Is the square root of -1 rational?

This is not a deep question, but if there is a definite answer then here is the place where I will find it.

Is it justified to say that $i =\sqrt{-1}$ is rational?

The origin of this question lies in a regular discussion I have over this t-shirt of mine:

While obvious $\pi$'s comment is completely legit, $\sqrt{-1}$'s might be hypocritical, if the rationality of $\sqrt{-1}$ is questionable.

Solution 1:

It is a Gaussian rational number, but it is not rational in the conventional sense of the word because rational numbers are real.

Solution 2:

The number $\sqrt{-1}$ is not real. Since the rationals are just a particular type of real number, it cannot be rational, either.

Another way to look at it: Were there integers $a$ and $b$ such that $\sqrt{-1} = \frac{a}{b}$ then $$ -1 = \frac{a^2}{b^2}, $$ and so $$ a^2 = -b^2. $$ Since $b^2$ is certainly positive, that means $a^2$ is certainly negative, which is impossible.

Solution 3:

$\sqrt{-1} = i$ is an imaginary number, not lying anywhere on the real number line. Therefore as others have said, it is neither rational nor irrational in the usual senses of those words.

To comment on the specific grammar of the shirt: $\pi$'s imperative is 'Get real' which I would say has the connotation of 'join [the speaker's] group'. Clearly this makes sense, as $\pi$ is a real number. $i$, on the other hand, says only 'be rational' which I would say lacks the connotation of joining the speakers group; instead, $i$ is only imploring $\pi$ to join the rationals, with no implication of $i$'s membership.

Solution 4:

Rather $\sqrt{-1}$ is the algebraic number, but not rational.