The complex numbers come with a few advantages in Haskell, though. One is that they have an additive structure, whereas tuples do not already have an addition provided. Another is that the complex numbers come with a norm (the magnitude function), which tuples don’t come with. All of this could be implemented for tuples, but why add code when we can just use a type that already has the desired functionality?

However, in addition to that, we have that multiplication of a complex number by a complex number on the unit circle will not change the magnitude, but will rotate according to the angle from the positive real axis. The complex numbers on the unit circle can be obtained nicely by applying the natural exponential function exp to imaginary numbers (complex numbers with zero real part). exp(t*i) is the unit complex number which is at angle t from the positive real axis, counterclockwise. For example, exp(pi*i) = -1, exp(pi/2 * i) = i, and exp(pi/4 * i) = (1 + i)/sqrt(2).

For 3 dimensions, you don’t get such a nice algebra, though the quaternions, a 4-dimensional algebra, can be used to represent orientations and scalings in 3D (via conjugation and a particular embedding of 3D space into the quaternions).

E.G. a common method for implementing unit movement is to have a per-tick fractional unit speed (scalar) that is multiplied by the normal of the units intended heading, which is then added to the unit position to determine the new location for that tick. This has the distinct advantage of not allowing diagonal movement to be faster than movement in a single direction. How would a movement system based on complex numbers handle this issue?