Awards and recognition

Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published by MIT Press, in four volumes.

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Development of Mathematics

The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C., it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step.

Greek Contributions

A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. B.C.), Pythagoras, Plato, and Aristotle, and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.

During the Golden Age (5th cent. B.C.), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as , also dates from this period. Eudoxus of Cnidus (4th cent. B.C.) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.

The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, Euclid's Elements (c.300 B.C.), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid's presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.

In the 3d cent. B.C., Archimedes, in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of Perga named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c.A.D. 100, spherical triangles), Heron of Alexandria (geometry), Ptolemy (A.D. 150, astronomy, geometry, cartography), Pappus (3d cent., geometry), and Diophantus (3d cent., arithmetic).

Chinese and Middle Eastern Advances

Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by Aryabhata and Brahmagupta (5th and 6th cent. A.D.). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-Khowarizmi (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-Battani worked on trigonometry. In Egypt, Ibn al-Haytham was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra.

Western Developments from the Twelfth to Eighteenth Centuries

Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò Tartaglia and Geronimo Cardano, in trigonometry by François Viète, and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Gérard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René Descartes and that of the calculus by Isaac Newton and, independently, by G. W. Leibniz. Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L'Hôpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822).

In the Nineteenth Century

The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by János Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab /= ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

In the Twentieth Century

In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as “self-evident truths” has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).

The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gödel and A. Church.
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The increasing cost of energy has caused many companies to make permanent changes in their offices. On a small scale, office managers are purchasing energy-efficient office machines and encouraging recycling programs to cut energy costs. On a larger scale, architects and builders are responding to the demands of companies for more energy-efficient buildings.

Buildings constructed or renovated in the last few years have incorporated energy-saving measures. Office maintenance workers have sealed cracks around windows and doors. Builders have installed sets of double doors to reduce the exchange of indoor and outdoor air when doors are opened.

This has reduced transfer of air in and out of the building. While it has had cost-saving benefits, it has caused personnel-related costs such as increased employee absences due to illness. Since the interior air is recirculated and little fresh air is allowed in, everyday contaminants such as dust and germs remain in the air. Employees in energy-efficient buildings breathe the same air again and again. They suffer from an increase in headaches, colds, dry skin, and dry throats, and in severe cases respiratory problems.

GRAMMAR TEST: CONDITIONALS

1. Choose the correct answer.

1) If she ... not so slowly she would enjoy the party. A) were B) is C) will be

2) If you ... my library book I will have to buy a new one. A) will lose B) lost C) loose

3) If she ... you were in hospital she would have visited you.

A) had known B) knew C) would have known

4) I wish I ... rich. A) would be B) were C) had been

5) I wish I ... his opinion before. A) would know B) had known C) knew

2. Match the two parts of the sentences.

6) He wouldn't have become so strong;... a) ... I wouldn't be worried now.

7)They would have come... b) ... I would have gone to the library.

8) If they had been ready the day before... c) ... we wouldn't have come so early.

9) If I hadn't needed the book... d) ... unless he had done sports.

10) If they had had a city map... e) ... they wouldn't have been lost.

11) If you had warned us... f) ... if Jane had invited them.

12) He wouldn't know much... g) ... unless you had agreed with us.

13) We wouldn't have wasted so much time... h) ... unless he had read much.

14) If you had sent me a telegram... i) ... they would have taken their exam.

15) We had never done this ... j) if you have bought everything beforehand

3. Correct the errors, if necessary.

16) If I knew her well I will visit her.

17) If I were you I would have visited Jane yesterday.

18) If I have a computer I would learn Computer Studies.

19) If the weather would be nice tomorrow we'll go on excursion.

20) You did not miss the plane if you had taken a taxi.

21) I wish you have a car.

22) I wish things were different in the past.

23) I wish the weather were warmer.

24) I wish I did not decide to work in New York.

25) I wish I did not go to bed early yesterday.

4. Complete the following radio programme by putting the verbs in brackets into the correct form.

Interviewer: Welcome once again to our weekly programme in which we ask the questions "If you (26) ___ (be) alone on a tropical island for a month, what two items (27) ___ you ___ (choose) to take with you and why?" My two guests are racing driver Charles Brown and journalist Helen Howk, Charles?

Charles: Well, I think (28) ___ (get) very bored on this island if I (29) ___ (not have) anything to do. So, I (30) ___ (take) a knife and a ball of string. Then I (31) ___ (be able) to make useful things to catch food, and, maybe, build some kind of house to live in.

Interviewer: (32) ___ you ___ (try) to escape from the island?

Charles: If I (33) ___ (manage) to make a boat, I think I (34) ___ (try).

Interviewer: Helen, what about you?

Helen: Well, I definitely (35) ___ (not try) to escape. I'm totally impractical. So, if I (36) ___ (try) to make anything, I'm sure it (37) ___ (fall) to pieces very quickly. No, if I (38) ___ (have) to spend a month on the island, I (39) ___ (want) to have a good book and a pair of sunglasses.

Charles: But how (40) ___ you ___ (catch) things to eat if you (41) ___ (not have) any tools?

Helen: Oh, I expect there (42) ___ (be) plenty of fruit on the island. And I'm sure it (43) ___ (not hurt) me if I (44) ___ (not eat) meat or fish for a month.

Interviewer: (45) ___ either of you ___ (be) lonely?

Charles: Definitely. I (46) ___ (find) it very difficult if I (47) ___ (not speak) to anyone for a month.

Helen: I think (48) ___ (enjoy) the peace and quiet at first, but after a couple of weeks, yes, I (49) ___ (begin) to feel lonely.

Interviewer: Charles and Helen, thank you very much.

GRAMMAR TEST: Reported Speech

1. Report the questions:

1 'What is your name?' he asked me.

...He asked me what my name was....

2 'Where are your parents?' Uncle Bill asked us.

3 'Will you help me carry the box, please?' Dad asked.

4 'What time will you be home?' Mum asked me.

5 'Can you play the guitar?' he asked her.

6 'Who was at the door?' David asked Janet.

7 'Where is the post office?' they asked us.

8 'When will you do your homework?' Meg asked me.

9 The boss asked me, 'Have you finished those reports?'

10 John asked Sam, 'Do you like computer games?'

11 'Will you give me a lift to work, please?' he asked her.

12 'Where is your jacket?' she asked him.

1. Write the sentences in reported speech:

1 'I'd do maths, Jo, if I were you.'

Ms Jennings advised ........

2 'Take a photo of the bank.'

The journalist ordered the photographer.........

3 'John and Laura, do the more difficult exercises.'

The teacher told .............

4 'Give the video to your sister, please.'

Peter asked ................

5 'Don't tell your grandmother yet.'

The doctor advised ...............

6 'Make the sandwiches'.

The cook told Jenny...............

7 'Don't drink the coffee yet!'

The doctor advised ...............

8 'Take the meat out of the fridge!'

Mum requested..............

9 'Don't look so serious'

Jim told me............

10 'Serve those customers, please!'

Mr Jones asked.............

11 'Help with the washing up now!'

Mum ordered..............

1. Fill in the correct form of say, tell, speak or talk.

1)"Could you ... me the time?", ... the old man.

2) She ... me to ... up because she couldn't hear me.

3) Mr. White ... he can ... French and Arabic.

4) I don't ... to Robert anymore; he always lies.

5) “I'd like to ... to the headmaster please", ... the client.

6) The judge ... the witness to ... the truth and nothing but the truth.

7) She ... us not to ... anything to her family.

8) " … your prayers and go to bed", Mum

9) " … louder, please.", the teacher ....

10) He ... to me: "Be serious! Don't laugh at her!"

11) The King ... : "How could I get rid of the mice?"

12) The students in the seventh form ... good English.

13) Mother ... her children not to open the front door.

14) My father ... us the good news.

Date: 2015-01-12; view: 824