IS IT RATIONAL OR IRRATIONAL Fillintheblank Rational or

IS IT 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 _____.

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

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

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.

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

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