Rational and Irrational Number Set proof.

[linuxux]

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.

 

[matt grime]

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.

 

[linuxux]

whoa, thanks, i would have never gotten that.

 

[HallsofIvy]

For a moment I thought you were trying to prove that the sum of a rational number and an integer was an integer!

 

[ZioX]

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.