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√2 + b√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 π (pi) is irrational or a proof that the number e (Euler's number)
is irrational (or both). UPDATE. Include a link or a picture of the text.
UPDATE. For 1 point of extra credit, type this homework in overleaf. To turn in your homework, just share your file with the instructor (mnogin@csufresno.edu).
All future homework assignments will have to use this method.
If you choose to do this assignment on paper, then scan or take a picture of your work and send it by email. Please make sure to write legibly and check the quality of
the pictures before sending them. Also make sure to attach all pictures.