Countability

(1) Prove the familiar expansion 1/3=0.333... as a repeating decimal. By long division, expand the fraction 2/17 as a repeating decimal. What is the length of the repeating part? For any fraction p/q, what is the maximal possible length of the repeating part?

(2) Inversely, convert the repeating decimal 0.543543543... into a fraction.

(3) As we saw in class and in the examples above, a rational number expressed as a decimal is always repeating. A finite decimal is a decimal with only a finite number of digits to the right of the decimal point; it can be considered as a repeating decimal whose repeating part is the digit 0. Describe the set of fractions which correspond to finite decimals.

(4) We saw in class how to prove that the set NxN of pairs (i,j), where i and j are both whole numbers, is countable. We did this by putting an order on the pairs and counting them off in that order. Prove that the set ZxZ of pairs (i,j) where i and j are both integers (i.e. they can be 0 or negative) is countable, by adapting the counting order for NxN$ to this case. (We mentioned how to do this briefly in class, but without writing it down.)

(5) Write a one-paragraph summary describing the major issues of the Cantor-Kronecker dispute. Explain the mathematical justification of each side, and give your own views.