Is it rational or irrational?

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Is it rational or irrational?
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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 _____.

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1) The sum of two rational numbers is rational.

• Example

• Proof

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

• Example

• Proof

Let and be two rational numbers.
Then, , which is a
rational number!
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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.
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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.
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5) The product of two irrational numbers is
sometimes rational and sometimes irrational.

• Examples