Imaginary numbers were useful as soon as they were invented. They are algebraically closed, unlike the real numbers, and are required for the fundamental theore

Imaginary numbers were useful as soon as they were invented. They are algebraically closed, unlike the real numbers, and are required for the fundamental theorem of algebra. https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algebra https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algeb... The first use of negative roots was to find the (real) roots of certain polynomials. They were considered a mathemathical hack: useful, but specious on their own. And then it turned out that if you take them on their own terms, they're fantastically useful.