$leftlfloor frac{a-b}{2} rightrfloor + leftlceil frac{a+b}{2} rightrceil = a$ when $a,b$ are integers?...












4












$begingroup$


Let $a$ and $b$ be positive integers.



If $b$ is even, then we have $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil frac{a+b}{2} rightrceil = a$$



I think the equality also hold when $b$ is odd. What could be a proof for it?










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$endgroup$



closed as off-topic by user21820, Lord_Farin, Did, RRL, Trevor Gunn Jan 28 at 16:37


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Lord_Farin, Did, RRL, Trevor Gunn

If this question can be reworded to fit the rules in the help center, please edit the question.
















  • $begingroup$
    Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
    $endgroup$
    – John Hughes
    Jan 25 at 12:58






  • 1




    $begingroup$
    @JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
    $endgroup$
    – The Great Duck
    Jan 26 at 1:32










  • $begingroup$
    You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
    $endgroup$
    – John Hughes
    Jan 26 at 3:23
















4












$begingroup$


Let $a$ and $b$ be positive integers.



If $b$ is even, then we have $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil frac{a+b}{2} rightrceil = a$$



I think the equality also hold when $b$ is odd. What could be a proof for it?










share|cite|improve this question











$endgroup$



closed as off-topic by user21820, Lord_Farin, Did, RRL, Trevor Gunn Jan 28 at 16:37


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Lord_Farin, Did, RRL, Trevor Gunn

If this question can be reworded to fit the rules in the help center, please edit the question.
















  • $begingroup$
    Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
    $endgroup$
    – John Hughes
    Jan 25 at 12:58






  • 1




    $begingroup$
    @JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
    $endgroup$
    – The Great Duck
    Jan 26 at 1:32










  • $begingroup$
    You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
    $endgroup$
    – John Hughes
    Jan 26 at 3:23














4












4








4


1



$begingroup$


Let $a$ and $b$ be positive integers.



If $b$ is even, then we have $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil frac{a+b}{2} rightrceil = a$$



I think the equality also hold when $b$ is odd. What could be a proof for it?










share|cite|improve this question











$endgroup$




Let $a$ and $b$ be positive integers.



If $b$ is even, then we have $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil frac{a+b}{2} rightrceil = a$$



I think the equality also hold when $b$ is odd. What could be a proof for it?







floor-function ceiling-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 26 at 10:12









user21820

39k543153




39k543153










asked Jan 25 at 12:49









Adam54Adam54

915




915




closed as off-topic by user21820, Lord_Farin, Did, RRL, Trevor Gunn Jan 28 at 16:37


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Lord_Farin, Did, RRL, Trevor Gunn

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by user21820, Lord_Farin, Did, RRL, Trevor Gunn Jan 28 at 16:37


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Lord_Farin, Did, RRL, Trevor Gunn

If this question can be reworded to fit the rules in the help center, please edit the question.












  • $begingroup$
    Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
    $endgroup$
    – John Hughes
    Jan 25 at 12:58






  • 1




    $begingroup$
    @JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
    $endgroup$
    – The Great Duck
    Jan 26 at 1:32










  • $begingroup$
    You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
    $endgroup$
    – John Hughes
    Jan 26 at 3:23


















  • $begingroup$
    Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
    $endgroup$
    – John Hughes
    Jan 25 at 12:58






  • 1




    $begingroup$
    @JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
    $endgroup$
    – The Great Duck
    Jan 26 at 1:32










  • $begingroup$
    You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
    $endgroup$
    – John Hughes
    Jan 26 at 3:23
















$begingroup$
Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
$endgroup$
– John Hughes
Jan 25 at 12:58




$begingroup$
Write $b$ as $2k + 1$, where $k$ is a nonnegative integer. Then look at two cases: $a = 2s$ ($s$ a positive integer), and $a = 2s + 1$ ($s$ a nonnegative integer). What do you get for the left-hand side in each case? [You can show your work by clicking "edit" beneath your question.]
$endgroup$
– John Hughes
Jan 25 at 12:58




1




1




$begingroup$
@JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
$endgroup$
– The Great Duck
Jan 26 at 1:32




$begingroup$
@JohnHughes why are you presuming the existence of work? Perhaps they have no work and no idea, and to now do work and retroactively add it to the question would be the equivalent of answering it, so they shouldn't edit to add new work that wasn't done before originally asking.
$endgroup$
– The Great Duck
Jan 26 at 1:32












$begingroup$
You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
$endgroup$
– John Hughes
Jan 26 at 3:23




$begingroup$
You might notice that my comment appeared a few minutes after the question was asked, and before lots of other folks posted answers. Either way, I generally like to lead people in the direction of answering their own questions when I can (esp. when they look to me like homework questions).
$endgroup$
– John Hughes
Jan 26 at 3:23










5 Answers
5






active

oldest

votes


















14












$begingroup$

If $a$ and $b$ are both odd (or both even), then $a-b$ and $a+b$ are both even, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2}.$$



Otherwise, if exactly one of $a$ and $b$ is odd, then $a-b$ and $a+b$ are both odd, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} - frac12 quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2} + frac12.$$



In either case, it's easy to check that your equation holds.





BTW, as noted by Gareth McCaughan, your equation in fact holds for all real numbers $b$, as long as $a$ is an integer. One fairly simple way to show this is to note that $frac{a+b}{2} = a - frac{a-b}{2}.$ Thus, we can rewrite your equation as $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil a - frac{a-b}{2} rightrceil = a.$$



Since $a$ is an integer (by assumption), and since $lceil k + x rceil = k + lceil x rceil$ for any integer $k$, we can extract $a$ from the ceiling term to get $$leftlfloor frac{a-b}{2} rightrfloor + a + leftlceil - frac{a-b}{2} rightrceil = a,$$ and finally, by applying the identity $lceil -x rceil = - lfloor x rfloor$, rewrite this as $$leftlfloor frac{a-b}{2} rightrfloor + a - leftlfloor frac{a-b}{2} rightrfloor = a.$$ Cancelling the floor terms then just leaves the identity $a = a$.



On the other hand, as also noted by Gareth, your equation cannot hold for any non-integer $a$, since its left-hand side is always an integer.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Thank you very much for the odd case, made very simple!
    $endgroup$
    – Adam54
    Jan 25 at 13:05



















6












$begingroup$

The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then



$$leftlfloorfrac02rightrfloor+leftlceilfrac02rightrceil=0$$
and
$$leftlfloorfrac12rightrfloor+leftlceilfrac12rightrceil=1$$
are enough as a proof.






share|cite|improve this answer









$endgroup$





















    6












    $begingroup$

    This actually works whatever the value of $b$ -- it doesn't need to be an integer. Clearly it's true when $b=0$. Now imagine changing $b$ smoothly from $0$ to its final value. When does the value of our expression change? Precisely when $(a+b)/2$ or $(a-b)/2$ passes an integer; that is, when $apm b$ is an even integer; that is, when $b$ differs from $a$ by an even integer. When this happens, both terms change in opposite ways, so the expression as a whole doesn't change its value. So by the time $b$ reaches its final value, our expression still hasn't changed.



    (On the other hand, if $a$ isn't an integer then the equation never holds because one side is an integer and the other isn't.)






    share|cite|improve this answer









    $endgroup$





















      4












      $begingroup$

      If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.



      Then, we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m+1)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m+1)+(2n+1)}2rightrceil\
      &=leftlfloorfrac{2m-2n}2rightrfloor+ leftlceilfrac{2m+2n+2}2rightrceil\
      &=lfloor m-nrfloor + lceil m+n+1rceil\
      &= m-n + m+n+1tag{$*$}\
      &= 2m+1\
      &=aend{align}



      We can get to $(*)$ because $m$ and $n$ are integers and so their floor (or ceiling) is just the number inside (two integers added or subtracted will always give an integer answer).



      If $a$ is even and $b$ is odd, then we have $a=2m$ and $b=2n+1$, again for $m$ and $n$ integers.



      So we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m)+(2n+1)}2rightrceil\
      &=leftlfloorfrac{2m-2n-1}2rightrfloor+ leftlceilfrac{2m+2n+1}2rightrceil\
      &=leftlfloor m-n-frac 12rightrfloor + leftlceil m+n+frac12rightrceil\
      &= m-n -1+ m+n+1tag{$dagger$}\
      &= 2m\
      &=aend{align}



      Here we get to $(dagger)$ because we round the first half down and the second half up to their respective nearest integers






      share|cite|improve this answer









      $endgroup$





















        2












        $begingroup$

        As $s:=a+b$ and $a-b$ have the same parity, we can write



        $$leftlfloorfrac{s-2b}2rightrfloor+leftlceilfrac s2rightrceil=leftlfloorfrac s2rightrfloor+leftlceilfrac s2rightrceil-b=s-b=a.$$






        share|cite|improve this answer









        $endgroup$




















          5 Answers
          5






          active

          oldest

          votes








          5 Answers
          5






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          14












          $begingroup$

          If $a$ and $b$ are both odd (or both even), then $a-b$ and $a+b$ are both even, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2}.$$



          Otherwise, if exactly one of $a$ and $b$ is odd, then $a-b$ and $a+b$ are both odd, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} - frac12 quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2} + frac12.$$



          In either case, it's easy to check that your equation holds.





          BTW, as noted by Gareth McCaughan, your equation in fact holds for all real numbers $b$, as long as $a$ is an integer. One fairly simple way to show this is to note that $frac{a+b}{2} = a - frac{a-b}{2}.$ Thus, we can rewrite your equation as $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil a - frac{a-b}{2} rightrceil = a.$$



          Since $a$ is an integer (by assumption), and since $lceil k + x rceil = k + lceil x rceil$ for any integer $k$, we can extract $a$ from the ceiling term to get $$leftlfloor frac{a-b}{2} rightrfloor + a + leftlceil - frac{a-b}{2} rightrceil = a,$$ and finally, by applying the identity $lceil -x rceil = - lfloor x rfloor$, rewrite this as $$leftlfloor frac{a-b}{2} rightrfloor + a - leftlfloor frac{a-b}{2} rightrfloor = a.$$ Cancelling the floor terms then just leaves the identity $a = a$.



          On the other hand, as also noted by Gareth, your equation cannot hold for any non-integer $a$, since its left-hand side is always an integer.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Thank you very much for the odd case, made very simple!
            $endgroup$
            – Adam54
            Jan 25 at 13:05
















          14












          $begingroup$

          If $a$ and $b$ are both odd (or both even), then $a-b$ and $a+b$ are both even, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2}.$$



          Otherwise, if exactly one of $a$ and $b$ is odd, then $a-b$ and $a+b$ are both odd, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} - frac12 quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2} + frac12.$$



          In either case, it's easy to check that your equation holds.





          BTW, as noted by Gareth McCaughan, your equation in fact holds for all real numbers $b$, as long as $a$ is an integer. One fairly simple way to show this is to note that $frac{a+b}{2} = a - frac{a-b}{2}.$ Thus, we can rewrite your equation as $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil a - frac{a-b}{2} rightrceil = a.$$



          Since $a$ is an integer (by assumption), and since $lceil k + x rceil = k + lceil x rceil$ for any integer $k$, we can extract $a$ from the ceiling term to get $$leftlfloor frac{a-b}{2} rightrfloor + a + leftlceil - frac{a-b}{2} rightrceil = a,$$ and finally, by applying the identity $lceil -x rceil = - lfloor x rfloor$, rewrite this as $$leftlfloor frac{a-b}{2} rightrfloor + a - leftlfloor frac{a-b}{2} rightrfloor = a.$$ Cancelling the floor terms then just leaves the identity $a = a$.



          On the other hand, as also noted by Gareth, your equation cannot hold for any non-integer $a$, since its left-hand side is always an integer.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Thank you very much for the odd case, made very simple!
            $endgroup$
            – Adam54
            Jan 25 at 13:05














          14












          14








          14





          $begingroup$

          If $a$ and $b$ are both odd (or both even), then $a-b$ and $a+b$ are both even, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2}.$$



          Otherwise, if exactly one of $a$ and $b$ is odd, then $a-b$ and $a+b$ are both odd, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} - frac12 quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2} + frac12.$$



          In either case, it's easy to check that your equation holds.





          BTW, as noted by Gareth McCaughan, your equation in fact holds for all real numbers $b$, as long as $a$ is an integer. One fairly simple way to show this is to note that $frac{a+b}{2} = a - frac{a-b}{2}.$ Thus, we can rewrite your equation as $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil a - frac{a-b}{2} rightrceil = a.$$



          Since $a$ is an integer (by assumption), and since $lceil k + x rceil = k + lceil x rceil$ for any integer $k$, we can extract $a$ from the ceiling term to get $$leftlfloor frac{a-b}{2} rightrfloor + a + leftlceil - frac{a-b}{2} rightrceil = a,$$ and finally, by applying the identity $lceil -x rceil = - lfloor x rfloor$, rewrite this as $$leftlfloor frac{a-b}{2} rightrfloor + a - leftlfloor frac{a-b}{2} rightrfloor = a.$$ Cancelling the floor terms then just leaves the identity $a = a$.



          On the other hand, as also noted by Gareth, your equation cannot hold for any non-integer $a$, since its left-hand side is always an integer.






          share|cite|improve this answer











          $endgroup$



          If $a$ and $b$ are both odd (or both even), then $a-b$ and $a+b$ are both even, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2}.$$



          Otherwise, if exactly one of $a$ and $b$ is odd, then $a-b$ and $a+b$ are both odd, and thus $$leftlfloor frac{a-b}{2} rightrfloor = frac{a-b}{2} - frac12 quadtext{and}quad leftlceil frac{a+b}{2} rightrceil = frac{a+b}{2} + frac12.$$



          In either case, it's easy to check that your equation holds.





          BTW, as noted by Gareth McCaughan, your equation in fact holds for all real numbers $b$, as long as $a$ is an integer. One fairly simple way to show this is to note that $frac{a+b}{2} = a - frac{a-b}{2}.$ Thus, we can rewrite your equation as $$leftlfloor frac{a-b}{2} rightrfloor + leftlceil a - frac{a-b}{2} rightrceil = a.$$



          Since $a$ is an integer (by assumption), and since $lceil k + x rceil = k + lceil x rceil$ for any integer $k$, we can extract $a$ from the ceiling term to get $$leftlfloor frac{a-b}{2} rightrfloor + a + leftlceil - frac{a-b}{2} rightrceil = a,$$ and finally, by applying the identity $lceil -x rceil = - lfloor x rfloor$, rewrite this as $$leftlfloor frac{a-b}{2} rightrfloor + a - leftlfloor frac{a-b}{2} rightrfloor = a.$$ Cancelling the floor terms then just leaves the identity $a = a$.



          On the other hand, as also noted by Gareth, your equation cannot hold for any non-integer $a$, since its left-hand side is always an integer.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 26 at 1:12

























          answered Jan 25 at 13:03









          Ilmari KaronenIlmari Karonen

          19.9k25186




          19.9k25186








          • 1




            $begingroup$
            Thank you very much for the odd case, made very simple!
            $endgroup$
            – Adam54
            Jan 25 at 13:05














          • 1




            $begingroup$
            Thank you very much for the odd case, made very simple!
            $endgroup$
            – Adam54
            Jan 25 at 13:05








          1




          1




          $begingroup$
          Thank you very much for the odd case, made very simple!
          $endgroup$
          – Adam54
          Jan 25 at 13:05




          $begingroup$
          Thank you very much for the odd case, made very simple!
          $endgroup$
          – Adam54
          Jan 25 at 13:05











          6












          $begingroup$

          The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then



          $$leftlfloorfrac02rightrfloor+leftlceilfrac02rightrceil=0$$
          and
          $$leftlfloorfrac12rightrfloor+leftlceilfrac12rightrceil=1$$
          are enough as a proof.






          share|cite|improve this answer









          $endgroup$


















            6












            $begingroup$

            The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then



            $$leftlfloorfrac02rightrfloor+leftlceilfrac02rightrceil=0$$
            and
            $$leftlfloorfrac12rightrfloor+leftlceilfrac12rightrceil=1$$
            are enough as a proof.






            share|cite|improve this answer









            $endgroup$
















              6












              6








              6





              $begingroup$

              The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then



              $$leftlfloorfrac02rightrfloor+leftlceilfrac02rightrceil=0$$
              and
              $$leftlfloorfrac12rightrfloor+leftlceilfrac12rightrceil=1$$
              are enough as a proof.






              share|cite|improve this answer









              $endgroup$



              The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then



              $$leftlfloorfrac02rightrfloor+leftlceilfrac02rightrceil=0$$
              and
              $$leftlfloorfrac12rightrfloor+leftlceilfrac12rightrceil=1$$
              are enough as a proof.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Jan 25 at 13:01









              Yves DaoustYves Daoust

              128k674226




              128k674226























                  6












                  $begingroup$

                  This actually works whatever the value of $b$ -- it doesn't need to be an integer. Clearly it's true when $b=0$. Now imagine changing $b$ smoothly from $0$ to its final value. When does the value of our expression change? Precisely when $(a+b)/2$ or $(a-b)/2$ passes an integer; that is, when $apm b$ is an even integer; that is, when $b$ differs from $a$ by an even integer. When this happens, both terms change in opposite ways, so the expression as a whole doesn't change its value. So by the time $b$ reaches its final value, our expression still hasn't changed.



                  (On the other hand, if $a$ isn't an integer then the equation never holds because one side is an integer and the other isn't.)






                  share|cite|improve this answer









                  $endgroup$


















                    6












                    $begingroup$

                    This actually works whatever the value of $b$ -- it doesn't need to be an integer. Clearly it's true when $b=0$. Now imagine changing $b$ smoothly from $0$ to its final value. When does the value of our expression change? Precisely when $(a+b)/2$ or $(a-b)/2$ passes an integer; that is, when $apm b$ is an even integer; that is, when $b$ differs from $a$ by an even integer. When this happens, both terms change in opposite ways, so the expression as a whole doesn't change its value. So by the time $b$ reaches its final value, our expression still hasn't changed.



                    (On the other hand, if $a$ isn't an integer then the equation never holds because one side is an integer and the other isn't.)






                    share|cite|improve this answer









                    $endgroup$
















                      6












                      6








                      6





                      $begingroup$

                      This actually works whatever the value of $b$ -- it doesn't need to be an integer. Clearly it's true when $b=0$. Now imagine changing $b$ smoothly from $0$ to its final value. When does the value of our expression change? Precisely when $(a+b)/2$ or $(a-b)/2$ passes an integer; that is, when $apm b$ is an even integer; that is, when $b$ differs from $a$ by an even integer. When this happens, both terms change in opposite ways, so the expression as a whole doesn't change its value. So by the time $b$ reaches its final value, our expression still hasn't changed.



                      (On the other hand, if $a$ isn't an integer then the equation never holds because one side is an integer and the other isn't.)






                      share|cite|improve this answer









                      $endgroup$



                      This actually works whatever the value of $b$ -- it doesn't need to be an integer. Clearly it's true when $b=0$. Now imagine changing $b$ smoothly from $0$ to its final value. When does the value of our expression change? Precisely when $(a+b)/2$ or $(a-b)/2$ passes an integer; that is, when $apm b$ is an even integer; that is, when $b$ differs from $a$ by an even integer. When this happens, both terms change in opposite ways, so the expression as a whole doesn't change its value. So by the time $b$ reaches its final value, our expression still hasn't changed.



                      (On the other hand, if $a$ isn't an integer then the equation never holds because one side is an integer and the other isn't.)







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Jan 25 at 13:39









                      Gareth McCaughanGareth McCaughan

                      3,4281113




                      3,4281113























                          4












                          $begingroup$

                          If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.



                          Then, we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m+1)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m+1)+(2n+1)}2rightrceil\
                          &=leftlfloorfrac{2m-2n}2rightrfloor+ leftlceilfrac{2m+2n+2}2rightrceil\
                          &=lfloor m-nrfloor + lceil m+n+1rceil\
                          &= m-n + m+n+1tag{$*$}\
                          &= 2m+1\
                          &=aend{align}



                          We can get to $(*)$ because $m$ and $n$ are integers and so their floor (or ceiling) is just the number inside (two integers added or subtracted will always give an integer answer).



                          If $a$ is even and $b$ is odd, then we have $a=2m$ and $b=2n+1$, again for $m$ and $n$ integers.



                          So we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m)+(2n+1)}2rightrceil\
                          &=leftlfloorfrac{2m-2n-1}2rightrfloor+ leftlceilfrac{2m+2n+1}2rightrceil\
                          &=leftlfloor m-n-frac 12rightrfloor + leftlceil m+n+frac12rightrceil\
                          &= m-n -1+ m+n+1tag{$dagger$}\
                          &= 2m\
                          &=aend{align}



                          Here we get to $(dagger)$ because we round the first half down and the second half up to their respective nearest integers






                          share|cite|improve this answer









                          $endgroup$


















                            4












                            $begingroup$

                            If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.



                            Then, we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m+1)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m+1)+(2n+1)}2rightrceil\
                            &=leftlfloorfrac{2m-2n}2rightrfloor+ leftlceilfrac{2m+2n+2}2rightrceil\
                            &=lfloor m-nrfloor + lceil m+n+1rceil\
                            &= m-n + m+n+1tag{$*$}\
                            &= 2m+1\
                            &=aend{align}



                            We can get to $(*)$ because $m$ and $n$ are integers and so their floor (or ceiling) is just the number inside (two integers added or subtracted will always give an integer answer).



                            If $a$ is even and $b$ is odd, then we have $a=2m$ and $b=2n+1$, again for $m$ and $n$ integers.



                            So we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m)+(2n+1)}2rightrceil\
                            &=leftlfloorfrac{2m-2n-1}2rightrfloor+ leftlceilfrac{2m+2n+1}2rightrceil\
                            &=leftlfloor m-n-frac 12rightrfloor + leftlceil m+n+frac12rightrceil\
                            &= m-n -1+ m+n+1tag{$dagger$}\
                            &= 2m\
                            &=aend{align}



                            Here we get to $(dagger)$ because we round the first half down and the second half up to their respective nearest integers






                            share|cite|improve this answer









                            $endgroup$
















                              4












                              4








                              4





                              $begingroup$

                              If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.



                              Then, we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m+1)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m+1)+(2n+1)}2rightrceil\
                              &=leftlfloorfrac{2m-2n}2rightrfloor+ leftlceilfrac{2m+2n+2}2rightrceil\
                              &=lfloor m-nrfloor + lceil m+n+1rceil\
                              &= m-n + m+n+1tag{$*$}\
                              &= 2m+1\
                              &=aend{align}



                              We can get to $(*)$ because $m$ and $n$ are integers and so their floor (or ceiling) is just the number inside (two integers added or subtracted will always give an integer answer).



                              If $a$ is even and $b$ is odd, then we have $a=2m$ and $b=2n+1$, again for $m$ and $n$ integers.



                              So we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m)+(2n+1)}2rightrceil\
                              &=leftlfloorfrac{2m-2n-1}2rightrfloor+ leftlceilfrac{2m+2n+1}2rightrceil\
                              &=leftlfloor m-n-frac 12rightrfloor + leftlceil m+n+frac12rightrceil\
                              &= m-n -1+ m+n+1tag{$dagger$}\
                              &= 2m\
                              &=aend{align}



                              Here we get to $(dagger)$ because we round the first half down and the second half up to their respective nearest integers






                              share|cite|improve this answer









                              $endgroup$



                              If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.



                              Then, we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m+1)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m+1)+(2n+1)}2rightrceil\
                              &=leftlfloorfrac{2m-2n}2rightrfloor+ leftlceilfrac{2m+2n+2}2rightrceil\
                              &=lfloor m-nrfloor + lceil m+n+1rceil\
                              &= m-n + m+n+1tag{$*$}\
                              &= 2m+1\
                              &=aend{align}



                              We can get to $(*)$ because $m$ and $n$ are integers and so their floor (or ceiling) is just the number inside (two integers added or subtracted will always give an integer answer).



                              If $a$ is even and $b$ is odd, then we have $a=2m$ and $b=2n+1$, again for $m$ and $n$ integers.



                              So we have begin{align}leftlfloorfrac{a-b}2rightrfloor+leftlceilfrac{a+b}2rightrceil &= leftlfloorfrac{(2m)-(2n+1)}2rightrfloor+ leftlceilfrac{(2m)+(2n+1)}2rightrceil\
                              &=leftlfloorfrac{2m-2n-1}2rightrfloor+ leftlceilfrac{2m+2n+1}2rightrceil\
                              &=leftlfloor m-n-frac 12rightrfloor + leftlceil m+n+frac12rightrceil\
                              &= m-n -1+ m+n+1tag{$dagger$}\
                              &= 2m\
                              &=aend{align}



                              Here we get to $(dagger)$ because we round the first half down and the second half up to their respective nearest integers







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Jan 25 at 13:03









                              lioness99alioness99a

                              3,8232727




                              3,8232727























                                  2












                                  $begingroup$

                                  As $s:=a+b$ and $a-b$ have the same parity, we can write



                                  $$leftlfloorfrac{s-2b}2rightrfloor+leftlceilfrac s2rightrceil=leftlfloorfrac s2rightrfloor+leftlceilfrac s2rightrceil-b=s-b=a.$$






                                  share|cite|improve this answer









                                  $endgroup$


















                                    2












                                    $begingroup$

                                    As $s:=a+b$ and $a-b$ have the same parity, we can write



                                    $$leftlfloorfrac{s-2b}2rightrfloor+leftlceilfrac s2rightrceil=leftlfloorfrac s2rightrfloor+leftlceilfrac s2rightrceil-b=s-b=a.$$






                                    share|cite|improve this answer









                                    $endgroup$
















                                      2












                                      2








                                      2





                                      $begingroup$

                                      As $s:=a+b$ and $a-b$ have the same parity, we can write



                                      $$leftlfloorfrac{s-2b}2rightrfloor+leftlceilfrac s2rightrceil=leftlfloorfrac s2rightrfloor+leftlceilfrac s2rightrceil-b=s-b=a.$$






                                      share|cite|improve this answer









                                      $endgroup$



                                      As $s:=a+b$ and $a-b$ have the same parity, we can write



                                      $$leftlfloorfrac{s-2b}2rightrfloor+leftlceilfrac s2rightrceil=leftlfloorfrac s2rightrfloor+leftlceilfrac s2rightrceil-b=s-b=a.$$







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Jan 25 at 13:15









                                      Yves DaoustYves Daoust

                                      128k674226




                                      128k674226















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