I would definitely not say that sqrt(2) is more controversial than sqrt(-1). With sqrt(2), we define it into existence, because with all the surrounding bits,

I would definitely not say that sqrt(2) is more controversial than sqrt(-1). With sqrt(2), we define it into existence, because with all the surrounding bits, it makes sense for it to exist. With sqrt(-1), we define it into existence, because under the assumption that it exists, you can do some calculations that you can't do otherwise, or you can do them more easily, and the results of those calculations happen to be correct.