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.

Fast Fourier Transformation.

FFT was invented in 1805, and it was applied here and there in the decades after it, but it wasn’t particularly widespread. However, in the age of computers FFT is used in audio editing, signal processing, scientific instruments, image processing etc. just to name a few examples. Currently, it’s practically everywhere.

FFT is a really complicated topic, so I won’t bore you with the details. If you want to read more, check the wikipedia article about it.

black swans?