is generally treated as a symbol of unity. Therefore, in monotheistic religions, it often symbolizes God or the universe. The Pythagoreans did not consider 1 to be a number at all because number means plurality and 1 is singular. However, they considered it to be the source of all numbers because adding many 1s together can create any other (positive whole) number. In their system, where odd numbers were male and even numbers female, the number 1 was neither; instead, it changed each to the other. If 1 is added to an even number, it becomes odd; similarly, if 1 is added to an odd number, it becomes even   The number 2 symbolizes many of the basic dualities: me/you, male/female, yes/no, alive/dead, left/right, yin/yang, and so on. Dualities are common in human approaches to the world, probably because of our preference for two-valued logic – yet another duality, true/false. Although 2 was female to the Pythagoreans, other numerological schemes viewed it as male. In Agrippa von Nettesheim’s De occulta philosophia (1533; ‘On the Philosophy of the Occult’), 2 is the symbol for man, sex, and evil. One reason that some have associated 2 with evil is that the biblical book of Genesis does not use the formula ‘and it was good’ when referring to the second day of Creation. Religions are dualistic, with two gods in place of the one God of monotheism. Examples include Zoroastrianism, where Ahura Mazdā (the god of light and goodness) battles with Ahriman (the god of darkness and evil). The number 2 is often associated with negatives, as in the words duplicity and two-faced. Northwest Coast Indians required the parents of twins to observe various taboos because they believed that supernatural powers would bring the wishes of twins to fruition
The number 3 is a very mystical and spiritual number featured in many folktales (three wishes, three guesses, three little pigs, three bears). In ancient Babylon the three primary gods were Anu, Bel (Baal), and Ea, representing Heaven, Earth, and the Abyss. Similarly, there were three aspects to the Egyptian sun god: Khepri (rising), Re (midday), and Atum (setting). In Christianity there is the Trinity of God the Father, God the Son, and God the Holy Spirit. Plato saw 3 as being symbolic of the triangle, the simplest spatial shape, and considered the world to have been built from triangles. In German folklore a paper triangle with a cross in each corner and a prayer in the middle was thought to act as protection against gout, as well as protecting a cradle from witches. Three black animals were often sacrificed when attempting to conjure up demons. On the other hand, a three-coloured cat was a protective spirit. In William Shakespeare’s Macbeth (1606–07) there are three witches, and their spell begins, ‘Thrice the brindled cat hath mewed,’ reflecting such superstitions. Also, 3 is the dimension of the smallest magic square in which every row, column, and diagonal sums to 15 The number of order in the universe is 4 – the four elements of earth, air, fire, and water; the four seasons; the four points of the compass; the four phases of the Moon (new, half-moon waxing, full, half-moon waning). The Four Noble Truths epitomize Buddhism. To the Pythagoreans 4 was the source of the tetracts 1 + 2 + 3 + 4 = 10, the most perfect number. In medieval times there were thought to be four humours (phlegm, blood, choler, and black bile—hence the adjectives phlegmatic, sanguine, choleric, and melancholic), and the body was bled at various places to bring these humours into balance. The number 4 is central in the world view of the Sioux, with four groups of gods (superior, ally, subordinate, and spirit), four types of animal (creeping, flying, four-legged, and two-legged), and four ages of humans (infant, child, mature, and elderly). Their medicine men instructed them to carry out all activities in groups of four. Because 4 is generally a practical, material number, few superstitions are associated with it. An exception is in China, where 4 is unlucky because she (‘four’) and shi (‘death’) sound similar. In the biblical Revelation to John the Four Horsemen of the Apocalypse wreak destruction upon humanity
The sum of the first even and odd numbers (2 + 3) is 5. (To the Pythagoreans 1 was not a number and was not odd.) It therefore symbolizes human life andin the Platonic and Pythagorean traditions – marriage, as the sum of the female 2 and the male 3. The Pythagoreans discovered the five regular solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron; now known as the Platonic solids). Early Pythagoreans acknowledged only four of these, so the discovery of the fifth (the dodecahedron, with 12 pentagonal faces) was something of an embarrassment. Perhaps for this reason 5 was often considered exotic and rebellious. The number 5 was associated with the Babylonian goddess Ishtar and her Roman parallel, Venus, and the symbol for both was the five-pointed star, or pentagram. In England a knot tied in the form of the pentagram is called a lover’s knot because of this association with the goddess of love. In Manichaeism 5 has a central position: the first man had five sons; there are five elements of light (ether, wind, water, light, and fire) and a further five of darkness. The body has five parts; there are five virtues and five vices. The number 5 was also important to the Maya, who placed a fifth point at the centre of the four points of the compass. The five fingers of the human hand lent a certain mystery to 5, as did the five extremities of the body (two arms, two legs, head). A human placed in a circle with outspread arms and legs approximates the five points of a pentagon, and if each point is joined to its second-nearest neighbour a pentagram results. This geometric figure is central to occultism, and it plays a prominent role in summoning spells whereby it is supposed to trap a demon, or devil, who can then be compelled to do the sorcerer’s bidding. The belief that 5 was sacred led to an extra element, augmenting the traditional four that made a human being. This fifth essence, or quintessence, is the origin of the word quintessential. In Islam 5 is a sacred number. Foremost are the five Pillars of Islam: declaration of faith (shahādah), prayer (ṣalāt), fasting during Ramadan, giving alms (zakāt), and making the pilgrimage to Mecca (the hajj). Prayers are said five times every day. There are five categories of Islamic law and five law-giving prophets (Noah, Abraham, Moses, Jesus, and Muhammad).   By a wonderful conjunction of mathematical coincidences, 6 is both the sum (1 + 2 + 3) and the product (1 × 2 × 3) of the first three numbers. It is therefore considered ‘perfect.’ In mathematics, a perfect number is one that equals the sum of its divisors (excluding itself). 6 is the first perfect number in this sense because its divisors are 1, 2, and 3. No odd perfect numbers are known, but it has not been proved that none exists. The perfection of 6 shows up in the six days of Creation in Genesis, with God resting on the seventh day. The structure of the Creation parallels the sum 1 + 2 + 3: on day 1 light is created; on days 2 and 3 Heaven and Earth appear; finally on days 4, 5 and 6 all living creatures are created. The sum of the spiritual 3 and the material 4 is 7. In medieval education, students pursued the trivium (grammar, rhetoric, and logic) and the quadrivium (music, arithmetic, geometry, and astronomy), a total of seven subjects, collectively known as the liberal arts. Pythagorean interest in the mathematical patterns in music gives 7 a privileged role, for there are seven distinct notes in the musical scale corresponding roughly to the white notes on a piano. Counting from 1, the eighth note up the scale is the exceedingly harmonious octave, which is how the name arose. The number 7 is often considered lucky, and it has a definite mystique, perhaps because it is a prime number, that is, it cannot be obtained by multiplying two smaller numbers together. There are seven days of the week, named after various ancient gods and planets (Sun-day, Moon-day, Tiw’s-day, Woden’s-day, Thor’s-day, Frigg’s-day, Saturn-day). Tiw was a Norse god of war, parallel to Mars in role but to Zeus in etymology, and Frigg was the Old English version of Frea (or Freya), wife of Woden (= Odin). Shakespeare wrote of the seven ages of man, an idea that goes back much earlier. In China 7 determines the stages of female life: a girl gets her ‘milk teeth’ at seven months, loses them at seven years, reaches puberty at 2 × 7 = 14 years, and reaches menopause at 7 × 7 = 49. The phases of the Moon last approximately seven days, with 4 × 7 = 28 days in a month and also in a female menstrual period. Many cultures recognized seven planets (Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn) in the sense of ‘wandering bodies,’ unlike the ‘fixed stars,’ which retain the same relative position in the night sky. The seven candles of the Jewish menorah that burned in the Tabernacle symbolized the Creation and, according to the English scholar Robert Graves, may be connected to the seven planets of antiquity. In ancient Egypt there were seven paths to heaven and seven heavenly cows; Osiris led his father through seven halls of the underworld. The seven deadly sins are well-known in Christian tradition. The number 7 was the fundamental number of the Rosicrucians, who used it as an organizational basis for their text Chymische Hochzeit Christiani Rosenkreutz (1459; Alchemical Wedding of Christian Rosycross). The number was also central to the cult of Mithra, which believed the soul rose to paradise through seven planetary spheres. The Christian idea of seven layers of purgatory may be related.                                The number 7 features prominently in folk sayings. Breaking a mirror leads to seven years of bad luck. In Iran a cat has seven lives, not the nine of Western myth. The most common numbers in the Indian Vedas are 3 and 7. Agni, the god of fire, has seven wives, mothers, or sisters and can produce seven flames. The sun god has seven horses to pull his heavenly chariot. In the Rigveda there are seven parts of the world, seven seasons, and seven heavenly fortresses. The cow has 21 = 3 × 7 names. In the Hippocratic tradition of medicine, the number 7 rules the illnesses of the body, with painful illnesses lasting 7, 14 or 21 days. In Germany it was believed that pigs would not contract hog cholera if they were treated for seven days with water containing asphodel. In Jewish magic a fever can be cured by taking seven prickles from seven palm trees, seven chips from seven beams, seven nails from seven bridges, seven ashes from seven ovens…terminating in seven hairs from the beard of an old dog

II. Match the words (1–7) with their definitions/explanations (a–g):

1 dual a the result that you wanted to achieve from a plan or idea
2 fruition b a shape with five sides
3 pentagon c a very frightening or dangerous situation, or one in which there seems to be no hope; a large deep hole that appears to have no bottom
4 pentagram d a star with five points that is often used as a magic symbol
5 abyss e used about things that have two similar aspects, parts or functions
6 female f
7 superstitions g a women

III. Match the number with its symbolic meaning:


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