1. Is it rational or irrational?

2. Rational or Irrational?
Fill-in-the-blank:
Rational or Irrational?
1) The sum of two rational numbers is ________.
2) The product of two rational numbers is ______.
3) The sum of a rational and an irrational is ____.
4) The product of a non-zero rational and an irrational is _____.
5) The product of two irrational numbers is _____.

3. 1) The sum of two rational numbers is rational.
Example:
Proof:
Let and be two rational numbers.
Then, , which is a rational number!

4. 2) The product of two rational numbers is rational.
Example:
Proof:
Let and be two rational numbers.
Then, , which is a rational number!

5. 3) The sum of a rational and an irrational is irrational.
Example:
Proof:
Let x be irrational and r be rational.
Assume x + r = t is rational. (So, t is rational.)
Then, x = t – r.
Since t and r are rational, their difference is rational.
However, this contradicts the assumption that x is irrational.
So, proof by contradiction, x + r is irrational.

6. 4) The product of a non-zero rational and an irrational is irrational.
Example:
Proof:
Let x be irrational and be rational.
Assume x  is rational. Then, x 
So, x = , which by definition is a rational number.
This contradicts the assumption that x is irrational.
Proof by contradiction … x  is irrational.

7. 5) The product of two irrational numbers is sometimes rational and sometimes irrational.
Examples: