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charmedbeauty
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Homework Statement
Prove that if a and b are rational numbers with a≠b then
a+(1/√2)(b-a) is irrational.
Homework Equations
The Attempt at a Solution
Assume that a+(1/√2)(b-a) is rational.
then by definition of rationality
a+(1/√2)(b-a) =p/q for some integers p&q
so a+(b-a)/√2 =p/q
a(1+(b-1)/√2) = p/q
so (p/qa) -1= (b-1)/√2
qa/p-1= √2/(b-1)
so √2 = (b-1)((qa/p) -1)
but (b-1)((qa/p) -1) is rational since b,1,q,a,p are all integers and the sum, difference and products of integers are integers.
but √2 is not an integer. Contradiction a+(1/√2)(b-a) must be irrational.