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Irrational Decision (Posted on 2015-11-15) Difficulty: 3 of 5
Let A be an irrational number and let N is an integer > 1.

Is N√(A + √(A2-1)) + N√( A - √(A2-1)) always an irrational number?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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most of a general proof | Comment 5 of 6 |
I think I'm most of the way to a proof for all N>1, however I just can't seem to get past the last hurdle.  I'm posting it here in hopes someone else might be able to finish it
lemma 1:

if x is irrational and n is an integer > 1, then x^(1/n) is irrational

proof:
assume x^(1/n)=r a rational number
then x=r^n, however r^n is rational and thus contradicts x being irrational

lemma 2:

if x is irrational then x-sqrt(x^2-1) is also irrational

proof:
assume x-sqrt(x^2-1)=r a rational number
sqrt(x^2-1)=(x-r)
x^2-1=(x-r)^2
x^2-1=x^2-2xr+r^2
r^2-2xr+1=0
2xr=r^2+1
x=(r^2+1)/2r
if r is rational then so is (r^2+1)/2r and thus contradicts x being irrational

lemma 3:

if x is irrational then x+sqrt(x^2-1) is also irrational

proof:
assume x+sqrt(x^2-1)=r a rational number
sqrt(x^2-1)=(r-x)
x^2-1=(x-r)^2
x^2-1=x^2-2xr+r^2
r^2-2xr+1=0
2xr=r^2+1
x=(r^2+1)/2r
if r is rational then so is (r^2+1)/2r and thus contradicts x being irrational

combining lemmas 1,2, and 3 we can show that if A is irrational then so is both
(A+sqrt(A^2-1))^(1/n) and (A-sqrt(A^2-1))^(1/n)

and this is where I am stuck.  I can show the two parts remain irrational, however I can not guarantee that their sum is also irrational.  For a simple example, imagine if they somehow simplified to 1+sqrt(5) and 1-sqrt(5) then they would both be irrational but their sum would be the rational value of 2.




  Posted by Daniel on 2015-11-17 15:26:00
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