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Topic: Irrational or Zero (Read 339 times) |
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ThudnBlunder
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Irrational or Zero
« on: Jan 18th, 2005, 2:35am » |
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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.
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| « Last Edit: Jan 18th, 2005, 2:37am by ThudnBlunder » |
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Sir Col
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Re: Irrational or Zero
« Reply #1 on: Jan 18th, 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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