Famously, the complex numbers were first hinted at in the 1500s (arguably even the greeks may have run into the problem of negative roots). Solving exponential equations pretty often will lead you to taking a root over a negative number, which was believed at one time to be as invalid as dividing by zero.
Although uses were found for imaginary numbers, they arguably didn’t have a “real world” use until quantum mechanics. They were useful for 2D rotations, which Hamilton pondered on until coming up with the Quaternions as a 4d analog for doing 3d rotations. Those also, arguably, wouldn’t find much use for decades until computers made using quaternions an efficient way to transform 3d shapes.
Famously, the complex numbers were first hinted at in the 1500s (arguably even the greeks may have run into the problem of negative roots). Solving exponential equations pretty often will lead you to taking a root over a negative number, which was believed at one time to be as invalid as dividing by zero.
Although uses were found for imaginary numbers, they arguably didn’t have a “real world” use until quantum mechanics. They were useful for 2D rotations, which Hamilton pondered on until coming up with the Quaternions as a 4d analog for doing 3d rotations. Those also, arguably, wouldn’t find much use for decades until computers made using quaternions an efficient way to transform 3d shapes.
See also prime numbers, which were considered a useless field of study before cryptography