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 $b = r - a$ where $b$ is irrational but $r$ isn't. With that we get $a + b = a - a + r = r$ which means the whole expression becomes rational.

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

With $b = \frac{ab}{a} = aba^{-1}$ 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 $a^2$ such that $a^4$ is rational?

Yes, for example $a = \sqrt[4]{2}$.

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

Yes, for $a = \sqrt{2}$ and $b = -\sqrt{2}$.

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