The complex numbers make a field out of pairs of real numbers. The quaternions almost make a field out of four-tuples of numbers, though multiplication is not commutatative. Technically, quaternions form a **division algebra**.

**Frobenius’s theorem** says only real vector spaces that can be made into division algebras are the real numbers, complex numbers, and quaternions.

So can you multiply triples of real numbers? Sure. **Cross product** is one way, but there’s no “division” analog to cross product. For example, the cross product of any vector with itself is zero, so cross product has lots of **zero divisors**, non-zero elements that multiply to zero.

Frobenius tells us we can’t form triples of real numbers into a division algebra, but can we come closer to a division algebra than we get with cross products? It turns out we can! We can define a multiplication that is **commutative**, has an identity element, and has no zero divisors. But what’s missing? The multiplication doesn’t have the distributive property you’d have in a division algebra.

The multiplication operation is complicated to describe. For details see “A Commutative Multiplication of Number Triplets” by Frank R. Pfaff, The American Mathematical Monthly, Vol. 107, No. 2 (Feb., 2000), pp. 156-162.

**Related posts**:

Does this object have any property relating multiplication to addition that replaces the distributive property? I ask because, without the distributive property, an algebra (or more generally a ring) is generally not a very interesting object, since the two operations could be completely unrelated.

Good question. The article gives an example to show that multiplication does not distribute in general, but it’s not clear whether you have a limited distributive property.

Without some sort of connection between multiplication and addition, you could just define multiplication to

beaddition! Et voilà, now multiplication has lots of nice properties. :)I really like these kinds of posts, introducing some theorems and seemingly fundamental results, to at least know of their existence.