Is it rational or irrational?

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Presentation transcript:

Is it rational or irrational?

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 and be two rational numbers. Then, , which is a rational number!

2) The product of two rational numbers is rational. Example: Proof: Let and be two rational numbers. Then, , 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 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.

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