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Which of the following statements is false?

The sum of two rational numbers is always rational.

The sum of a rational number and an irrational number is always rational.

The product of a nonzero rational number and an irrational number is always irrational.

The product of two irrational numbers is either rational or irrational.

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2 answers
  1. "The sum of two rational numbers is always rational." - true, for example, 2+0.5=2.5 (decimals and fractions are rational)
    So statement is true.

    "The product of a nonzero rational number and an irrational number is always irrational." - true, 5*√2=5(√2) remains irrational. But watch 0*√2=0 (rational). However, 0 is not "nonzero", so the statement remains true.

    "The product of two irrational numbers is either rational or irrational." √2*√3=√6 - irrational
    √2*√2=√4=2 - rational.
    So statement is true.

    Finally,
    "The sum of a rational number and an irrational number is always rational."
    I can easily find examples where the sum of rational and irrational is irrational, as in:
    2+√2 : irrational
    but unable to find a case where the sum is rational.
    Since the statement says "always rational", one single counter-example (as I gave above) invalidates the statement.

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  2. so the answer is D?

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