History of Mathematics

Homework Assignment 2

Prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

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

Note the following interesting and useful identity for fractions whose numerators are 1:

1. Use the splitting algorithm and the Fibonacci/Sylvester algorithm for writing the fractions below as as Egyptian fractions (i.e. sums of unit fractions with distinct denominators).

(Apply the splitting algorithm for only parts (a) and (b). The splitting algorithm refers to using the fact that a/b can be written as the sum of 1/b (a times), and then using the identity above as needed.)

(a) 3/10

(b) 4/5

(c) 4/49

(d) 521/1050

2. Verify that given a/b and c/d that (a + c )'(b + d) satisfies:

(*)

3. Write down the reduced fractions for 0/1 to 1/1 with denominator 9 and construct the "medial fractions" associated with these (i.e. use (*) above for adjacent fractions in your list).

Example: given 2/9 and 3/9 = 1/3 one constructs (2+1)/(9 + 3) = 3/12 = 1/4.

4. a. Apply the "Farey process" one line at a times starting with only 0/1 and 1/1. What this means is that one uses (*) to constuct all the medial fractions starting with 0'1 and 1/1.

Thus, after 3 steps you should have: 0/1, 1/4, 1/3, 2/5/1/2, 3/5, 2/3, 3/4, 1/1. The sequence after 1 step is 0/1, 1/2, 1/1.

b. Take two adjacent fractions x/y and u/v (obtained from the nth Farey sequence and compute xv and yu; what do you notice?

5. What fraction is represented by: .1111111111.... base 3.