Problem 12

(a) If a is rational and b is irrational, is a+b necessary irrational? What if both a and b are irrational?

To answer the first question, we can construct the following equation: b=(a+b)a. If a+b would be rational then (a+b)a would be rational too, which is a contradiction. Regarding the second question, we can construct a b=ra where b is irrational but r isn't. With that we get a+b=aa+r=r which means the whole expression becomes rational.

(b) If a is rational and b is irrational is ab necessary irrational?

With b=aba=aba1 we show that ab cannot be rational, otherwise, we would get a contradiction here. The only exception is the case a=0.

(c) Is there a irrational number a2 such that a4 is rational?

Yes, for example a=24.

(d) Are the 2 irrational numbers whose sum and product is rational?

Yes, for a=2 and b=2.


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