Triple fractions (and more complex fractions)Usually a b c = a c b i.e. b / c is seen as the denominator, and a is the numerator.If you have a / b / c / d, what do you choose to take as the denominator? b c d , c d , or d ?and why?And what about for a / b / c / d / e?

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Triple fractions (and more complex fractions)Usually      a          b      c        =            a      c        b  i.e.   b      /    c is seen as the denominator, and   a is the numerator.If you have   a      /    b      /    c      /    d, what do you choose to take as the denominator?      b          c      d        ,      c    d    ,   or   d  ?and why?And what about for   a      /    b      /    c      /    d      /    e?

karburitc · Accepted Answer

The trouble is  a      /    (  b      /    c  )  ≠  (  a      /    b  )      /    csince   a      /    (  b      /    c  )  =            a      c        b   and   (  a      /    b  )      /    c  =      a          b      c      . So, without brackets, the notation is ambiguous. Introducing brackets, for example, we could choose between  a      /    b      /    c      /    d  =  a      (    b          /        c          /        d          )              −        1            and  a      /    b      /    c      /    d  =  (  a            /        b          /        c    )          d              −        1            The first leads to  a      (    b          (      c              /            d              )                  −          1                            )              −        1              =  a      (    b                  c                  −          1                      d          )              −        1              =            a      c              b      d      The second leads to  (  a      /    b  )            c              −        1                        d              −        1              =  a            b              −        1                        c              −        1                        d              −        1              =      a          b      c      d      which is less interesting, it seems to me. The first expression leads to  a      /    b      /    c      /    d      /    e  =  a      (    b    (    c          /        d          /        e    )          )              −        1              =  a                    (                                            b              d                                      c              e                                      )                    −        1              =            a      c      e              b      d      Collecting results, the following interesting pattern emerges:  a      /    b  =  a            b              −        1              =      a    b    a      /    b      /    c  =  a            b              −        1              c  =            a      c        b    a      /    b      /    c      /    d  =  a            b              −        1              c            d              −        1              =            a      c              b      d        a      /    b      /    c      /    d      /    e  =  a            b              −        1              c            d              −        1              e  =            a      c      e              b      d        a      /    b      /    c      /    d      /    e      /    f  =  a            b              −        1              c            d              −        1              e            f              −        1              =            a      c      e              b      d      f        .  .  .and so on.

Triple fractions (and more complex fractions) Usually a b c </

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