Messrs. Huynh, Meeks, and Wesby, MthT 430 Fall 2005, proposed a
simplification of the proof for the irrationality of Ö{N2 -1} except in the obvious cases.
Irrationality of Ö{N2 - 1}
Use the same picture:
In the above picture note that DEDC @ DABC, so
that
CD
CB
=
DE
AB
=
CE
AC
.
Also note that DE = EB since DADE @ DABE -
or use the Pythagorean Theorem.
If q2 (N2 - 1) = p2 for natural numbers q > 1, p,
we may construct DABC with integer sides so that
AB
= q
Ö
N2 - 1
,
AC
= q N,
CB
= q.
Then
CE
= CD ·
AC
CB
= integer ·
qN
q
= integer.
DE
= EB
= CB - CE
= integer.
Thus we have a smaller triangle with integer sides which is similar to the triangle ABC.
