let 0 &lt; x &lt; y and x , y be irrational number .Does it exist ?

Question

let   0  &amp;lt;  x  &amp;lt;  y and   x  ,  y be irrational number .Does it exist ?

Cristian Hamilton · Accepted Answer

x    y    =      y    x    ,by dividing both sides by       x    x   you&#039;ll arrive at            x      y              x      x        =      x          y      −      x        =      x          x      (      y              /            x      −      1      )        =            y      x              x      x        =            (              y        x            )        x    .Under your assumption   x  &amp;gt;  0, raising both sides to power   1      /    x gives      x          y              /            x      −      1        =      y    x    .You assume   y  &amp;gt;  x, so you have                    (1)                              y          x                =        1        +                              1            u                              with some   u  &amp;gt;  0, and you obtain      x          1              /            u        =  1  +      1    u    .Raising both sides to power   u implies                    (2)                    x        =                              (            1            +                          1              u                        )                    u                .            Combining   (  1  ) and   (  2  )  you get                    (3)                    y        =                              (            1            +                          1              u                        )                                u            +            1                          .            So you see that   (  2  )  and   (  3  )  with some real   u  &amp;gt;  0 give all real solutions   0  &amp;lt;  x  &amp;lt;  y of your equation, indeed.

let 0 < x <

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