Quaternions can be used to solve some of the limitations of transformation matrices in the real world, for instance, they can mitigate Gimbal Lock

BryceSummers

The rotation properties of Complex numbers are of foundation importance for the Fast Fourier Transform. Complex numbers help transform the Slow or Naive Fourier Transform which runs in $\mathcal{O}(n^{2})$ time into the Fast variety that runs in $\mathcal{O}(n \cdot log(n))$ time.

Quaternions can be used to solve some of the limitations of transformation matrices in the real world, for instance, they can mitigate Gimbal Lock

The rotation properties of Complex numbers are of foundation importance for the

FastFourier Transform. Complex numbers help transform theSlowor Naive Fourier Transform which runs in $\mathcal{O}(n^{2})$ time into theFastvariety that runs in $\mathcal{O}(n \cdot log(n))$ time.Please See: Primitive Roots of Unity and Cooley-Tukey FFT.