Solving A+b*sqrt2=c with Positive Rationals

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Finding two positive rational numbers a and b such that a + b*sqrt(2) equals an irrational number c is generally impossible. The discussion highlights that multiplying a rational number by an irrational number results in an irrational product, complicating the solution. It is noted that there are no rational a and b that satisfy equations like a + b*sqrt(2) = π. The participants question the necessary mathematical background to prove these assertions. Ultimately, the consensus is that such problems typically do not have solutions.
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Homework Statement



what's the best way solving question of finding two positive rational a and b given c
where a +b*sqrt2= an irrational number c


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The Attempt at a Solution

 
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when you multiply an irrational number by a rational one, as you are doing here, which type of number is/can be the product? When you add...

Prove it, using the definitions.
 


In general, such a problem will NOT have a solution. For example, there exist no rational a, b, such that a+ b\sqrt{2}= \pi.
 


HallsofIvy said:
In general, such a problem will NOT have a solution. For example, there exist no rational a, b, such that a+ b\sqrt{2}= \pi.

I was taking "an irrational number c" to mean some irrational number c in which case there is quite some scope. What math level would be necessary to be able to prove your example?
 

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