My question is perhaps as much about the philosophy of math as it is about the specific tools of math: is perpendicularity and rotation integral and fundamental to the concept of multiplication - in all number spaces? As I understand it, the product of complex numbers x = (a, ib) and y = (c, id) can be calculated as: ( (ac-bd), i(ad+bc) ) I noticed that this expression uses a particular mixing of the real and imaginary components of x and y, and it includes a negation in the real part. And I understand that this negation is related to the square root of -1. I have noticed a similarity between the multiplication of complex numbers and the operation for finding the perpendicular of a 2D vector (x, y) as (-y, x). The other perpendicular is (y, -x). It appears that negation of one component is critical for calculating perpendicularity. Is there a way to express the concept of multiplication in a general way such that the behaviors of complex numbers is consistent with that of real (1-dimensional) numbers? Or do these properties of perpendicularity, negation, and rotation apply only to the higher number spaces (complex, quaternion, octonian, etc)? Might there be higher-order behaviors in the higher-dimensioned number spaces that are supersets of rotation and perpendicularity that our 3D brains cannot visualize or even comprehend? Finally, what higher-order principle dictates that multiplication in higher-order number spaces should involve rotational effects? Some philosophers of mathematics would argue that these techniques are an invention of humans - an extrapolation of the concept of multiplication of real numbers to higher dimensions, which is arbitrary (although extremely useful and fully-consistent with all other mathematical concepts and operations).