Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions (“triernions”).

Yet no one uses these. Why is this?

Answer

It’s because there isn’t one! (Indeed, Hamilton was originally searching for such a thing, and found the quaternions instead; it was only later that people understood why he hadn’t been successful, initially.)

The quaternions – along with the real numbers and the complex numbers – have a number of nice properties: specifically, they form a real division algebra. This is a mouthful, but basically amounts to:

  • Addition/multiplication of quaternions satisfy the ring axioms.

  • We can divide by quaternions.

  • We can multiply a quaternion by a real (and this “scalar multiplication” satisfies the basic properties it should).

It turns out the only finite-dimensional real division algebras are R, C, and the quaternions; see this. (I include associativity in the definition of algebra: if we allow non-associative algebras, then the octonions also count.)


By the way, there is a way to (sort of) keep going past the quaternions: the Cayley-Dickson construction. This produces things like the octonions and the sedenions, and other delightfully weird algebraic structures. However, it has a couple drawbacks:

  • Each time you apply Cayley-Dickson, the dimension of the starting algebra doubles. So this won’t help us get to 3.

  • Also, you keep losing nice properties. Passing from the reals to the complex numbers, we lose order; going from the complexes to the quaternions, we lose commutativity of multiplication. If we keep going, we lose associativity of multiplication, in increasing degrees: the sedenions are even less associative than the octonions, etc.

Attribution
Source : Link , Question Author : thecat , Answer Author : J. W. Tanner

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