Show that 2 is an irrational number is an irrational number with strong mathematical induction.

Question

Show that       2      is an irrational number is an irrational number with strong mathematical induction.

Zayden Wiley · Accepted Answer

P  (  n  )  ≡       2     ≠            n      b                           ,  ∀  b  ∈      N  .We show by induction that for any   n  ≥  1, there is no integer   b such that       2    =      n    b  .The result is true for   n  =  1, since       2   is not the reciprocal of an integer. Suppose the result holds for all   k  &amp;lt;  n. We show the result is true for   n.Suppose to the contrary that there is an integer   b such that       2    =      n    b  . Then       n    2    =  2      b    2  , so n is even, say   n  =  2      n    ′  . Then quickly we find that       b    2    =  2  (      n    ′        )    2  , so   b is even, say   b  =  2      b    ′  . It follows that       2    =            n      ′              b      ′      .This contradicts the induction assumption that       2   cannot be expressed as a ratio of integers, with numerator positive and less than   n.

Show that <msqrt> 2

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