# Generalizing complex numbers: Is there a mathematical system isomorphic to 3 dimensional space?

As I understand it, complex numbers: $ax+i$ are isomorphic to two-dimensional space.

Quaternions consist of $4$ dimensions. Is that right? Wikipedia says "quaternions form a four-dimensional associative normed division algebra over the real numbers."

Is there a mathematical system generalizing complex numbers that consists of $3$ dimensions and is isomorphic to 3-dimensional space?

• FYI - Hamilton was inspired by how the complex numbers illuminated the geometry of the plane, and tried for many years to find a 3-dimensional field to accomplish the same for space. When he finally realized on a walk that a 4-dimensional division algebra could be obtained, he grew so excited that he carved the relations between $i,j,k$ into the stone railing of a bridge he was crossing. This was the invention of Quaternions. Later, the study of quaternions introduced the idea of vectors, which gave us the ability to study other dimensions that Hamilton was looking for. Jan 13, 2016 at 18:58
• You might also look at Clifford Algebras. Jan 13, 2016 at 21:41
• It seems, no. But 4-dimensional tessarines are much closer in their properties to complex numbers than quaternions (commutative, etc). Mar 2, 2021 at 22:13

Not completely an answer to your question, but this might be interesting to you:

The Frobenius theorem states that up to isomorphism there are three finite-dimensional (unital) associative division algebras over the reals: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).

If you're willing to give up associativity you can also add the octonions (dimension 8) to that list. See Hurwitz's theorem for that.

So depending on how similar to the complex numbers you want it to be, the answer might be a definitive no.

• Amazing and counter intuitive. Thanks! I will need to read up on the Frobenius Theorem. :-) Jan 13, 2016 at 15:56
• The progression of powers of two in the dimensions of these algebras is hard to miss. Are there any further properties of octonions one could abandon, that would then allow a dimension-16 algebra? Jan 13, 2016 at 18:49
• Absolutely. In fact you can keep creating Cayley-Dickson tables ad infinitum, but the number systems just tend to get less and less useful. The $16$-dimensional algebra you're looking for is called the sedenions. The property they lose is division.
– user137731
Jan 13, 2016 at 18:54
• Am I missing something, but it does not look that the question asked for division algebras? For instance, tessarines are much closer to complex numbers by their properties than quaternions. But yes, they are still 4-dimensional. Mar 2, 2021 at 22:10