Many people think of Complex Numbers as vectors, and I think of negative numbers as vectors too. They have both magnitude and orientation, even if the orientation is only 1-dimensional. When you multiply complex numbers, the resulting "vector" has magnitude equal to the product of the magnitudes, and angle (from the positive x-axis) equal to the sum of the angles. So you can actually calculate these products without actually multiplying the numbers themselves, but instead only considering magnitude and angle (this only really matters with complex numbers). But the definition of a vector, that is has magnitude and direction, doesn't apply to 0. It doesn't have any direction, so it's technically not a vector. Adding angles doesn't make sense to something that just doesn't have angles. Zero is actually a mess when considering multiplication, and is generally excluded from the set of numbers you get to play with under multiplication. Another number you don't get to use for the same reason is infinity. A professor once told me an interesting insight, that zero is messed up in the same way that complex infinity is messed up. Neither has orientation, both have magnitudes that consume the other number when multiplying. And if you think about it, you could `s/0/infinity/g` in integer multiplication tables and they'd still work exactly the same.
[0] https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim...