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Is it rational or irrational?

Learn about the rationality of number operations, including sums and products of rational and irrational numbers. Explore examples and proofs to grasp concepts better.

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Is it rational or irrational?

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  1. Is it rational or irrational?

  2. 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:

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