Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)? 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?
 A: Have you heard of the Frobenius theorem?
https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)
Triernions would not be an associative division algebra.
A: Assume $A$ is a three-dimensional (associative) algebra over $\mathbb{R}$. We can assume $\mathbb{R}$ is embedded in $A$. If $a\in A$, $a\notin\mathbb{R}$ the map $l_a\colon A\to A$, $l_a(x)=ax$, is an endomorphism of $A$ as a vector space over $\mathbb{R}$.
Let $\lambda$ be a real eigenvalue of $l_a$, with eigenvector $b\ne0$, so $ab=\lambda b$. Such an eigenvalue exists, because the characteristic polynomial of $l_a$ has degree $3$. Then $(a-\lambda)b=0$. Note that $a-\lambda\ne0$, so $A$ has zero divisors, in particular $A$ is not a division algebra.
It's a bit more complicated showing that a finite-dimensional division algebra over $\mathbb{R}$ can only have dimension $1$, $2$ or $4$ and it is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (the quaternions); this is known as Frobenius' theorem.
On the other hand, three-dimensional algebras over $\mathbb{R}$ exist (but they have zero divisors, as shown above). A simple example is $\mathbb{R}[X]/(X^3-1)$, but they can be non-commutative as well.
A: Among the vectorial spaces $\mathbb R^n$, only $\mathbb R$ and $\mathbb R^2\space ( \approx \mathbb C)$ admit a multiplication that gives them a structure of field. For the other values, only for $\mathbb R^4$ we can have a multiplication without divisors of zero and associative seeming a field; actually,with this multiplication, $\mathbb R^4$  becomes a division ring or, also called, skew field and is named quaternions.
According to a celebrated theorem of Wedderburn all finite division rings are necessarily commutative so quaternions are the first example of a non-commutative skew field. French mathematicians used the terminology “corps” for both “fields and skew fields” so there are for them commutative and non-commutative corps. By the theorem of Wedderburn, quaternions give the first example of a non-commutative “corp”.
A: There is an algebra in dimension 6, halfway between 4 and 8, that is not a division algebra but correctly interpolates various constructions.
http://arxiv.org/abs/math/0411428
Nothing like that is known for dimension 3 sitting between 2 and 4.
A: The closest thing to triterniums would be the structure $[\mathbb R^3, +, \times]$ where $``\times"$ represents the cross product
$$(a_1 \mathbf i + b_1 \mathbf j + c_1 \mathbf k) \times 
  (a_2 \mathbf i + b_2 \mathbf j + c_2 \mathbf k) =
\left| \begin{matrix}
    \mathbf i & \mathbf j & \mathbf k \\
          a_1 &       b_1 &       c_1 \\
          a_2 &       b_2 &       c_2 \\
\end{matrix} \right|$$
It distributes over $``+"$,
is anti commutative,
and isn't associative, but
$[a \times (b \times c)] + [b \times (c \times a)] + [c \times (a \times b)] = 0$
A: 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 $\mathbb{R}$, $\mathbb{C}$, 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. 
A: All of these other answers go way over my head, but this is the way I think about it.
The complex numbers can be represented on a 2-dimensional plane, but they are an extension of the one-dimensional real number line. They are not really 2-dimensional, it's just a convenient way for us to represent them.  When we expand the reals from 1 dimension into 2 dimensions, the corresponding complex numbers must double their dimension as well, going from 2 to 4. That is why they appear to "skip" 3.
Perhaps I am wrong, or this argument was folded into the other explanations, and I just didn't see it.
