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        riddles >> putnam exam (pure math) >> Irrational or Zero
        
(Message started by: THUDandBLUNDER on Jan 18^th, 2005, 2:35am)

Title: Irrational or Zero
Post by THUDandBLUNDER on Jan 18^th, 2005, 2:35am

Given that a,b are rational numbers and that p,q are integers which are not perfect squares,
prove that a[smiley=surd.gif]p + b[smiley=surd.gif]q is either irrational or equal to zero.

Title: Re: Irrational or Zero
Post by Sir Col on Jan 18^th, 2005, 7:38am

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Let us assume that the sum is rational. So multiplying through by 1/a we get:
sqrt(p)+c*sqrt(q)=r [c,r are rational]
c*sqrt(q)=r-sqrt(p)
c²q=r²+p-2r*sqrt(p)

As LHS is rational, RHS is rational iff b is a perfect square. As b is not a perfect square, the sum cannot be rational as we first assumed.

Similarly by multiplying through by 1/b we can show the same result.

Hence the sum is either irrational or zero.
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