Page 1 of 2 12 LastLast
Results 1 to 15 of 18
Like Tree6Thanks

Math Help - Prove (square root 2) is irrational - I have a problem with this proof.

  1. #1
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Prove (square root 2) is irrational - I have a problem with this proof.

    I believe that this is a standard proof but I have a problem with it.

    4d383176cd06283c92755eafb4b9586c.gif

    The end statement appears to me to be

    root2 = a/b where a and b are both even therefore a and b are not coprime which contradicts the initial condition that a and b must be coprime therefore root2 is irrational.
    -------------------------------------------------
    I ran through a different proof to attempt to prove that root4 is irrational
    (this is not the full proof, I have truncated it a little)
    suppose root4 were irrational
    then root4 = a/b
    4b^2=a^2
    therefore a is even
    Let a=2k
    4b^2=4k^2
    b^2=k^2
    k=+-b
    therefore
    a=2k and b=+/-k
    Therefore a and b are not coprime
    Therefore root4 is irrational.

    Obviously I realize that 2k/k can be reduced to 2/1 which then shows root4 to be rational

    BUT

    when the first proof was looking at root2 and it got down to (an even number/an even number) so why is it impossible for that fraction also be reduced to some p/q where p and q are coprime. To me, the proof just doesn't seem to be finished and does not prove anything (not to me).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by Melody2 View Post
    I believe that this is a standard proof but I have a problem with it.

    4d383176cd06283c92755eafb4b9586c.gif

    The end statement appears to me to be

    root2 = a/b where a and b are both even therefore a and b are not coprime which contradicts the initial condition that a and b must be coprime therefore root2 is irrational.
    -------------------------------------------------
    I ran through a different proof to attempt to prove that root4 is irrational
    (this is not the full proof, I have truncated it a little)
    suppose root4 were irrational
    then root4 = a/b
    4b^2=a^2
    therefore a is even
    Let a=2k
    4b^2=4k^2
    b^2=k^2
    k=+-b
    therefore
    a=2k and b=+/-k
    Therefore a and b are not coprime
    what if b=1, as it does here?
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Thanks but I don't get what you are trying to tell me Romsek.
    b=+/-k and
    k can equal any whole number it doesn't have to be 1. (or does it?)
    Does the answer have implied restrictions on k?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by Melody2 View Post
    Thanks but I don't get what you are trying to tell me Romsek.
    b=+/-k and
    k can equal any whole number it doesn't have to be 1. (or does it?)
    Does the answer have implied restrictions on k?
    you used the fact that a=2k and b=+/-k to say that a and b are not coprime. But this isn't true if b=k=1.
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Ok so you are telling me that they must be coprime for some particular value of k
    whereas I was thinking that they must be coprime for all values of k.
    I guess that helps. Thanks.
    Last edited by Melody2; November 30th 2013 at 09:50 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    no, it just doesn't apply if k=1. Take 2 and 1. you can't reduce 2/1 any further. I don't know that you'd call 2 and 1 coprime but certainly the fraction 2/1 is in fully reduced form. Here k=1.
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by romsek View Post
    no, it just doesn't apply if k=1. Take 2 and 1. you can't reduce 2/1 any further. I don't know that you'd call 2 and 1 coprime but certainly the fraction 2/1 is in fully reduced form. Here k=1.
    It was sqrt4 = a/b=2k/k if k is anything other than 1 then 2k and k are not coprime.
    so this particular proof only works if k=1
    Last edited by Melody2; November 30th 2013 at 10:07 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by Melody2 View Post
    It was sqrt4 = a/b=2k/k if k is anything other than 1 then 2k and k are not coprime.
    so this particular proof only works if k=1
    the key to the proof of sqrt[2] being irrational though was that b=2k and was thus even. Which since a was found to be even earlier gives a contradiction that a/b was in fully reduced form.

    you can't make that statement here.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    The whole reason that that the factor of 2 was relevant was that it meant a and b were not coprime.
    so although I hear frustration in your post I am not going to give way completely. Sorry
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    right but in your proof b = +/- k not +/- 2k, so you can't claim b is even here like the other proof did. No frustration here and I certainly don't view it as a fight :P
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    4f5e972eeed7eb1158dfe6eeadc9fc49.gif

    We don't know exactly what k is although it does have a specific value. We do know that 'a' has a factor of 2 so therefore 'b' cannot be even.
    etc
    --------------------------------------------------
    If I make this change to the wording of the proof then I am happy and I think it all makes sense.

    Thankyou for your help Romsec. I have enjoyed our communications this afternoon and I appreciate your help.
    ----------------------------------------------------

    I just a had a brainwave. Of course k has a specific value because 'a' has to have a specific value and k is half of a.
    Now I have it. In the sqrt4 proof k has to be 1, there is not other possibliliy.
    Thankyou so much for helping me reach this understanding.
    ---------------------------------------------------------
    I know you kept telling me that k=1 but the real reason for this was not sinking into my brain.
    Sometimes students (that's me) can be very frustrating. I didn't mean any offence by the comment.
    Last edited by Melody2; November 30th 2013 at 11:22 PM.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Glad you got it squared away.

    Romsek (with a k!)
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by Melody2 View Post
    I believe that this is a standard proof but I have a problem with it.
    4d383176cd06283c92755eafb4b9586c.gif.
    I realize that you have solved this for P=2. But is is a general proof.

    Suppose that P is a positive integer that is not a square. We prove that \sqrt P is irrational.

    For a contradiction suppose that \sqrt P is rational. Then the is a smallest positive integer K for which it is true that K\sqrt P is a positive integer.

    One of the properties of the floor function is that  0<\sqrt P-\left\lfloor {\sqrt P } \right\rfloor < 1

    But that means  0<K\sqrt P-K\left\lfloor {\sqrt P } \right\rfloor < K which means K\sqrt P-K\left\lfloor {\sqrt P } \right\rfloor  is a positive integer smaller that K
    but (K\sqrt P-K\left\lfloor {\sqrt P } \right\rfloor)\sqrt P is also a positive integer.

    That contradicts the minimal nature of K.
    Last edited by Plato; December 1st 2013 at 06:02 AM.
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Member
    Joined
    Nov 2013
    From
    Australia
    Posts
    215
    Thanks
    49

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Thanks Plato,
    I at least followed your argument right up to the last line.
    It then dissipated into the realm of mystery for me.

    also,
    how do you implant things like you implanted the 'floor function'.
    I had seen these floor and ceiling symbols before but i had no idea what they were.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,789
    Thanks
    1147

    Re: Prove (square root 2) is irrational - I have a problem with this proof.

    Quote Originally Posted by Melody2 View Post
    Thanks Plato,
    I at least followed your argument right up to the last line.
    It then dissipated into the realm of mystery for me.

    also,
    how do you implant things like you implanted the 'floor function'.
    I had seen these floor and ceiling symbols before but i had no idea what they were.
    Thanks.
    Embedded TeX can do all sorts of things!

    \lceil y\rceil  \lfloor x\rfloor
    Thanks from Melody2
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Square root Irrational
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 26th 2010, 03:51 AM
  2. Replies: 11
    Last Post: October 25th 2009, 07:45 PM
  3. Replies: 2
    Last Post: May 26th 2009, 10:09 AM
  4. Proof of square root being irrational
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 14th 2007, 02:07 PM
  5. How to prove square root 2 is irrational?
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: June 24th 2007, 08:40 AM

Search Tags


/mathhelpforum @mathhelpforum