Results 1 to 5 of 5

Math Help - Prove the following statement: If r3 is irrational, then r is irrational. disprove

  1. #1
    Super Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    508
    Thanks
    72

    Prove the following statement: If r3 is irrational, then r is irrational. disprove

    9. a. Prove the following statement: If r3 is irrational, then r is irrational. (by contrapositive)


    b. Disprove the converse of the statement in part (a).

    a) let r be rational number

    r=\frac{a}{b} for a,b \in \mathbb{Z};b \neq 0

    r^3=\frac{a^3}{b^3}

    a^3r^3=b^3 \rightarrow a*a*a*r*r*r=b*b*b

    by definition integers are closed under multiplication, all integers are rational

    therefore r^3 is rational

    b) converse is if r is irrational then r^3 is irrational

    counterexample??
    Last edited by Jonroberts74; July 6th 2014 at 01:18 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,662
    Thanks
    1481

    Re: Prove the following statement: If r3 is irrational, then r is irrational. disprov

    If you are told to use the contrapositive, you are NOT trying to reach a contradiction.

    These two statements are equivalent: p ->q and ~q -> ~p. So if you can prove the second statement, then you prove the first one.

    So since you are trying to show that if r^3 is irrational then r is irrational, one way to do this is to prove its contrapositive, that if r is rational, then r^3 is rational.

    There is no disproof involved!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: Prove the following statement: If r3 is irrational, then r is irrational. disprov

    To disprove part (b), yes you need a counter-example. So we need to find an irrational number, whose cube is rational.

    The obvious candidate is $\sqrt[3]{2}$. Its cube is clearly rational. Can you prove it is irrational?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    508
    Thanks
    72

    Re: Prove the following statement: If r3 is irrational, then r is irrational. disprov

    Quote Originally Posted by Prove It View Post
    If you are told to use the contrapositive, you are NOT trying to reach a contradiction.

    These two statements are equivalent: p ->q and ~q -> ~p. So if you can prove the second statement, then you prove the first one.

    So since you are trying to show that if r^3 is irrational then r is irrational, one way to do this is to prove its contrapositive, that if r is rational, then r^3 is rational.

    There is no disproof involved!
    didn't I prove it by contrapositive? I took p --> q and made it ~q-->~p, I let r be rational then showed r^3 is rational


    Quote Originally Posted by Deveno View Post
    To disprove part (b), yes you need a counter-example. So we need to find an irrational number, whose cube is rational.

    The obvious candidate is $\sqrt[3]{2}$. Its cube is clearly rational. Can you prove it is irrational?
    assume the cuberoot of 2 is rational
    \sqrt[3]{2} = \frac{a}{b}; a,b \in \mathbb{Z}, b \neq 0 and a/b is lowest terms


    2=\frac{a^3}{b^3}

    2a^3=b^3 the left hand side is even, thus the right side must also be even

    2a^3=(2m)^3; (2m)=b

    a^3=\frac{8}{2}m^3

    both have a factor of 2, both are even therefore they were not in there lowest terms, it is not rational, therefore is irrational
    Last edited by Jonroberts74; July 6th 2014 at 09:27 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    508
    Thanks
    72

    Re: Prove the following statement: If r3 is irrational, then r is irrational. disprov

    Quote Originally Posted by Jonroberts74 View Post
    9. a. Prove the following statement: If r3 is irrational, then r is irrational. (by contrapositive)


    b. Disprove the converse of the statement in part (a).

    a) let r be rational number

    r=\frac{a}{b} for a,b \in \mathbb{Z};b \neq 0

    r^3=\frac{a^3}{b^3}

    a^3r^3=b^3 \rightarrow a*a*a*r*r*r=b*b*b

    by definition integers are closed under multiplication, all integers are rational

    therefore r^3 is rational

    b) converse is if r is irrational then r^3 is irrational

    counterexample??
    r^3=\frac{a^3}{b^3}

    a^3r^3=b^3 \rightarrow a*a*a*r*r*r=b*b*b


    this was wrong meant


    b^3r^3=a^3 \Rightarrow b*b*b*r*r*r=a*a*a
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: July 19th 2010, 05:04 PM
  2. Replies: 1
    Last Post: February 23rd 2010, 05:54 PM
  3. Replies: 0
    Last Post: February 16th 2010, 05:04 AM
  4. Replies: 2
    Last Post: January 31st 2010, 05:40 AM
  5. Replies: 7
    Last Post: January 29th 2009, 03:26 AM

Search Tags


/mathhelpforum @mathhelpforum