Mathematics 479

History and Philosophy of Science

Review for Final Examination (Fall 2008)

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

1. Name some famous mathematicians who were also famous physicists.

2. Can one define every term one uses in a mathematical system? What are the common terms that are left undefined in the axiomatic development of Euclidean Geometry.

3. Find the value in decimal notation of the following numbers expressed in other bases or notations:

a. DCCCXLXXIV

b. MMDCLVXXI

c. (1101110)2 =

d. (1001)2

e. (1001)3

f. (213)4

g. (1202)3

h.
g. (1202)4

4. Write the following fractions as Egyptian Fractions (use the Sylvester-Fibonacci method):

a. 11/14

b. 24/31

c. 10/19

5. Find, if any, the rational roots of:

a.

b.

c. Apply Descartes Rule of signs (or its extension) to the above equations to try to determine how many:

i. Positive real roots the equation has.

ii. Negative real roots the equation has:

6. Use synthetic division to find the quotient and remainder when:

a. Is divided by x - 4

b. Is divided by x + 3.

4. Use synthetic division and "direct computation" to find the value of:

a. f(6)

b. f(-3)

c. f(3)

where:

7. Give a brief description of the role of the mathematical understanding of the parallelism concept in shaping the development of non-Euclidean geometry. Give a statement of the the modern version of Euclid's 5th Postulate.

8. Briefly discuss the difference between progress in science versus progress in mathematics. (You may wish to use such terms as inductive investigations, deductive investigations, experimentation, verifiability, a theory vs. a theorem, etc.)

9. a. Find the line which joins (1,1, 0) and (-2, 1, 3) in the real projective plane.

b. Find the point where x - y + z and 2x + z = 0 meet in the real projective plane.

10. What is the Cayley-Klein Model for the Bolyai-Lobachevsky plane?

11. Is 1/4 + 1/4 = 1/2 an Egyptian fraction decomposition of 1/2?

12. Compute the value of the following determinants:

a.

b.

c. Use determinants to find the equation of the Euclidean line through:

i. (2, 3) and (-3, 5)

ii. (-1, 0) and (2, 7)

13.

14. Compute the Euclidean, taxicab, and max distance between the following points:

a. (2, 2, 3) and (3,2, -1)

b. (3,9) and (2, -4)

c. (-1, 3) and (-7, -4)

d. (2, -4, 6) and (11, 4, -5)

15. Compute the Hamming distance between:

a. (111001110) and (111010111)

b. TTCGGCTAT and TTCCATTGT

d. attends and retains

16. Give a modern statement equivalent to the 5th postulate of Euclid.

17. Give an example of an infinite projective geometry.

18. Is the sphere with diametrically opposed points an example of a Euclidean, projective, or hyperbolic geometry?

19. Briefly describe the difference between geometry thought of as a branch of mathematics and geometry thought of as a branch of physics.

20. What topics did you enjoy in this course?

21. What topics were you disappointed because they were not treated?

22. What role do axioms play in a mathematical system?

23. What mathematicians were you hoping to learn more about but were not mentioned in the course?