# The Significance of Complex Numbers: Frobenius Theorem

By Xah Lee. Date:

Learned a new thing today. This answered me indirectly a important question, namely, why are complex numbers special (among algebras). It also helped in understanding the relative importance of quaternions.

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. (Division algebra is a algebra over a field where division is possible.) According to the theorem, every such algebra is isomorphic to one of the following:

• ℝ (the real numbers)
• ℂ (the complex numbers)
• ℍ (the quaternions).

These algebras have dimensions 1, 2, and 4, respectively. Of these three algebras, the real and complex numbers are commutative, but the quaternions are not.

This theorem is closely related to Hurwitz's theorem, which states that the only normed division algebras over the real numbers are ℝ, ℂ, ℍ, and the (non-associative) algebra 𝕆 of octonions.