Mathematics 479

History and Philosophy of Science

Review for Final Examination (Spring, 2005)

Prepared by:

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

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

2. Can one define every term one uses in a mathematical system?

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

a. MCCCXXXIV

b. CLVIII

c. (110001)2 =

d. (1101)2

e. (1021)3

f. (111)4

g. (1202)3

4. Write the following fractions as Egyptian Fractions

a. 3/10

b. 25/31

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 + 1.

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

a. f(-5)

b. 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.

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. Are there problems which are so hard that a computer can not solve them? Are there statements that can not be proven? (What is the difference between the issue of a statement being true or false versus whether or not it can be proven?)

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

11. Draw a diagram that illustrates Desargues Theorem.

12. What made Kurt Gödel famous?

13. What made Alan Turing famous?

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

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

b. (3, 8) 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 (101010111)

b. ATCTGCTAT and TTCCATTAT

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. In the real projective plane

a. What points are equivalent to the Euclidean points (1, 14) and (-3, 7)?

b. What lines are equivalent to the Euclidean lines x-3y = 4 and -y + 2x +5=0?

c. Find the projective line which joins the points (2, 4, 0) and (1, -1, 1).

d. Find the projective line which joints the points (2, 4, 3) and (2, 1, -1).

e. Find the intersection of the projective lines x1 + 2x2 - x3 = 0 and -x1 - 2x2 + 2x3 = 0

f. Find the intersection of the projective lines x1 - x2 - x3 = 0 and x1 - 3x2 + 1x3 = 0

21. What topics did you enjoy in this course?

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

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

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

25. Evaluate the determinants below:

a.

b.