  A finite-dimensional unital and associative algebra over $\mathbb{R}$, or
what we shall call simply "an algebra" in this paper for short, generalities
the construction by which we derive the complex numbers by "adjoining an
element $i$" to $\mathbb{R}$ and imposing the relation $i^2 = -1$. In this
paper, we examine some of the elementary algebraic properties of such algebras,
how they break-down when compared to standard grade-school algebra, and discuss
how such properties are relevant to other areas of our research regarding
algebras, such as the $\mathcal{A}$-calculus and the theory of
$\mathcal{A}$-ODEs.
