Rationally irrational?

Number Theory Level 3

Given that $a$ is rational , and $b$ is irrational , can $ab$ be rational?

No Yes

1 solution

Geoff Pilling
Aug 2, 2016

If $a = 0$ and $b =\sqrt{2},$ , then $ab = 0$ , which is rational. So the answer is $\boxed{\mbox{Yes}}$

Bonus: Prove that this is only possible when $a=0.$

Eli Ross Staff - 4 years, 10 months ago

Assume $a \neq 0$ . Let $ab=r$ where $r$ is rational:

$ab=r \implies b=\dfrac{r}{a}$

But $\dfrac{r}{a}$ is rational as $a \neq 0$ but this contradicts $b$ being irrational so $a=0$ .

Sam Bealing - 4 years, 10 months ago

Suppose that a is rational and not equal 0; then a = e/f for integers e,f with e not = 0. If ab is rational, b irrational, then ab = k/m, and b = (k/m)*((f/e) = kf/me, a contradiction.

Edwin Gray - 2 years, 3 months ago

Rationally irrational?

1 solution

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