Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra.

The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics.

The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse.

It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.

This is a very formal/technical and not very enlightening (and often quite misleading to novices) way of interpreting what the complex numbers “are”. It is basically just a declaration “we have this particular set of symbolic manipulations which we are making up; deal with it.” It’s also kind of ironic because historically the invention of linear algebra came out of attempts to generalize complex numbers.

For me, the key point is that the complex numbers are isomorphic to the transformations of rotation and scaling centered at a point in a plane, with composition of such transformations corresponding to complex multiplication. That is, a complex number can be associated 1:1 with such a transformation. So to understand complex arithmetic, what we really need to understand is the composition of plane rotation and scaling.

In other words, a complex number is not a vector (in the sense of the word “vector” used in physics; it is a “vector” in the sense of an abstract mathematical object satisfying certain axioms).

Personally I would say that the complex numbers “are” quotients of two-dimensional vectors (using the Clifford product, and with non-zero divisor) which include a given planar orientation (the bivector i, with i^2 = –1) but are otherwise unitless and need not be defined in terms of any particular basis though splitting them into the sum of a scalar and bivector part or into the product of a rotation and a scaling is often convenient. But doing this explanation justice requires spending a considerable amount of time explaining the difference between affine points and vectors, talking about how two-dimensional vectors work and how to think about vector multiplication, etc.

This makes generalization to multiplication of vectors in higher dimensional Euclidean (or pseudo-Euclidean) spaces very natural without bothering with any discussion of classifying Lie groups or whatever.

But you could also say that complex numbers are linear combinations of the two matrices [1 0; 0 1] and [0 –1; 1 0]; which when right-multiplied by arbitrary column vectors are respectively the identity and a 90° rotation.

Or you could say that complex numbers are the quotient ring R[X]/(X^2 + 1). [This is closest to the historical approach, where we just declare by fiat that √–1 will be a meaningful symbol.]

Or you could define complex numbers purely in terms of classical trigonometry and analytic geometry.

Etc.

Could you comment in more detail on which mathematical skill you have found useful as data scientist? Which resources did you use to teach yourself? Very appreciative of any help.

Sounds familiar.