Review for Examination (Fall, 2007)

Mathematics 479 (History and Philosophy of Mathematics)

Prepared by:

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

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

a. CLXXXIV and MMCCXLVI

b. (110101)2 =

c. (1001)2

d. (10011)3

e. (123)4

2. Write the following fractions as Egyptian Fractions

a. 3/7

b. 7/23

c. 27/74

(for a. use both the splitting method and the Fibonacci-Sylvester method. For the other parts use the Fibonacci-Sylvester method).

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

a.

b.

c. Apply Descartes Rule of signs to the above equations to try to determine how many:

i. Positive real roots the equation has.

ii. Negative real roots the equation has:

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

a. Is divided by x + 5

b. Is divided by x - 2.

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

a. f(2)

b. f(-2)

where:

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

a. (1, 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)

6. Compute the Hamming distance between:

a. (111001110) and (101010101)

b. ATCGGCTAT and TTCCATCGT

d. YORK COLLEGE and CITY COLLEGE

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

8. Give an example of an infinite projective geometry.

9. What are the points and lines of the Klein model for the Bolyai-Lobachevsky plane?

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

11. In the real projective plane:

a. What points are equivalent to the Euclidean points (2, 3) and (-3, -7)?

b. What lines are equivalent to the Euclidean lines x-3y = -4 and 4y + 3x -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, 6, 3) and (2, 1, -1).

e. Find the intersection of the projective lines x1 + 3x2 - x3 = 0 and x1 - 4x2 + 3x3 = 0

f. Find the intersection of the projective lines x1 + 1x2 - x3 = 0 and x1 - 2x2 + 3x3 = 0

12. Name several famous ancient Greek mathematicians.

13. What is meant by a place notation system for notating numbers?

14. What are some of the bases that were used by various civilizations for their number systems?

15. What is the number system we currently used known as?

16. If 2 + 5i is a root of a polynomial with integer coefficients, what other number must also be a root of that equation?

17. Can be the root of a polynomial equation with integer coefficients?

18. Some of the contributors to mathematical insight into the nature of physical space, astronomy, cosmology, or geometry were:

a. Janos Bolyai b. Claudius Ptolemy

c. Johannes Kepler d. N. I. Lobachevsky

e. Isaac Newton f. Galileo

g. Rene Descartes h. James Joseph Sylvester

i. Fibonacci

For each person state i. what "country" he was associated with ii. approximately when did he lived iii. Name one or more of his accomplishments.

19. What is meant by the heliocentric theory of cosmology?

20. Evaluate the determinants below:

a. b. c.