I recently read a fascinating article about string theory, which discussed higher-dimensional algebras and their applications to supersymmery. The author mentioned that there were only four algebras in which this was possible:
$\mathbb{R}$ (dimension 1),
$\mathbb{C}$ (dimension 2),
$\mathbb{H}$ (Quaternions, dimension 4),
$\mathbb{O}$ (Octonions, dimension 8).
My question is why does division not work in other dimensional algebras? What's so special about 1, 2, 4, and 8 that make this possible? I am looking for a high-level answer to this question, as I don't have much familiarity with algebra.
