i'm not expert on mathematics at all, but think about it this way:
first we had no numbers at all, we then started using our fingers to count things like livestock, rocks, whatever nature had to offer us and as such the set of natural numbers
but then you started needing more and more types of numbers, i.e traders started to need to subtract, divide and multiply to calculate their earnings. at first we only had Euclid as a reference for our mathematical knowledge, but then came other scientists who gave birth to other mathematical system containing new models, elements, etc etc (Lobachevsky, Euler, Fermat, etc) . that hints to the fact that math isn't an objective science, you can't simply say there is one way to solve an equation or that all equations can only have one solution, so there CAN be other sets independent from what exists materially, since math relies solely on human logic
so say you're in the middle of solving a math problem and you get to 1=-1. well, obviously that's impossible, at least in the set of real numbers, so you create another set that includes imaginary numbers, where 1=i2 to be able to work with equations that don't give you a logical result. equations of the 2nd degree sometimes need imaginary numbers so that they can be solved.
last but not least, just because something doesn't exist in our material realm doesn't mean it doesn't exist at all. there is no such thing as a perfect circle, triangle or sphere in space, sure you might be reminded of a circle/sphere when you look at the sun but geometrically speaking the sun isn't a sphere at all. spheres still do exist as a concept, they just don't exist in our realm because, again, math doesn't care for what exists materially and what doesn't, it relies on logic and brain power. that doesn't mean it can't be used in real-life situations.
i feel like this question is more philosophical than mathematical