Now that we know the irrational numbers exist, we should prove a few facts about

Question

Now that we know the irrational numbers exist, we should prove a few facts about them. You can use the following useful facts in your proofs. You do not have to prove them. Fact 1: The sum of rational numbers x=a/b and y=c/d is (ad+bc)/(bd).
Fact 2: If a is rational, then -a is rational; if a is irrational, then -a is irrational.

Prove the following theorems:

(a) The sum of two rational numbers is a rational number.
(b) The sum of a rational number and an irrational number is an irrational number.

(c) The product of an irrational number and a non-zero rational number is an irrational number. (d) The sum of two irrational numbers may be a rational number.

Explanation / Answer

I have written the answer on a paper and uploaded it as 2 pictures at the following links:

Part-1: http://i.imgur.com/rFXjhxz.jpg

Part-2: http://i.imgur.com/8eNCSSg.jpg

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