Modern mathematics 103-107 pages

III. Give full answers:

Model: Where did modern mathematics begin? (in Ancient Greece)

Modern mathematics began in Ancient Greece.

1.What mathematicians were the most distingulished in early Greek mathematics? (Undoubtedly Pythagoras and his followers) 2. Whose pupil was Pythagoras? (Thales, the founder of deductive methods) 3. What doctrines did the Pythagoreans teach? (A mixture of morality, astrology, music and genuine mathematics) 4. What ancient cultures did the Pythagoreans inherit mathematical knowledge from? (Babylonians, Egyptians and Indians) 5. What contribution did the Pythagoreans make to mathematics? (Abstracted the concepts of Number, Geometric Form and Figure; Developed and applied deductive reasoning) 6. What were their profoundest discoveries? ( Pythagorean theorem, Incommen - surables) 7. Who proved the general theorem? ( Pythagoras is credited with; the proof is attributed to) 8. How did the Pythagoreans prove their great theorem? (Under the unchallenged assumption that “Numbers rule the Universe”) 9. What was the Pythagoreans contribution to number theory? (The creation of the classical theory of natural numbers) 10. How did the Pythagoreans treat natural numbers? (The essence of all the existing things) 11. How did the Pythagoreans come to recognize incommensurable quantities? (Failed to find a rational number for √2) 12. How did they regard the discovery of incommensurables? (Logical scandal in the foundations of mathematics) 13. What did they call numbers √2, √3, etc.? (Irrationals) 14. How did they deal with irrationals? (By approximating them by means of ratios) 15. What did the discovery of incommensurables result in? (A need to establish a new theory of proportions independent of commensurability) 16. Who developed a theory of incommensurables? (Eudoxus) 17. Who made this theory popular? (Euclid) 18. How did mathematicians learn about the theory? (Euclid presented it in geometrical form in his “Elements”) 19. How can Eudoxus’ theory be estimated? (A masterpiece of Greek mathematics) 20. When did Eudoxus’s theory become fully appreciated? (In the late nineteenth century) 21. Who constructed the first truly rigorous theory of irrational numbers? (Dedikind, Cantor, Weierstrass) 22. How did Greek mathematics benefit from its first classical crisis in the foundations of mathematics? (Ultimate influence was beneficial and considerable) 23. What did the Pythagoreans’ discovery of “incommensurable” quantities bring about? (Ultimately dispelled the belied that the Universe was built on natural numbers)

IV. Ask questions in the model using the words suggested:

Model: The word “geometry” was derived from the Greek words for “earth measure”. (Where … from?)

1.The ancients believed that the earth was flat (What?) 2. The early geometers dealt with measurements of line segments, angles, and other figures in a plane. (What … with?) 3. Gradually the meaning of “geometry” was extended to the ordinary space of solids (How?) 4. Greek mathematicians considered geometry as a logical system. (Who?) 5.They assumed certain properties and try to deduce other properties from these assumptions. (How?) 6. During the last century geometry was still further extended to include the study of abstract spaces.(Why?) 7. Nowadays Geometry has to be defined in an entirely new way. (How?) 8. In contemporary science geometrical imagery (points, lines, planes, etc.) may be represented in many ways. (What?) 9. Any modern geometric discourse starts with a list of undefined terms and relations. (What … with?) 10. The set of relations to which the points are subjected is called the structure of the space. 11. Geometry today is the theory of any space structure. (What?) 12. Geometry multiplied from one to many. (How?) 13. Some very general geometries came into being. (What?) 14. Each geometry has its own underlying controlling transformation group. (What?) 15. New geometries find invaluable application in the modern development of analysis (Where?)