MthT 430 Apostol's Irrationality - More
MthT 430 Apostol's Irrationality - More
Apostol remarks that a similar argument shows that Ö{N2 +1} and Ö{N2 - 1} are irrational except in the obvious cases.

apopic02.gif

In the above picture note that DEDC @ DABC, so that
CD
CB
 = DE
AB
 = CE
AC
 .
Irrationality of Ö{N2 + 1}
Suppose that Ö{N2 + 1} is a rational number.
If q2 (N2 + 1) = p2 for natural numbers q > 1, p > 1, we may construct DABC with integer sides so that
AC
= q
Ö
 
N2 + 1
 
 ,
AB
= q N,
CB
= q.
Choose the smallest such q . Then
CD
= q æ
è
Ö
 
N2 + 1
 
 - N ö
ø 		, an integer.
Define
a
= CD
CB
= æ
è
Ö
 
N2 + 1
 
 - N ö
ø 		.
Then
CD
= a CB = a q = q¢, an integer,
DE
= a AB = a q  N = q¢ N, an integer,
CE
= BC - BE = BC - DE
= a AC = q¢ 
Ö
 
N2 + 1
 
 , an integer.


Irrationality of Ö{N2 - 1}
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.
CD
= q æ
è 		N -
Ö
 
N2 - 1
 
 ö
ø 		, an integer,
CD
CB
= æ
è 		N -
Ö
 
N2 - 1
 
 ö
ø 		.
= b.
Then
CD
= b CB = b q = q¢, an integer,
DE
= b AB = b q  
Ö
 
N2 - 1
 
 = q¢ 
Ö
 
N2 - 1
 
 , an integer,
CE
= b AC = b q N = q¢ N, an integer.


Tom M. Apostol, Irrationality of The Square Root of Two - A Geometric Proof, American Mathematical Monthly 107, No. 9 (Nov., 2000), pp. 841-842.


File translated from TEX by TTH, version 3.74.
On 19 Sep 2006, 16:25.