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Math Help - product of rational and irrational is always irrational

  1. #1
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    product of rational and irrational is always irrational



    Am I assuming too much here?
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  2. #2
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    I don't understand why you're setting t=\frac{r}{s} in your proof if t is irrational.

    To prove these, you just use the fact that \mathbb{Q} is a field, i.e, closed under addition, subtraction, multiplication, and division.

    If a+t \in \mathbb{Q}, then by closure, (a+t)-a=t \in \mathbb{Q}. Contradiction.

    If at \in \mathbb{Q}, then by closure, \frac{at}{a}=t \in \mathbb{Q}. Contradiction.
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  3. #3
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    Quote Originally Posted by Black View Post
    I don't understand why you're setting t=\frac{r}{s} in your proof if t is irrational.
    He did NOT assume t= \frac{r}{s}. He showed, first, that if you assume a/t is a rational number, then you arrive at t= r/s, a ration number and so a contradiction.

    He then showed that if you assume at is rational, [b]then[b] you arrive at t= r/s, again a contradiction. His proof is perfectly valid.

    To prove these, you just use the fact that \mathbb{Q} is a field, i.e, closed under addition, subtraction, multiplication, and division.

    If a+t \in \mathbb{Q}, then by closure, (a+t)-a=t \in \mathbb{Q}. Contradiction.

    If at \in \mathbb{Q}, then by closure, \frac{at}{a}=t \in \mathbb{Q}. Contradiction.
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