Explain why the sum, the difference, and the product of two rational numbers are rational numbers. Is the product of two irrational numbers necessarily irrational? What about the sum?

Question

Explain why the sum, the difference, and the product of two rational numbers are rational numbers. Is the product of two irrational numbers necessarily irrational?  What about the sum?

moongazer · Answer

-Because when you add and multiply rational numbers you will still obatain a fraction or ratio of two integers.
-No
-Yes
Is this the best explanation?

parthkohli · Answer

Yes.

moongazer · Answer

If not , please give me some hints :)

parthkohli · Answer

If you need counter-proofs:
1) No counter-proof.
2) $\rm i\times i = 1$
3) No counter-proof.