How is rearranging 56 x 100 ÷ 8 into 56 ÷8 x100 allowed by the commutative property?












1












$begingroup$


So according to the commutative property for multiplication:



$a * b = b * a$



However this does not hold for division



$a ÷ b$ $!= b ÷a$



Why is it that in the following case:



$56 * 100 ÷ 8 = 56 ÷ 8* 100$



It seems like division is breaking the rule. There is something I am misunderstanding here.



Is it because $a*b÷c=a÷c*b$ is allowed since $b÷c$ are not being rearranged so that $c÷b$?



If this is the case are you allowed to rearrange values in equations so long as no values have the form $a ÷ b$ $= b ÷a$ and $a - b$ $= b -a$ ?






arithmetic







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    So according to the commutative property for multiplication:



    $a * b = b * a$



    However this does not hold for division



    $a ÷ b$ $!= b ÷a$



    Why is it that in the following case:



    $56 * 100 ÷ 8 = 56 ÷ 8* 100$



    It seems like division is breaking the rule. There is something I am misunderstanding here.



    Is it because $a*b÷c=a÷c*b$ is allowed since $b÷c$ are not being rearranged so that $c÷b$?



    If this is the case are you allowed to rearrange values in equations so long as no values have the form $a ÷ b$ $= b ÷a$ and $a - b$ $= b -a$ ?






    arithmetic







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      So according to the commutative property for multiplication:



      $a * b = b * a$



      However this does not hold for division



      $a ÷ b$ $!= b ÷a$



      Why is it that in the following case:



      $56 * 100 ÷ 8 = 56 ÷ 8* 100$



      It seems like division is breaking the rule. There is something I am misunderstanding here.



      Is it because $a*b÷c=a÷c*b$ is allowed since $b÷c$ are not being rearranged so that $c÷b$?



      If this is the case are you allowed to rearrange values in equations so long as no values have the form $a ÷ b$ $= b ÷a$ and $a - b$ $= b -a$ ?






      arithmetic







      share|cite|improve this question











      $endgroup$




      So according to the commutative property for multiplication:



      $a * b = b * a$



      However this does not hold for division



      $a ÷ b$ $!= b ÷a$



      Why is it that in the following case:



      $56 * 100 ÷ 8 = 56 ÷ 8* 100$



      It seems like division is breaking the rule. There is something I am misunderstanding here.



      Is it because $a*b÷c=a÷c*b$ is allowed since $b÷c$ are not being rearranged so that $c÷b$?



      If this is the case are you allowed to rearrange values in equations so long as no values have the form $a ÷ b$ $= b ÷a$ and $a - b$ $= b -a$ ?






      arithmetic




      arithmetic






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      share|cite|improve this question













      share|cite|improve this question




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      edited 58 mins ago







      Sphygmomanometer

















      asked 1 hour ago









      SphygmomanometerSphygmomanometer

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          5 Answers
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          active

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          1












          $begingroup$

          Notice that you have always $div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $div c$ as a multiplication with $d=1/c$. Then everything would look easier:
          $$atimes bdiv c=atimes btimes d=atimes dtimes b=adiv ctimes b$$






          share|cite|improve this answer









          $endgroup$





















            2












            $begingroup$

            The non-zero real numbers form an abelian group under multiplication. The notation $adiv b$ is shorthand for $ab^{-1}$.



            So $ab^{-1}ne ba^{-1}$ but $56times100 times8^{-1}=56times 8^{-1}times 100$






            share|cite|improve this answer









            $endgroup$





















              1












              $begingroup$

              Hope this makes sense.



              $$ atimes b ÷ c $$



              $$=atimesdfrac{b}{c}$$



              $$=atimes btimesdfrac{1}{c}$$



              $$=(atimesdfrac{1}{c})times b$$



              $$=dfrac{a}{c}times b$$



              $$=a÷ctimes b$$






              share|cite|improve this answer








              New contributor




              Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$





















                1












                $begingroup$

                Because division is the inverse of multiplication, that is: $$X div Y =Xcdot frac 1Y$$



                So you have: $$56cdot 100 cdot frac18 =56cdotfrac18cdot100$$
                Which is obvious.






                share|cite|improve this answer









                $endgroup$





















                  0












                  $begingroup$

                  Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are carried out.



                  So $adiv b times c$ is being interpreted from left to right as $(adiv b)times c=cfrac {ac}b$, but done from right to left $adiv (btimes c)=cfrac a{bc}$ and the two results are not the same.



                  Likewise with $adiv b div c$ we have $(adiv b)div c=cfrac a{bc} neq adiv (bdiv c)=cfrac {ac}b$.



                  So the conventional assumption that multiplication and division are operations of equal status and are carried out from left to right does make a difference in these cases and changes the result. This may be what is feeding your intuition that there is a potential problem with order.






                  share|cite|improve this answer









                  $endgroup$













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                    5 Answers
                    5






                    active

                    oldest

                    votes








                    5 Answers
                    5






                    active

                    oldest

                    votes









                    active

                    oldest

                    votes






                    active

                    oldest

                    votes









                    1












                    $begingroup$

                    Notice that you have always $div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $div c$ as a multiplication with $d=1/c$. Then everything would look easier:
                    $$atimes bdiv c=atimes btimes d=atimes dtimes b=adiv ctimes b$$






                    share|cite|improve this answer









                    $endgroup$


















                      1












                      $begingroup$

                      Notice that you have always $div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $div c$ as a multiplication with $d=1/c$. Then everything would look easier:
                      $$atimes bdiv c=atimes btimes d=atimes dtimes b=adiv ctimes b$$






                      share|cite|improve this answer









                      $endgroup$
















                        1












                        1








                        1





                        $begingroup$

                        Notice that you have always $div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $div c$ as a multiplication with $d=1/c$. Then everything would look easier:
                        $$atimes bdiv c=atimes btimes d=atimes dtimes b=adiv ctimes b$$






                        share|cite|improve this answer









                        $endgroup$



                        Notice that you have always $div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $div c$ as a multiplication with $d=1/c$. Then everything would look easier:
                        $$atimes bdiv c=atimes btimes d=atimes dtimes b=adiv ctimes b$$







                        share|cite|improve this answer












                        share|cite|improve this answer



                        share|cite|improve this answer










                        answered 51 mins ago









                        AndreiAndrei

                        12.1k21127




                        12.1k21127























                            2












                            $begingroup$

                            The non-zero real numbers form an abelian group under multiplication. The notation $adiv b$ is shorthand for $ab^{-1}$.



                            So $ab^{-1}ne ba^{-1}$ but $56times100 times8^{-1}=56times 8^{-1}times 100$






                            share|cite|improve this answer









                            $endgroup$


















                              2












                              $begingroup$

                              The non-zero real numbers form an abelian group under multiplication. The notation $adiv b$ is shorthand for $ab^{-1}$.



                              So $ab^{-1}ne ba^{-1}$ but $56times100 times8^{-1}=56times 8^{-1}times 100$






                              share|cite|improve this answer









                              $endgroup$
















                                2












                                2








                                2





                                $begingroup$

                                The non-zero real numbers form an abelian group under multiplication. The notation $adiv b$ is shorthand for $ab^{-1}$.



                                So $ab^{-1}ne ba^{-1}$ but $56times100 times8^{-1}=56times 8^{-1}times 100$






                                share|cite|improve this answer









                                $endgroup$



                                The non-zero real numbers form an abelian group under multiplication. The notation $adiv b$ is shorthand for $ab^{-1}$.



                                So $ab^{-1}ne ba^{-1}$ but $56times100 times8^{-1}=56times 8^{-1}times 100$







                                share|cite|improve this answer












                                share|cite|improve this answer



                                share|cite|improve this answer










                                answered 51 mins ago









                                Angela RichardsonAngela Richardson

                                5,28911733




                                5,28911733























                                    1












                                    $begingroup$

                                    Hope this makes sense.



                                    $$ atimes b ÷ c $$



                                    $$=atimesdfrac{b}{c}$$



                                    $$=atimes btimesdfrac{1}{c}$$



                                    $$=(atimesdfrac{1}{c})times b$$



                                    $$=dfrac{a}{c}times b$$



                                    $$=a÷ctimes b$$






                                    share|cite|improve this answer








                                    New contributor




                                    Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                    Check out our Code of Conduct.






                                    $endgroup$


















                                      1












                                      $begingroup$

                                      Hope this makes sense.



                                      $$ atimes b ÷ c $$



                                      $$=atimesdfrac{b}{c}$$



                                      $$=atimes btimesdfrac{1}{c}$$



                                      $$=(atimesdfrac{1}{c})times b$$



                                      $$=dfrac{a}{c}times b$$



                                      $$=a÷ctimes b$$






                                      share|cite|improve this answer








                                      New contributor




                                      Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                      Check out our Code of Conduct.






                                      $endgroup$
















                                        1












                                        1








                                        1





                                        $begingroup$

                                        Hope this makes sense.



                                        $$ atimes b ÷ c $$



                                        $$=atimesdfrac{b}{c}$$



                                        $$=atimes btimesdfrac{1}{c}$$



                                        $$=(atimesdfrac{1}{c})times b$$



                                        $$=dfrac{a}{c}times b$$



                                        $$=a÷ctimes b$$






                                        share|cite|improve this answer








                                        New contributor




                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.






                                        $endgroup$



                                        Hope this makes sense.



                                        $$ atimes b ÷ c $$



                                        $$=atimesdfrac{b}{c}$$



                                        $$=atimes btimesdfrac{1}{c}$$



                                        $$=(atimesdfrac{1}{c})times b$$



                                        $$=dfrac{a}{c}times b$$



                                        $$=a÷ctimes b$$







                                        share|cite|improve this answer








                                        New contributor




                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.









                                        share|cite|improve this answer



                                        share|cite|improve this answer






                                        New contributor




                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.









                                        answered 47 mins ago









                                        AdityaAditya

                                        112




                                        112




                                        New contributor




                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.





                                        New contributor





                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.






                                        Aditya is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                        Check out our Code of Conduct.























                                            1












                                            $begingroup$

                                            Because division is the inverse of multiplication, that is: $$X div Y =Xcdot frac 1Y$$



                                            So you have: $$56cdot 100 cdot frac18 =56cdotfrac18cdot100$$
                                            Which is obvious.






                                            share|cite|improve this answer









                                            $endgroup$


















                                              1












                                              $begingroup$

                                              Because division is the inverse of multiplication, that is: $$X div Y =Xcdot frac 1Y$$



                                              So you have: $$56cdot 100 cdot frac18 =56cdotfrac18cdot100$$
                                              Which is obvious.






                                              share|cite|improve this answer









                                              $endgroup$
















                                                1












                                                1








                                                1





                                                $begingroup$

                                                Because division is the inverse of multiplication, that is: $$X div Y =Xcdot frac 1Y$$



                                                So you have: $$56cdot 100 cdot frac18 =56cdotfrac18cdot100$$
                                                Which is obvious.






                                                share|cite|improve this answer









                                                $endgroup$



                                                Because division is the inverse of multiplication, that is: $$X div Y =Xcdot frac 1Y$$



                                                So you have: $$56cdot 100 cdot frac18 =56cdotfrac18cdot100$$
                                                Which is obvious.







                                                share|cite|improve this answer












                                                share|cite|improve this answer



                                                share|cite|improve this answer










                                                answered 47 mins ago









                                                Rhys HughesRhys Hughes

                                                6,7101530




                                                6,7101530























                                                    0












                                                    $begingroup$

                                                    Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are carried out.



                                                    So $adiv b times c$ is being interpreted from left to right as $(adiv b)times c=cfrac {ac}b$, but done from right to left $adiv (btimes c)=cfrac a{bc}$ and the two results are not the same.



                                                    Likewise with $adiv b div c$ we have $(adiv b)div c=cfrac a{bc} neq adiv (bdiv c)=cfrac {ac}b$.



                                                    So the conventional assumption that multiplication and division are operations of equal status and are carried out from left to right does make a difference in these cases and changes the result. This may be what is feeding your intuition that there is a potential problem with order.






                                                    share|cite|improve this answer









                                                    $endgroup$


















                                                      0












                                                      $begingroup$

                                                      Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are carried out.



                                                      So $adiv b times c$ is being interpreted from left to right as $(adiv b)times c=cfrac {ac}b$, but done from right to left $adiv (btimes c)=cfrac a{bc}$ and the two results are not the same.



                                                      Likewise with $adiv b div c$ we have $(adiv b)div c=cfrac a{bc} neq adiv (bdiv c)=cfrac {ac}b$.



                                                      So the conventional assumption that multiplication and division are operations of equal status and are carried out from left to right does make a difference in these cases and changes the result. This may be what is feeding your intuition that there is a potential problem with order.






                                                      share|cite|improve this answer









                                                      $endgroup$
















                                                        0












                                                        0








                                                        0





                                                        $begingroup$

                                                        Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are carried out.



                                                        So $adiv b times c$ is being interpreted from left to right as $(adiv b)times c=cfrac {ac}b$, but done from right to left $adiv (btimes c)=cfrac a{bc}$ and the two results are not the same.



                                                        Likewise with $adiv b div c$ we have $(adiv b)div c=cfrac a{bc} neq adiv (bdiv c)=cfrac {ac}b$.



                                                        So the conventional assumption that multiplication and division are operations of equal status and are carried out from left to right does make a difference in these cases and changes the result. This may be what is feeding your intuition that there is a potential problem with order.






                                                        share|cite|improve this answer









                                                        $endgroup$



                                                        Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are carried out.



                                                        So $adiv b times c$ is being interpreted from left to right as $(adiv b)times c=cfrac {ac}b$, but done from right to left $adiv (btimes c)=cfrac a{bc}$ and the two results are not the same.



                                                        Likewise with $adiv b div c$ we have $(adiv b)div c=cfrac a{bc} neq adiv (bdiv c)=cfrac {ac}b$.



                                                        So the conventional assumption that multiplication and division are operations of equal status and are carried out from left to right does make a difference in these cases and changes the result. This may be what is feeding your intuition that there is a potential problem with order.







                                                        share|cite|improve this answer












                                                        share|cite|improve this answer



                                                        share|cite|improve this answer










                                                        answered 15 mins ago









                                                        Mark BennetMark Bennet

                                                        81.3k983180




                                                        81.3k983180






























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