perplexus.info :: Just Math : Irrational Decision

Irrational Decision (Posted on 2015-11-15)

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.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

N=2 only

| Comment 1 of 6

I don't know yet if this will generalize to all N, but if N=2:

X=√(A + √(A²-1)) + √( A - √(A²-1))

X^2 = (A + √(A²-1)) +2√(A + √(A²-1)( A - √(A²-1)) +( A - √(A²-1))

X^2= 2A+2√(A² - (A²-1))

X^2= 2A+2

X=√(2A+2)

Which is clearly not rational if A is not rational.

Posted by Jer on 2015-11-16 07:26:45

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