Chapter 8.6, Problem 10P

Carl’s calculator displays only 10 digits. Carl punches some numbers and some operations on his calculator and shows you his calculator’s display: 0.034482758. Carl has to figure out whether the number whose first 9 digits behind the decimal point are displayed on his calculator is rational or irrational.

a. If you had to make a guess, what would you guess about Carl’s number: Do you think it is rational or irrational? Why?

b. Find a fraction of whole numbers whose first nine digits behind the decimal point in its decimal representation agree with Carl’s calculator’s display. (You do not have to put the fraction in simplest form.)

c. What if Carl’s number was actually 0.034483758003448375800034483758000034483758..

where the pattern of putting in more and more zeros before a block of 34483758 continues forever? In that case, is Carl’s number rational or irrational? Why?

d. Without any knowledge of how Carl got his number to display on his calculator, is it possible to say for sure whether or not it is rational? Explain.