Sum of Rational and Irrational

[harpazo]

I asked my friend the following question:

Why is the sum of a rational number and irrational number irrational?

Here is his reply.

Let's assume that the sum of the rational number a/b (where a and b are integers) and irrational number x is also a rational number c/d (where c and d are some integer), i.e.

(a/b) + x = c/d

Then x = (c/d) - (a/b)

RHS is clearly rational while LHS is irrational. This is a contradiction.

So, the sum must be irrational.

Question:

Does this reply make sense?

 

[HallsofIvy]

Yes, that is a pretty standard "proof by contradiction". To prove "if P then Q", assume Q is false and that this leads to a contradiction. Here the conclusion is "the sum is irrational". We assume it is a rational number and use the properties of rational numbers (that they are closed under subtraction) to arrive that the result that a number is also rational contradicting the hypothesis that it is irrational.

 

[Deleted member 4993]

I asked my friend the following question:

Why is the sum of a rational number and irrational number irrational?

Here is his reply.

Let's assume that the sum of the rational number a/b (where a and b are integers) and irrational number x is also a rational number c/d (where c and d are some integer), i.e.

(a/b) + x = c/d

Then x = (c/d) - (a/b)

RHS is clearly rational while LHS is irrational. This is a contradiction.

So, the sum must be irrational.

Question:

Does this reply make sense?

Yes.... This type of logic (argument) is called "reductio ad absurdum".

 

[harpazo]

Thank you everyone.