When talking about numbers, there are many very different contexts that one could have in mind.
Here are the four most familiar ones:
Concepts that exist in one of these contexts may not exist in another. The question "does there exist a number between 1 and 2?" has the answer no in the first two contexts (you cannot go to the beach and pick up more than one but fewer than two pebbles), but yes in the last two contexts (you could eat three cookie halves, which is in between one whole cookie and two whole cookies).
Although in the first two contexts there does not exist a number between 1 and 2, most people are quite comfortable with the fact that such numbers do exist in other contexts. For instance, people don't usually have trouble accepting the existence of the fraction 3/2. Why then is it so hard to believe that the concept of "a number whose square is -1", though it does not exist in any of the four contexts mentioned above, might nonetheless exist in some other context?
It is because we usually forget the fact that we already have four quite different meanings for the word "number". We have become so familiar with each of the four contexts that we have jumbled them together in our mind as if they were a single concept. When we encounter a notion like "square root of -1" which does not exist in any of these four contexts, we think that it cannot exist at all, because we think the word "number" is a single concept that embodies just these four contexts.
Instead, what we should be thinking is something like this:
Okay, I know about four different number systems: one in which "number" means a measurement of how many items are in a set, a second one in which "number" means a relative measurement of the sizes of two sets, a third one in which "number" means a ratio of sizes of two sets, and a fourth one in which "number" means a measurement of a continuous quantity.The answer to that final question is "yes, there is". It is called the Complex Number System. Although it will involve a notion of "number" that is something different from what we are used to, the difference is not fundamentally any greater than is the difference between the concepts of "number of elements in a set" (natural number) and "ratio of sizes of two sets". In other words, the complex numbers are not that much more different from familiar numbers than rational numbers (fractions) are from natural numbers.
In neither of these four number systems does there exist a square root of -1.
Might there be a fifth context, a number system (where "number" means something different from any of the above four things) in which there does exist a square root of -1?