Can the product of two rational numbers be an irrational number? (Kindly see the example in description)












-1












$begingroup$


I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = frac{1}{7} ; ;;; b = frac{22}{1} ;$$



both a, b are individually rational (either repeating or terminating decimal vlaues) $$ a = 0.overline{142857} ; ;;;b = 22.0 ; $$
but their product $$ frac{22}{7}=3.14159265359
...$$ which is clearly irrational .



Then how is multiplication closed on rational numbers??






irrational-numbers rational-numbers decimal-expansion







share|cite|improve this question











$endgroup$












  • $begingroup$
    $ab =frac{22}{7}$ which is rational.
    $endgroup$
    – lightxbulb
    Jan 21 at 17:57






  • 1




    $begingroup$
    If you have expressed it as a fraction, how can that be irrational?
    $endgroup$
    – idriskameni
    Jan 21 at 17:58






  • 2




    $begingroup$
    If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
    $endgroup$
    – Keith Backman
    Jan 21 at 18:00










  • $begingroup$
    You seem to think that $pi={22over7}$ This is false.
    $endgroup$
    – saulspatz
    Jan 21 at 18:00










  • $begingroup$
    $dfrac{22}{7}=3.overline{142857}$
    $endgroup$
    – Henry
    Jan 21 at 18:00


















-1












$begingroup$


I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = frac{1}{7} ; ;;; b = frac{22}{1} ;$$



both a, b are individually rational (either repeating or terminating decimal vlaues) $$ a = 0.overline{142857} ; ;;;b = 22.0 ; $$
but their product $$ frac{22}{7}=3.14159265359
...$$ which is clearly irrational .



Then how is multiplication closed on rational numbers??






irrational-numbers rational-numbers decimal-expansion







share|cite|improve this question











$endgroup$












  • $begingroup$
    $ab =frac{22}{7}$ which is rational.
    $endgroup$
    – lightxbulb
    Jan 21 at 17:57






  • 1




    $begingroup$
    If you have expressed it as a fraction, how can that be irrational?
    $endgroup$
    – idriskameni
    Jan 21 at 17:58






  • 2




    $begingroup$
    If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
    $endgroup$
    – Keith Backman
    Jan 21 at 18:00










  • $begingroup$
    You seem to think that $pi={22over7}$ This is false.
    $endgroup$
    – saulspatz
    Jan 21 at 18:00










  • $begingroup$
    $dfrac{22}{7}=3.overline{142857}$
    $endgroup$
    – Henry
    Jan 21 at 18:00
















-1












-1








-1





$begingroup$


I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = frac{1}{7} ; ;;; b = frac{22}{1} ;$$



both a, b are individually rational (either repeating or terminating decimal vlaues) $$ a = 0.overline{142857} ; ;;;b = 22.0 ; $$
but their product $$ frac{22}{7}=3.14159265359
...$$ which is clearly irrational .



Then how is multiplication closed on rational numbers??






irrational-numbers rational-numbers decimal-expansion







share|cite|improve this question











$endgroup$




I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = frac{1}{7} ; ;;; b = frac{22}{1} ;$$



both a, b are individually rational (either repeating or terminating decimal vlaues) $$ a = 0.overline{142857} ; ;;;b = 22.0 ; $$
but their product $$ frac{22}{7}=3.14159265359
...$$ which is clearly irrational .



Then how is multiplication closed on rational numbers??






irrational-numbers rational-numbers decimal-expansion




irrational-numbers rational-numbers decimal-expansion






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 21 at 18:04









José Carlos Santos

163k22131234




163k22131234










asked Jan 21 at 17:55









Krishna SaiKrishna Sai

62




62












  • $begingroup$
    $ab =frac{22}{7}$ which is rational.
    $endgroup$
    – lightxbulb
    Jan 21 at 17:57






  • 1




    $begingroup$
    If you have expressed it as a fraction, how can that be irrational?
    $endgroup$
    – idriskameni
    Jan 21 at 17:58






  • 2




    $begingroup$
    If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
    $endgroup$
    – Keith Backman
    Jan 21 at 18:00










  • $begingroup$
    You seem to think that $pi={22over7}$ This is false.
    $endgroup$
    – saulspatz
    Jan 21 at 18:00










  • $begingroup$
    $dfrac{22}{7}=3.overline{142857}$
    $endgroup$
    – Henry
    Jan 21 at 18:00




















  • $begingroup$
    $ab =frac{22}{7}$ which is rational.
    $endgroup$
    – lightxbulb
    Jan 21 at 17:57






  • 1




    $begingroup$
    If you have expressed it as a fraction, how can that be irrational?
    $endgroup$
    – idriskameni
    Jan 21 at 17:58






  • 2




    $begingroup$
    If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
    $endgroup$
    – Keith Backman
    Jan 21 at 18:00










  • $begingroup$
    You seem to think that $pi={22over7}$ This is false.
    $endgroup$
    – saulspatz
    Jan 21 at 18:00










  • $begingroup$
    $dfrac{22}{7}=3.overline{142857}$
    $endgroup$
    – Henry
    Jan 21 at 18:00


















$begingroup$
$ab =frac{22}{7}$ which is rational.
$endgroup$
– lightxbulb
Jan 21 at 17:57




$begingroup$
$ab =frac{22}{7}$ which is rational.
$endgroup$
– lightxbulb
Jan 21 at 17:57




1




1




$begingroup$
If you have expressed it as a fraction, how can that be irrational?
$endgroup$
– idriskameni
Jan 21 at 17:58




$begingroup$
If you have expressed it as a fraction, how can that be irrational?
$endgroup$
– idriskameni
Jan 21 at 17:58




2




2




$begingroup$
If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
$endgroup$
– Keith Backman
Jan 21 at 18:00




$begingroup$
If you think $frac{22}{7}=pi$ that is not the case. $frac{22}{7}$ is only an approximation of $pi$.
$endgroup$
– Keith Backman
Jan 21 at 18:00












$begingroup$
You seem to think that $pi={22over7}$ This is false.
$endgroup$
– saulspatz
Jan 21 at 18:00




$begingroup$
You seem to think that $pi={22over7}$ This is false.
$endgroup$
– saulspatz
Jan 21 at 18:00












$begingroup$
$dfrac{22}{7}=3.overline{142857}$
$endgroup$
– Henry
Jan 21 at 18:00






$begingroup$
$dfrac{22}{7}=3.overline{142857}$
$endgroup$
– Henry
Jan 21 at 18:00












3 Answers
3






active

oldest

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0












$begingroup$

$frac {22} 7 = 3.overline{142857}$, which is a rational approximation of $pi$ but not exactly $pi$.






share|cite|improve this answer











$endgroup$





















    2












    $begingroup$

    You are wrong: $displaystylefrac{22}7=3.142857142857142857ldots$ and this periodic. Not to mention that by definition $dfrac{22}7$ is rational.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Yes I understand now, 22/7 is mere approximation of π. Thanks.
      $endgroup$
      – Krishna Sai
      Jan 21 at 18:02





















    0












    $begingroup$

    Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $frac{22}{7}$ is rational. The fact that this ratio approximates $pi$ is just an interesting coincidence.






    share|cite|improve this answer









    $endgroup$













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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

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      active

      oldest

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      0












      $begingroup$

      $frac {22} 7 = 3.overline{142857}$, which is a rational approximation of $pi$ but not exactly $pi$.






      share|cite|improve this answer











      $endgroup$


















        0












        $begingroup$

        $frac {22} 7 = 3.overline{142857}$, which is a rational approximation of $pi$ but not exactly $pi$.






        share|cite|improve this answer











        $endgroup$
















          0












          0








          0





          $begingroup$

          $frac {22} 7 = 3.overline{142857}$, which is a rational approximation of $pi$ but not exactly $pi$.






          share|cite|improve this answer











          $endgroup$



          $frac {22} 7 = 3.overline{142857}$, which is a rational approximation of $pi$ but not exactly $pi$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 21 at 18:15

























          answered Jan 21 at 18:03









          J. W. TannerJ. W. Tanner

          2,4581117




          2,4581117























              2












              $begingroup$

              You are wrong: $displaystylefrac{22}7=3.142857142857142857ldots$ and this periodic. Not to mention that by definition $dfrac{22}7$ is rational.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Yes I understand now, 22/7 is mere approximation of π. Thanks.
                $endgroup$
                – Krishna Sai
                Jan 21 at 18:02


















              2












              $begingroup$

              You are wrong: $displaystylefrac{22}7=3.142857142857142857ldots$ and this periodic. Not to mention that by definition $dfrac{22}7$ is rational.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Yes I understand now, 22/7 is mere approximation of π. Thanks.
                $endgroup$
                – Krishna Sai
                Jan 21 at 18:02
















              2












              2








              2





              $begingroup$

              You are wrong: $displaystylefrac{22}7=3.142857142857142857ldots$ and this periodic. Not to mention that by definition $dfrac{22}7$ is rational.






              share|cite|improve this answer









              $endgroup$



              You are wrong: $displaystylefrac{22}7=3.142857142857142857ldots$ and this periodic. Not to mention that by definition $dfrac{22}7$ is rational.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Jan 21 at 18:01









              José Carlos SantosJosé Carlos Santos

              163k22131234




              163k22131234












              • $begingroup$
                Yes I understand now, 22/7 is mere approximation of π. Thanks.
                $endgroup$
                – Krishna Sai
                Jan 21 at 18:02




















              • $begingroup$
                Yes I understand now, 22/7 is mere approximation of π. Thanks.
                $endgroup$
                – Krishna Sai
                Jan 21 at 18:02


















              $begingroup$
              Yes I understand now, 22/7 is mere approximation of π. Thanks.
              $endgroup$
              – Krishna Sai
              Jan 21 at 18:02






              $begingroup$
              Yes I understand now, 22/7 is mere approximation of π. Thanks.
              $endgroup$
              – Krishna Sai
              Jan 21 at 18:02













              0












              $begingroup$

              Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $frac{22}{7}$ is rational. The fact that this ratio approximates $pi$ is just an interesting coincidence.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $frac{22}{7}$ is rational. The fact that this ratio approximates $pi$ is just an interesting coincidence.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $frac{22}{7}$ is rational. The fact that this ratio approximates $pi$ is just an interesting coincidence.






                  share|cite|improve this answer









                  $endgroup$



                  Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $frac{22}{7}$ is rational. The fact that this ratio approximates $pi$ is just an interesting coincidence.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 21 at 18:11









                  GnumbertesterGnumbertester

                  623113




                  623113






























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