Solving A+b*sqrt2=c with Positive Rationals

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Homework Help Overview

The discussion revolves around finding two positive rational numbers, a and b, such that the equation a + b*sqrt(2) equals an irrational number c. The problem is situated within the context of number theory and rationality of expressions involving irrationals.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of multiplying and adding rational and irrational numbers, questioning the nature of the resulting products. Some express skepticism about the existence of solutions for certain irrational values of c, while others inquire about the mathematical foundations necessary to prove these assertions.

Discussion Status

The discussion is ongoing, with participants examining different interpretations of the problem and its constraints. Some have provided examples to illustrate their points, but there is no clear consensus on the existence of solutions for the given equation.

Contextual Notes

There is a mention of the need for specific mathematical knowledge to address the problem, indicating that the level of understanding required may vary among participants.

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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 [itex]a+ b\sqrt{2}= \pi[/itex].
 


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

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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