Square root of 6 proof rationality











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I was proving $sqrt 6 notin Bbb Q$, by assuming its negation and stating that: $exists (p,q) in Bbb Z times Bbb Z^*/ gcd(p,q) = 1$, and $sqrt 6 = (p/q)$.



$implies p^2 = 2 times 3q^2 implies exists k in Bbb Z; p = 2k implies 2k^2 = 3q^2$ and found two possible cases, either $q$ is even or odd, if even we get contradiction that $gcd(p, q) neq 1$, if odd we get contradiction that $2k^2 = 3q^2$.



Is it a right path for reasoning it?










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  • Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
    – fleablood
    Nov 15 at 2:13










  • Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
    – fleablood
    Nov 15 at 2:20












  • I rearranged it!
    – J.Moh
    Nov 15 at 3:51















up vote
2
down vote

favorite












I was proving $sqrt 6 notin Bbb Q$, by assuming its negation and stating that: $exists (p,q) in Bbb Z times Bbb Z^*/ gcd(p,q) = 1$, and $sqrt 6 = (p/q)$.



$implies p^2 = 2 times 3q^2 implies exists k in Bbb Z; p = 2k implies 2k^2 = 3q^2$ and found two possible cases, either $q$ is even or odd, if even we get contradiction that $gcd(p, q) neq 1$, if odd we get contradiction that $2k^2 = 3q^2$.



Is it a right path for reasoning it?










share|cite|improve this question
























  • Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
    – fleablood
    Nov 15 at 2:13










  • Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
    – fleablood
    Nov 15 at 2:20












  • I rearranged it!
    – J.Moh
    Nov 15 at 3:51













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I was proving $sqrt 6 notin Bbb Q$, by assuming its negation and stating that: $exists (p,q) in Bbb Z times Bbb Z^*/ gcd(p,q) = 1$, and $sqrt 6 = (p/q)$.



$implies p^2 = 2 times 3q^2 implies exists k in Bbb Z; p = 2k implies 2k^2 = 3q^2$ and found two possible cases, either $q$ is even or odd, if even we get contradiction that $gcd(p, q) neq 1$, if odd we get contradiction that $2k^2 = 3q^2$.



Is it a right path for reasoning it?










share|cite|improve this question















I was proving $sqrt 6 notin Bbb Q$, by assuming its negation and stating that: $exists (p,q) in Bbb Z times Bbb Z^*/ gcd(p,q) = 1$, and $sqrt 6 = (p/q)$.



$implies p^2 = 2 times 3q^2 implies exists k in Bbb Z; p = 2k implies 2k^2 = 3q^2$ and found two possible cases, either $q$ is even or odd, if even we get contradiction that $gcd(p, q) neq 1$, if odd we get contradiction that $2k^2 = 3q^2$.



Is it a right path for reasoning it?







rational-numbers






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edited Nov 17 at 23:00









Mr. Brooks

27311237




27311237










asked Nov 15 at 1:12









J.Moh

395




395












  • Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
    – fleablood
    Nov 15 at 2:13










  • Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
    – fleablood
    Nov 15 at 2:20












  • I rearranged it!
    – J.Moh
    Nov 15 at 3:51


















  • Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
    – fleablood
    Nov 15 at 2:13










  • Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
    – fleablood
    Nov 15 at 2:20












  • I rearranged it!
    – J.Moh
    Nov 15 at 3:51
















Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
– fleablood
Nov 15 at 2:13




Is the $k$ a typo? What is $p, k$ and $q$? Why is $q$ being even a condradiction with $gcd(p,q) = 1$? And why is $2k^2 = 3q^2$ a contradiction? You just said $2k^2 = 3q^2$ so that isn't a contradiction? You are on the right track but you haven't explained any of your arguments.
– fleablood
Nov 15 at 2:13












Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
– fleablood
Nov 15 at 2:20






Why is $q$ even a contradiction? What if $k$ or $p$ (whats the difference) is odd?
– fleablood
Nov 15 at 2:20














I rearranged it!
– J.Moh
Nov 15 at 3:51




I rearranged it!
– J.Moh
Nov 15 at 3:51










4 Answers
4






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up vote
2
down vote













How about a proof by descent?



First show that $2^2<6<3^2$. Then if $sqrt{6}$ is to be rational it must have a form $p/q$ where $p,qin mathbb{Z}, q>0, 2q<p<3q$. By simple algebra the square root is also equal to $6q/p$, thus



$p/q=6q/ptext{.....Eq. 1}$.



Now if $a/b=c/d$ then also



$a/b=(ma+nc)/(mb+nd)$



for any coefficients $m,n$ where the denominator is nonzero. In particular, Eq. 1 implies



$p/q=(3p-6q)/(3q-p)text{.....Eq. 2}$



where we already have $2q<p<3q$ and thus $0<3q-p<q$. So the proposed rational fraction $p/q$ must be equal to an alternative rational fraction with a smaller positive denominator. This causes an infinite descent contradiction forcing the assumption of a rational value to be false.



We can form a similar proof for the square root of any natural number that is not a squared integer. "Not a squared integer" is needed because the square root must be strictly between two adjacent integers to obtain a descent of positive denominators.






share|cite|improve this answer






























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    1
    down vote













    Assume $sqrt{6} = {a over b}$ with $a, b in mathbb{Z}$ . This implies



    $$2 cdot 3 b^2 = a^2$$



    Use the fundamental theorem of algebra to decompose both sides into a unique prime factorization. There are an even number of factors of $2$ on the rhs and an odd number of factors of $2$ on the lhs... a contradition. Same for factors of $3$.



    Thus $sqrt{6} neq {a over b}$, i.e., is irrational.






    share|cite|improve this answer























    • Yeah, I tried to reach it without prime factorisation. A second look?
      – J.Moh
      Nov 15 at 2:42












    • You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
      – David G. Stork
      Nov 15 at 2:47












    • Our professor used the value way, so I am following his steps for this, though, thanks!
      – J.Moh
      Nov 15 at 2:50












    • Show him/her the better way! And stand out as a student!
      – David G. Stork
      Nov 15 at 2:51










    • Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
      – J.Moh
      Nov 15 at 3:49




















    up vote
    0
    down vote













    I'll play more generally
    and see what happens.



    Suppose
    $sqrt{m}
    =dfrac{u}{v}
    $

    where
    $m = prod_{p in P} p^{m_i},
    u = prod_{p in P} p^{u_i},
    v = prod_{p in P} p^{v_i}
    $
    .



    Then
    $prod_{p in P} p^{m_i}
    =dfrac{u^2}{v^2}
    =dfrac{ prod_{p in P} p^{2u_i}}{prod_{p in P} p^{2v_i}}
    $

    so
    $prod_{p in P} p^{m_i}prod_{p in P} p^{2v_i}
    =prod_{p in P} p^{2u_i}
    $

    or
    $prod_{p in P} p^{m_i+2v_i}
    =prod_{p in P} p^{2u_i}
    $
    .



    By unique factorization,
    $m_i+2v_i
    =2u_i
    $
    ,
    so
    $m_i
    =2u_i-2v_i
    =2(u_i-v_i)
    $
    .



    Therefore
    $m
    =prod_{p in P} p^{m_i}
    =prod_{p in P} p^{2(u_i-v_i)}
    =left(prod_{p in P} p^{u_i-v_i}right)^2
    $

    so $m$ is a perfect square.



    Therefore
    the square root of an integer
    is rational
    only if
    it is a square.



    I'll now try to generalize this
    to $k$-th roots,
    with as much cut-and-paste
    as possible.



    Suppose
    $sqrt[k]{m}
    =dfrac{u}{v}
    $

    where
    $m = prod_{p in P} p^{m_i},
    u = prod_{p in P} p^{u_i},
    v = prod_{p in P} p^{v_i}
    $
    .



    Then
    $prod_{p in P} p^{m_i}
    =dfrac{u^k}{v^k}
    =dfrac{ prod_{p in P} p^{ku_i}}{prod_{p in P} p^{kv_i}}
    $

    so
    $prod_{p in P} p^{m_i}prod_{p in P} p^{kv_i}
    =prod_{p in P} p^{ku_i}
    $

    or
    $prod_{p in P} p^{m_i+kv_i}
    =prod_{p in P} p^{ku_i}
    $
    .



    By unique factorization,
    $m_i+kv_i
    =ku_i
    $
    ,
    so
    $m_i
    =ku_i-kv_i
    =k(u_i-v_i)
    $
    .



    Therefore
    $m
    =prod_{p in P} p^{m_i}
    =prod_{p in P} p^{k(u_i-v_i)}
    =left(prod_{p in P} p^{u_i-v_i}right)^k
    $

    so $m$ is a perfect $k$-th power.



    Therefore
    the $k$-th root of an integer
    is rational
    only if
    it is a $k$-th power.






    share|cite|improve this answer




























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      0
      down vote













      When you get $2p^2 = 3q^2$ that means that $2|q^2$ so $2|q$ and that $3|p^2$ so $3|p$.



      Then if you replace $p = 3p'$ for some integer $p'$ and $q=2q'$ for some integer $q'$ you get



      $2(3p')^2 = 3(2q')^2$



      $18p'^2 = 12q'^2$



      $3p'^3 = 2q'^2$.



      Can you finish from there?



      This assumes you know Euclid's lemma that 1) prime numbers exist and 2) if $p$ is prime and $p|a*b$ then either $p|a$ or $p|b$ (or both).






      share|cite|improve this answer





















      • I tried to reach it without prime factorisation, can you have a second look?
        – J.Moh
        Nov 15 at 2:42











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      4 Answers
      4






      active

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      4 Answers
      4






      active

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      active

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      active

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      up vote
      2
      down vote













      How about a proof by descent?



      First show that $2^2<6<3^2$. Then if $sqrt{6}$ is to be rational it must have a form $p/q$ where $p,qin mathbb{Z}, q>0, 2q<p<3q$. By simple algebra the square root is also equal to $6q/p$, thus



      $p/q=6q/ptext{.....Eq. 1}$.



      Now if $a/b=c/d$ then also



      $a/b=(ma+nc)/(mb+nd)$



      for any coefficients $m,n$ where the denominator is nonzero. In particular, Eq. 1 implies



      $p/q=(3p-6q)/(3q-p)text{.....Eq. 2}$



      where we already have $2q<p<3q$ and thus $0<3q-p<q$. So the proposed rational fraction $p/q$ must be equal to an alternative rational fraction with a smaller positive denominator. This causes an infinite descent contradiction forcing the assumption of a rational value to be false.



      We can form a similar proof for the square root of any natural number that is not a squared integer. "Not a squared integer" is needed because the square root must be strictly between two adjacent integers to obtain a descent of positive denominators.






      share|cite|improve this answer



























        up vote
        2
        down vote













        How about a proof by descent?



        First show that $2^2<6<3^2$. Then if $sqrt{6}$ is to be rational it must have a form $p/q$ where $p,qin mathbb{Z}, q>0, 2q<p<3q$. By simple algebra the square root is also equal to $6q/p$, thus



        $p/q=6q/ptext{.....Eq. 1}$.



        Now if $a/b=c/d$ then also



        $a/b=(ma+nc)/(mb+nd)$



        for any coefficients $m,n$ where the denominator is nonzero. In particular, Eq. 1 implies



        $p/q=(3p-6q)/(3q-p)text{.....Eq. 2}$



        where we already have $2q<p<3q$ and thus $0<3q-p<q$. So the proposed rational fraction $p/q$ must be equal to an alternative rational fraction with a smaller positive denominator. This causes an infinite descent contradiction forcing the assumption of a rational value to be false.



        We can form a similar proof for the square root of any natural number that is not a squared integer. "Not a squared integer" is needed because the square root must be strictly between two adjacent integers to obtain a descent of positive denominators.






        share|cite|improve this answer

























          up vote
          2
          down vote










          up vote
          2
          down vote









          How about a proof by descent?



          First show that $2^2<6<3^2$. Then if $sqrt{6}$ is to be rational it must have a form $p/q$ where $p,qin mathbb{Z}, q>0, 2q<p<3q$. By simple algebra the square root is also equal to $6q/p$, thus



          $p/q=6q/ptext{.....Eq. 1}$.



          Now if $a/b=c/d$ then also



          $a/b=(ma+nc)/(mb+nd)$



          for any coefficients $m,n$ where the denominator is nonzero. In particular, Eq. 1 implies



          $p/q=(3p-6q)/(3q-p)text{.....Eq. 2}$



          where we already have $2q<p<3q$ and thus $0<3q-p<q$. So the proposed rational fraction $p/q$ must be equal to an alternative rational fraction with a smaller positive denominator. This causes an infinite descent contradiction forcing the assumption of a rational value to be false.



          We can form a similar proof for the square root of any natural number that is not a squared integer. "Not a squared integer" is needed because the square root must be strictly between two adjacent integers to obtain a descent of positive denominators.






          share|cite|improve this answer














          How about a proof by descent?



          First show that $2^2<6<3^2$. Then if $sqrt{6}$ is to be rational it must have a form $p/q$ where $p,qin mathbb{Z}, q>0, 2q<p<3q$. By simple algebra the square root is also equal to $6q/p$, thus



          $p/q=6q/ptext{.....Eq. 1}$.



          Now if $a/b=c/d$ then also



          $a/b=(ma+nc)/(mb+nd)$



          for any coefficients $m,n$ where the denominator is nonzero. In particular, Eq. 1 implies



          $p/q=(3p-6q)/(3q-p)text{.....Eq. 2}$



          where we already have $2q<p<3q$ and thus $0<3q-p<q$. So the proposed rational fraction $p/q$ must be equal to an alternative rational fraction with a smaller positive denominator. This causes an infinite descent contradiction forcing the assumption of a rational value to be false.



          We can form a similar proof for the square root of any natural number that is not a squared integer. "Not a squared integer" is needed because the square root must be strictly between two adjacent integers to obtain a descent of positive denominators.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 21 at 0:03

























          answered Nov 18 at 2:40









          Oscar Lanzi

          11.8k11936




          11.8k11936






















              up vote
              1
              down vote













              Assume $sqrt{6} = {a over b}$ with $a, b in mathbb{Z}$ . This implies



              $$2 cdot 3 b^2 = a^2$$



              Use the fundamental theorem of algebra to decompose both sides into a unique prime factorization. There are an even number of factors of $2$ on the rhs and an odd number of factors of $2$ on the lhs... a contradition. Same for factors of $3$.



              Thus $sqrt{6} neq {a over b}$, i.e., is irrational.






              share|cite|improve this answer























              • Yeah, I tried to reach it without prime factorisation. A second look?
                – J.Moh
                Nov 15 at 2:42












              • You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
                – David G. Stork
                Nov 15 at 2:47












              • Our professor used the value way, so I am following his steps for this, though, thanks!
                – J.Moh
                Nov 15 at 2:50












              • Show him/her the better way! And stand out as a student!
                – David G. Stork
                Nov 15 at 2:51










              • Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
                – J.Moh
                Nov 15 at 3:49

















              up vote
              1
              down vote













              Assume $sqrt{6} = {a over b}$ with $a, b in mathbb{Z}$ . This implies



              $$2 cdot 3 b^2 = a^2$$



              Use the fundamental theorem of algebra to decompose both sides into a unique prime factorization. There are an even number of factors of $2$ on the rhs and an odd number of factors of $2$ on the lhs... a contradition. Same for factors of $3$.



              Thus $sqrt{6} neq {a over b}$, i.e., is irrational.






              share|cite|improve this answer























              • Yeah, I tried to reach it without prime factorisation. A second look?
                – J.Moh
                Nov 15 at 2:42












              • You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
                – David G. Stork
                Nov 15 at 2:47












              • Our professor used the value way, so I am following his steps for this, though, thanks!
                – J.Moh
                Nov 15 at 2:50












              • Show him/her the better way! And stand out as a student!
                – David G. Stork
                Nov 15 at 2:51










              • Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
                – J.Moh
                Nov 15 at 3:49















              up vote
              1
              down vote










              up vote
              1
              down vote









              Assume $sqrt{6} = {a over b}$ with $a, b in mathbb{Z}$ . This implies



              $$2 cdot 3 b^2 = a^2$$



              Use the fundamental theorem of algebra to decompose both sides into a unique prime factorization. There are an even number of factors of $2$ on the rhs and an odd number of factors of $2$ on the lhs... a contradition. Same for factors of $3$.



              Thus $sqrt{6} neq {a over b}$, i.e., is irrational.






              share|cite|improve this answer














              Assume $sqrt{6} = {a over b}$ with $a, b in mathbb{Z}$ . This implies



              $$2 cdot 3 b^2 = a^2$$



              Use the fundamental theorem of algebra to decompose both sides into a unique prime factorization. There are an even number of factors of $2$ on the rhs and an odd number of factors of $2$ on the lhs... a contradition. Same for factors of $3$.



              Thus $sqrt{6} neq {a over b}$, i.e., is irrational.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Nov 16 at 5:11

























              answered Nov 15 at 2:32









              David G. Stork

              9,25721232




              9,25721232












              • Yeah, I tried to reach it without prime factorisation. A second look?
                – J.Moh
                Nov 15 at 2:42












              • You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
                – David G. Stork
                Nov 15 at 2:47












              • Our professor used the value way, so I am following his steps for this, though, thanks!
                – J.Moh
                Nov 15 at 2:50












              • Show him/her the better way! And stand out as a student!
                – David G. Stork
                Nov 15 at 2:51










              • Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
                – J.Moh
                Nov 15 at 3:49




















              • Yeah, I tried to reach it without prime factorisation. A second look?
                – J.Moh
                Nov 15 at 2:42












              • You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
                – David G. Stork
                Nov 15 at 2:47












              • Our professor used the value way, so I am following his steps for this, though, thanks!
                – J.Moh
                Nov 15 at 2:50












              • Show him/her the better way! And stand out as a student!
                – David G. Stork
                Nov 15 at 2:51










              • Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
                – J.Moh
                Nov 15 at 3:49


















              Yeah, I tried to reach it without prime factorisation. A second look?
              – J.Moh
              Nov 15 at 2:42






              Yeah, I tried to reach it without prime factorisation. A second look?
              – J.Moh
              Nov 15 at 2:42














              You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
              – David G. Stork
              Nov 15 at 2:47






              You are considering the value of the rhs and lhs of the equation. I am instead looking at the number of prime factors. Frankly, I think that is so much more elegant... but I suppose this is a matter of taste. Mine applies to $sqrt{5/3}$ whereas yours doesn't... at least not as directly.
              – David G. Stork
              Nov 15 at 2:47














              Our professor used the value way, so I am following his steps for this, though, thanks!
              – J.Moh
              Nov 15 at 2:50






              Our professor used the value way, so I am following his steps for this, though, thanks!
              – J.Moh
              Nov 15 at 2:50














              Show him/her the better way! And stand out as a student!
              – David G. Stork
              Nov 15 at 2:51




              Show him/her the better way! And stand out as a student!
              – David G. Stork
              Nov 15 at 2:51












              Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
              – J.Moh
              Nov 15 at 3:49






              Well, that doesn t apply in Morocco. Pray for me I win visa lottery so I can join a better faculty in the US :)
              – J.Moh
              Nov 15 at 3:49












              up vote
              0
              down vote













              I'll play more generally
              and see what happens.



              Suppose
              $sqrt{m}
              =dfrac{u}{v}
              $

              where
              $m = prod_{p in P} p^{m_i},
              u = prod_{p in P} p^{u_i},
              v = prod_{p in P} p^{v_i}
              $
              .



              Then
              $prod_{p in P} p^{m_i}
              =dfrac{u^2}{v^2}
              =dfrac{ prod_{p in P} p^{2u_i}}{prod_{p in P} p^{2v_i}}
              $

              so
              $prod_{p in P} p^{m_i}prod_{p in P} p^{2v_i}
              =prod_{p in P} p^{2u_i}
              $

              or
              $prod_{p in P} p^{m_i+2v_i}
              =prod_{p in P} p^{2u_i}
              $
              .



              By unique factorization,
              $m_i+2v_i
              =2u_i
              $
              ,
              so
              $m_i
              =2u_i-2v_i
              =2(u_i-v_i)
              $
              .



              Therefore
              $m
              =prod_{p in P} p^{m_i}
              =prod_{p in P} p^{2(u_i-v_i)}
              =left(prod_{p in P} p^{u_i-v_i}right)^2
              $

              so $m$ is a perfect square.



              Therefore
              the square root of an integer
              is rational
              only if
              it is a square.



              I'll now try to generalize this
              to $k$-th roots,
              with as much cut-and-paste
              as possible.



              Suppose
              $sqrt[k]{m}
              =dfrac{u}{v}
              $

              where
              $m = prod_{p in P} p^{m_i},
              u = prod_{p in P} p^{u_i},
              v = prod_{p in P} p^{v_i}
              $
              .



              Then
              $prod_{p in P} p^{m_i}
              =dfrac{u^k}{v^k}
              =dfrac{ prod_{p in P} p^{ku_i}}{prod_{p in P} p^{kv_i}}
              $

              so
              $prod_{p in P} p^{m_i}prod_{p in P} p^{kv_i}
              =prod_{p in P} p^{ku_i}
              $

              or
              $prod_{p in P} p^{m_i+kv_i}
              =prod_{p in P} p^{ku_i}
              $
              .



              By unique factorization,
              $m_i+kv_i
              =ku_i
              $
              ,
              so
              $m_i
              =ku_i-kv_i
              =k(u_i-v_i)
              $
              .



              Therefore
              $m
              =prod_{p in P} p^{m_i}
              =prod_{p in P} p^{k(u_i-v_i)}
              =left(prod_{p in P} p^{u_i-v_i}right)^k
              $

              so $m$ is a perfect $k$-th power.



              Therefore
              the $k$-th root of an integer
              is rational
              only if
              it is a $k$-th power.






              share|cite|improve this answer

























                up vote
                0
                down vote













                I'll play more generally
                and see what happens.



                Suppose
                $sqrt{m}
                =dfrac{u}{v}
                $

                where
                $m = prod_{p in P} p^{m_i},
                u = prod_{p in P} p^{u_i},
                v = prod_{p in P} p^{v_i}
                $
                .



                Then
                $prod_{p in P} p^{m_i}
                =dfrac{u^2}{v^2}
                =dfrac{ prod_{p in P} p^{2u_i}}{prod_{p in P} p^{2v_i}}
                $

                so
                $prod_{p in P} p^{m_i}prod_{p in P} p^{2v_i}
                =prod_{p in P} p^{2u_i}
                $

                or
                $prod_{p in P} p^{m_i+2v_i}
                =prod_{p in P} p^{2u_i}
                $
                .



                By unique factorization,
                $m_i+2v_i
                =2u_i
                $
                ,
                so
                $m_i
                =2u_i-2v_i
                =2(u_i-v_i)
                $
                .



                Therefore
                $m
                =prod_{p in P} p^{m_i}
                =prod_{p in P} p^{2(u_i-v_i)}
                =left(prod_{p in P} p^{u_i-v_i}right)^2
                $

                so $m$ is a perfect square.



                Therefore
                the square root of an integer
                is rational
                only if
                it is a square.



                I'll now try to generalize this
                to $k$-th roots,
                with as much cut-and-paste
                as possible.



                Suppose
                $sqrt[k]{m}
                =dfrac{u}{v}
                $

                where
                $m = prod_{p in P} p^{m_i},
                u = prod_{p in P} p^{u_i},
                v = prod_{p in P} p^{v_i}
                $
                .



                Then
                $prod_{p in P} p^{m_i}
                =dfrac{u^k}{v^k}
                =dfrac{ prod_{p in P} p^{ku_i}}{prod_{p in P} p^{kv_i}}
                $

                so
                $prod_{p in P} p^{m_i}prod_{p in P} p^{kv_i}
                =prod_{p in P} p^{ku_i}
                $

                or
                $prod_{p in P} p^{m_i+kv_i}
                =prod_{p in P} p^{ku_i}
                $
                .



                By unique factorization,
                $m_i+kv_i
                =ku_i
                $
                ,
                so
                $m_i
                =ku_i-kv_i
                =k(u_i-v_i)
                $
                .



                Therefore
                $m
                =prod_{p in P} p^{m_i}
                =prod_{p in P} p^{k(u_i-v_i)}
                =left(prod_{p in P} p^{u_i-v_i}right)^k
                $

                so $m$ is a perfect $k$-th power.



                Therefore
                the $k$-th root of an integer
                is rational
                only if
                it is a $k$-th power.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  I'll play more generally
                  and see what happens.



                  Suppose
                  $sqrt{m}
                  =dfrac{u}{v}
                  $

                  where
                  $m = prod_{p in P} p^{m_i},
                  u = prod_{p in P} p^{u_i},
                  v = prod_{p in P} p^{v_i}
                  $
                  .



                  Then
                  $prod_{p in P} p^{m_i}
                  =dfrac{u^2}{v^2}
                  =dfrac{ prod_{p in P} p^{2u_i}}{prod_{p in P} p^{2v_i}}
                  $

                  so
                  $prod_{p in P} p^{m_i}prod_{p in P} p^{2v_i}
                  =prod_{p in P} p^{2u_i}
                  $

                  or
                  $prod_{p in P} p^{m_i+2v_i}
                  =prod_{p in P} p^{2u_i}
                  $
                  .



                  By unique factorization,
                  $m_i+2v_i
                  =2u_i
                  $
                  ,
                  so
                  $m_i
                  =2u_i-2v_i
                  =2(u_i-v_i)
                  $
                  .



                  Therefore
                  $m
                  =prod_{p in P} p^{m_i}
                  =prod_{p in P} p^{2(u_i-v_i)}
                  =left(prod_{p in P} p^{u_i-v_i}right)^2
                  $

                  so $m$ is a perfect square.



                  Therefore
                  the square root of an integer
                  is rational
                  only if
                  it is a square.



                  I'll now try to generalize this
                  to $k$-th roots,
                  with as much cut-and-paste
                  as possible.



                  Suppose
                  $sqrt[k]{m}
                  =dfrac{u}{v}
                  $

                  where
                  $m = prod_{p in P} p^{m_i},
                  u = prod_{p in P} p^{u_i},
                  v = prod_{p in P} p^{v_i}
                  $
                  .



                  Then
                  $prod_{p in P} p^{m_i}
                  =dfrac{u^k}{v^k}
                  =dfrac{ prod_{p in P} p^{ku_i}}{prod_{p in P} p^{kv_i}}
                  $

                  so
                  $prod_{p in P} p^{m_i}prod_{p in P} p^{kv_i}
                  =prod_{p in P} p^{ku_i}
                  $

                  or
                  $prod_{p in P} p^{m_i+kv_i}
                  =prod_{p in P} p^{ku_i}
                  $
                  .



                  By unique factorization,
                  $m_i+kv_i
                  =ku_i
                  $
                  ,
                  so
                  $m_i
                  =ku_i-kv_i
                  =k(u_i-v_i)
                  $
                  .



                  Therefore
                  $m
                  =prod_{p in P} p^{m_i}
                  =prod_{p in P} p^{k(u_i-v_i)}
                  =left(prod_{p in P} p^{u_i-v_i}right)^k
                  $

                  so $m$ is a perfect $k$-th power.



                  Therefore
                  the $k$-th root of an integer
                  is rational
                  only if
                  it is a $k$-th power.






                  share|cite|improve this answer












                  I'll play more generally
                  and see what happens.



                  Suppose
                  $sqrt{m}
                  =dfrac{u}{v}
                  $

                  where
                  $m = prod_{p in P} p^{m_i},
                  u = prod_{p in P} p^{u_i},
                  v = prod_{p in P} p^{v_i}
                  $
                  .



                  Then
                  $prod_{p in P} p^{m_i}
                  =dfrac{u^2}{v^2}
                  =dfrac{ prod_{p in P} p^{2u_i}}{prod_{p in P} p^{2v_i}}
                  $

                  so
                  $prod_{p in P} p^{m_i}prod_{p in P} p^{2v_i}
                  =prod_{p in P} p^{2u_i}
                  $

                  or
                  $prod_{p in P} p^{m_i+2v_i}
                  =prod_{p in P} p^{2u_i}
                  $
                  .



                  By unique factorization,
                  $m_i+2v_i
                  =2u_i
                  $
                  ,
                  so
                  $m_i
                  =2u_i-2v_i
                  =2(u_i-v_i)
                  $
                  .



                  Therefore
                  $m
                  =prod_{p in P} p^{m_i}
                  =prod_{p in P} p^{2(u_i-v_i)}
                  =left(prod_{p in P} p^{u_i-v_i}right)^2
                  $

                  so $m$ is a perfect square.



                  Therefore
                  the square root of an integer
                  is rational
                  only if
                  it is a square.



                  I'll now try to generalize this
                  to $k$-th roots,
                  with as much cut-and-paste
                  as possible.



                  Suppose
                  $sqrt[k]{m}
                  =dfrac{u}{v}
                  $

                  where
                  $m = prod_{p in P} p^{m_i},
                  u = prod_{p in P} p^{u_i},
                  v = prod_{p in P} p^{v_i}
                  $
                  .



                  Then
                  $prod_{p in P} p^{m_i}
                  =dfrac{u^k}{v^k}
                  =dfrac{ prod_{p in P} p^{ku_i}}{prod_{p in P} p^{kv_i}}
                  $

                  so
                  $prod_{p in P} p^{m_i}prod_{p in P} p^{kv_i}
                  =prod_{p in P} p^{ku_i}
                  $

                  or
                  $prod_{p in P} p^{m_i+kv_i}
                  =prod_{p in P} p^{ku_i}
                  $
                  .



                  By unique factorization,
                  $m_i+kv_i
                  =ku_i
                  $
                  ,
                  so
                  $m_i
                  =ku_i-kv_i
                  =k(u_i-v_i)
                  $
                  .



                  Therefore
                  $m
                  =prod_{p in P} p^{m_i}
                  =prod_{p in P} p^{k(u_i-v_i)}
                  =left(prod_{p in P} p^{u_i-v_i}right)^k
                  $

                  so $m$ is a perfect $k$-th power.



                  Therefore
                  the $k$-th root of an integer
                  is rational
                  only if
                  it is a $k$-th power.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 15 at 1:37









                  marty cohen

                  71.5k546123




                  71.5k546123






















                      up vote
                      0
                      down vote













                      When you get $2p^2 = 3q^2$ that means that $2|q^2$ so $2|q$ and that $3|p^2$ so $3|p$.



                      Then if you replace $p = 3p'$ for some integer $p'$ and $q=2q'$ for some integer $q'$ you get



                      $2(3p')^2 = 3(2q')^2$



                      $18p'^2 = 12q'^2$



                      $3p'^3 = 2q'^2$.



                      Can you finish from there?



                      This assumes you know Euclid's lemma that 1) prime numbers exist and 2) if $p$ is prime and $p|a*b$ then either $p|a$ or $p|b$ (or both).






                      share|cite|improve this answer





















                      • I tried to reach it without prime factorisation, can you have a second look?
                        – J.Moh
                        Nov 15 at 2:42















                      up vote
                      0
                      down vote













                      When you get $2p^2 = 3q^2$ that means that $2|q^2$ so $2|q$ and that $3|p^2$ so $3|p$.



                      Then if you replace $p = 3p'$ for some integer $p'$ and $q=2q'$ for some integer $q'$ you get



                      $2(3p')^2 = 3(2q')^2$



                      $18p'^2 = 12q'^2$



                      $3p'^3 = 2q'^2$.



                      Can you finish from there?



                      This assumes you know Euclid's lemma that 1) prime numbers exist and 2) if $p$ is prime and $p|a*b$ then either $p|a$ or $p|b$ (or both).






                      share|cite|improve this answer





















                      • I tried to reach it without prime factorisation, can you have a second look?
                        – J.Moh
                        Nov 15 at 2:42













                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      When you get $2p^2 = 3q^2$ that means that $2|q^2$ so $2|q$ and that $3|p^2$ so $3|p$.



                      Then if you replace $p = 3p'$ for some integer $p'$ and $q=2q'$ for some integer $q'$ you get



                      $2(3p')^2 = 3(2q')^2$



                      $18p'^2 = 12q'^2$



                      $3p'^3 = 2q'^2$.



                      Can you finish from there?



                      This assumes you know Euclid's lemma that 1) prime numbers exist and 2) if $p$ is prime and $p|a*b$ then either $p|a$ or $p|b$ (or both).






                      share|cite|improve this answer












                      When you get $2p^2 = 3q^2$ that means that $2|q^2$ so $2|q$ and that $3|p^2$ so $3|p$.



                      Then if you replace $p = 3p'$ for some integer $p'$ and $q=2q'$ for some integer $q'$ you get



                      $2(3p')^2 = 3(2q')^2$



                      $18p'^2 = 12q'^2$



                      $3p'^3 = 2q'^2$.



                      Can you finish from there?



                      This assumes you know Euclid's lemma that 1) prime numbers exist and 2) if $p$ is prime and $p|a*b$ then either $p|a$ or $p|b$ (or both).







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Nov 15 at 2:28









                      fleablood

                      66.5k22684




                      66.5k22684












                      • I tried to reach it without prime factorisation, can you have a second look?
                        – J.Moh
                        Nov 15 at 2:42


















                      • I tried to reach it without prime factorisation, can you have a second look?
                        – J.Moh
                        Nov 15 at 2:42
















                      I tried to reach it without prime factorisation, can you have a second look?
                      – J.Moh
                      Nov 15 at 2:42




                      I tried to reach it without prime factorisation, can you have a second look?
                      – J.Moh
                      Nov 15 at 2:42


















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