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 irrational (or both). Be
prepared to share the proof (or at least the idea of the proof) with the class.