Assignment V Math for Poets 2007

Question 1. Write as a fraction: 0./571428/ (Read this as .571428571428571428571428... ; the common notation has a slash through the 5 and the 8).

Question 2. In this question the numbers p and q are rational, the number r is irrational.
a) What can you say about the numbers p+q (i.e. is it rational, is it irrational or is it impossible to decide)?
b) What can you say about p+r ?
c) And about pr (i.e. p times r)?
d) And about r^2 (i.e. the square of r)?
In each case (try to) prove your statement.

Question 3. The 'infinite doubter' just cannot decide. If I would ask her "Shall we go to the movies tonight?", she would first say "No, I am tired." and then say "No, I have to work on my maths." and then say "Yes, I have not been to the cinema for ages." etc. etc. One of her answers could be written down like this:
NNYNYNYYNYNNNYNNYN ... and another one like this
NYNYNYNYNYNYNYNYNY ...
It is clear that the 'infinite doubter' can give an infinite number of answers.
a) Is it possible to put all possible answers in a numbered list?
b) Compare this question with the Stories in Chapter 1. Is there any resemblance?