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Math Help - semi-urgent: rational and irrational numbers!

  1. #1
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    semi-urgent: rational and irrational numbers!

    Okay I don't get this =( I think I've made a decent attempt but I need some help getting there all the way. This is due tomorrow. can someone help me? =(

    a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational?
    b. If a is rational and b is irrational, is ab necessarily irrational?
    c. Is there a number of a such that a^2 is irrational, but a^4 is rational?
    a. I think I have this first part. You can prove it by contradiction.

    R= some rational number

    a+b = R
    b = R-a
    A rational number minus a rational number is a rational number. This would mean b = rational, which is not true. therefore a+b is irrational.

    This second part, if both are irrational? I was thinking:

    a+b = R

    a = R-b, or b = R-a. I'm not sure how this helps me x(
    -----
    b. If a is rational and b is irrational, is ab necessarily irrational?

    No idea, but here's my attempt:

    Two cases. If a = 0, ab = 0, and 0 is rational so ab is rational.

    if a =/= 0...proof by contradiction maybe?

    a*b = rational
    a*b = a*b
    b = a*b*a^-1?

    But then b=b? And that doesn't help me.
    -----
    c Is there a number a such that a^2 is irrational, but a^4 is rational?

    Well again I have no idea but here's my attempt:

    a=b
    a^2 = ab
    ab = x
    b = x/a
    b = x * a^-1

    can someone help me finish these? =( thanks in advance.
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  2. #2
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    Quote Originally Posted by Sven View Post
    a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational?
    b. If a is rational and b is irrational, is ab necessarily irrational?
    c. Is there a number of a such that a^2 is irrational, but a^4 is rational?
    a. No. All you need is a specific counterexample. eg. a = -1 and b = 1 + \sqrt{2}.

    b. Assume ab = c where c is rational and a \neq 0. Then b = \frac{c}{a}. What does this mean about b ....?

    c. Yes. eg. a = 2^{1/4}.
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