If mn m n is an approximation to 2 then prove that m 2 n + n m is a better approximation to 2 ..(where mn is a rational number)

Question

If mn      m    n   is an approximation to       2   then prove that       m          2      n        +      n    m   is a better approximation to       2    ..(where mn is a rational number)

Shawn Castaneda · Accepted Answer

The claim is false without additional assumptions on The claim is false without additional assumptions on mn. For example if       m    n    =      1    10   then your next guess is   10.05 which is further away from       2  . The claim is true, however, if we impose a condition like       m    n    &amp;gt;  1. Write   x  =      m    n  . You want to show that       x    2    +      1    x   is closer to       2   than   x is. In other words, show:                    (1)                              |                                                                      (                                      x                    2                                    +                                      1                    x                                    )                                −                                  2                                                            x                −                                  2                                                              |                &amp;lt;        1.            By algebra we have:                    (2)                              |                                                                      (                                      x                    2                                    +                                      1                    x                                    )                                −                                  2                                                            x                −                                  2                                                              |                =                  |                                                    x                −                                  2                                                            2                x                                              |                    If we assume   x  &amp;gt;  1, we can argue by cases that the RHS of   (  2  ) is less than one:If   1  &amp;lt;  x  ≤      2   then       |    x  −      2        |    =      2    −  x  &amp;lt;  2  &amp;lt;  2  x  =      |    2  x      |  .If   x  &amp;gt;      2   then by the triangle inequality   x  &amp;gt;      2  .

If mn m n </mfrac> is an approximation to <msqrt> 2 </msqrt> then p

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