>>5010>Or does it not make sense to give a counter-example that involves i because of what you said in your post (excluding them)?
This. i was invented to resolve the contradiction.
>What does "gives a positive square" mean?
The square is the number B you get by multiplying the number A by itself. The square of 2 is 4. Etc.
It's often mixed up with square root, which is the reverse - the number B you must multiply by itself to get the number A. The square root of 4 is 2.
It just so happens that both the square and square root of 1 are also 1, because 1 is the identity quantity. I think this may be part of the confusion.
A "positive square" means a positive number you get from multiplying two numbers together. Unless you include i (and by extension the complex numbers), you can only have a positive square, because a negative times a negative cancels to being a positive.
>If it's because you exclude them, I think that makes much more sense to me but that would seem to suggest that i is really a concept invented to "hide" this contradiction that exists at the simple level of maths.
It's only a "contradiction" because the system was constructed assuming that this was impossible. Basically, the assumption that because we don't know any "real" number that can be squared and give a negative, there is no such thing. But the fact that you can express the idea of a square root of a negative makes it possible to make a mathematical construct representing that. Once you have that much you can extend the math. The fact that complex numbers are pretty widely applicable IRL means that the "contradiction" was more like a limit on the original model.