Rational and Irrational Number Set proof.

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Discussion Overview

The discussion revolves around proving properties of sums and products involving rational and irrational numbers. Participants explore the implications of combining these number sets, specifically focusing on whether the sum or product of a rational number and an irrational number remains irrational. The conversation includes theoretical reasoning and attempts to construct proofs.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes proving that if \( a \in \mathbf{Q} \) and \( t \in \mathbf{I} \), then both \( a+t \in \mathbf{I} \) and \( at \in \mathbf{I} \), questioning the validity of their approach.
  • Another participant notes that the rational numbers form a field and points out that the notation for irrational numbers is not standard.
  • A different participant suggests that nothing can be definitively stated about the sum or product of two irrational numbers \( s \) and \( t \) without specific examples, as they can yield rational results.
  • One participant humorously misinterprets the initial problem as relating to the sum of a rational number and an integer being an integer.
  • Another participant outlines a proof by contradiction, arguing that if \( a \) is rational and \( t \) is irrational, then both \( a+t \) and \( at \) must be irrational based on the assumption that they could be rational.

Areas of Agreement / Disagreement

Participants express differing views on the properties of sums and products involving rational and irrational numbers. There is no consensus on the implications of combining these sets, and the discussion remains unresolved regarding the general case of two irrational numbers.

Contextual Notes

Some participants note the need for specific examples to illustrate points, and there are references to standard proofs by contradiction, indicating a reliance on established mathematical reasoning without resolving the underlying assumptions.

linuxux
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Hello, here is my problem:

how can i prove that if a\in\mathbf{Q} and t\in\mathbf{I}, then a+t\in\mathbf{I} and at\in\mathbf{I}?

My original thought was to show that neither a+t or at can be belong to N, Z, or Q, thus they must belong to I. However I'm not certain if that train of thought is correct.

Also, i have a question that says given two irrational numbers s and t, what can be said about s+t and st.

My original thought he was that nothing can be shown, since it is possible to create numbers that belong to N, Z, Q, or I.

thanks for clarification.
 
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The rational numbers are a field. Oh, and I is not standard notation, by the way.

As for the second one, then you can't say anythingabout s or t's rationality. Just construct some examples.
 
whoa, thanks, i would have never gotten that.
 
For a moment I thought you were trying to prove that the sum of a rational number and an integer was an integer!
 
The first set of problems are standard proofs by contradiction.

Suppose a is rational and t is irrational and at is rational and a+t is rational.

Since at is rational, at=m/n for appropriate integral m & n.

Then, t=m/na, which is rational. But t is irrational by our hypothesis. Therefore, at cannot be rational, hence it is irrational.

The proof for a+t is similar.
 
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