What is Non-vector?

Answer 1

Numbers come in two flavors real r and vector v. Vectors are defined by their square being negative v^2<0. Non-vectors are defined by their square being non-negative. Real is the non-vector. Real is defined by the square being non-negative r^2>=0.

The numbers in the universe are quaternions which consist of a real r and three vectors v= ix +jy +kz where i^2=j^2=k^2=ijk= -1.

Quaternions q=r + v = Q(cos(q) + v sin(q)) where Q^2= r^2 + x^2 + y^2 +z^2 and cos(q)= r/Q. Quaternions are a superset of real numbers and complex numbers so real and complex numbers are a subset of quaternions. Numbers have a very important feature called commutativity AB=BA. ALl real and Complex numbers are commutative or follow that rule. Quaternions because it has more than one vector "i", does not follow that rule, thus AB does not equal BA in quaternions. The reason is the vectors. AvxBv = -BvxAv unless Av is parallel to Bv then AvxBv= BvxAv=0 for parallel vectors. Because complex numbers only have i, then ixi=0 there is always parallelism and thus always commutativity.

The real numbers are closely related to the origin. Establishing the origin is arbitrary in many problems, attentions must be paid to signs when the problem is set up.