Write a detailed proof for the statement: Between every two rational numbers there exists an irrational number.

Suppose you divide a natural number n by a positive rational number less than 1. Will theanswer always be less than n, sometimes less than n, or never less than n? Why?

a) Simplify the following: the set from the photo b)  Based on your “experimental” results from part a), make a general conjecture, in the form of an equation that you think will be true for all positive integers n, whose left-hand side for n = 1,2,3,4 gives you the four fractions listed in part a) (and whose right-hand side agrees with the results you found). Do not try to prove your conjecture at this stage. c)  Assuming your conjecture from part b) is true, what expression do you get for 1+4+9+···+n^2 ? Again, don’t worry about proving this yet.

Give a (direct) proof of the following statement. Include a full argument using the definition of rational numbers and complete sentences. For all rational numbers r and s, 2r – s is a rational number. Be sure to use the definition of rational number! 3.