Homework 5

• Which of the following statements are true? Prove or disprove each statement.
  • The sum of two rational numbers is always rational.
  • The sum of two irrational numbers is always irrational.
  • The sum of a rational number and an irrational number is always irrational.
  • The product of a rational number and an irrational number is always irrational.
  • For any positive integers a and b, the number a * sqrt(2) + b * sqrt(3) is irrational.
  Hints: use the definition of a rational number (the one that is usually taught in high school, i.e. a real number is called rational if it can be written as a quotient of integers). You may want to use a proof by contradiction for some of these statements. To disprove a statement, give a specific counterexample.

• Find and read (and try to understand) either a proof that the number pi is irrational or a proof that the number e is irratioanl (or both). Be prepared to share the proof (or at least the idea of the proof) with the class.