A (finite-dimensional unital) division algebra is an algebra with multiplicative identity where unique right and left inverses exist for every non-zero element. For an associative division algebra, these inverses are equal, turning the non-zero vectors into a group under multiplication. Ignoring scalar multiplication, an associative algebra is a ring, and a commutative associative division algebra is a field; thus an associative division algebra with scalar multiplication ignored is sometimes called a division ring or skew field. A module over a non-commutative skew field (such as ${\mathbb{H}}$) can be seen to have much of the same features as a vector space, including a basis.

In a division algebra, the existence of the left inverse ${u_{L}^{-1}}$ of ${u}$ allows us to “divide” elements in the sense that for any non-zero ${u}$ and ${v}$, ${xu=v}$ has the solution ${x=vu_{L}^{-1}}$, which we can regard as the “left” version of ${v/u}$; similarly, ${ux=v}$ has the solution ${x=u_{R}^{-1}v}$. This is equivalent to requiring that there be no zero divisors, i.e. ${uv=0\Rightarrow u=0}$ or ${v=0}$. A normed division algebra has a vector space norm that additionally satisfies ${\left\Vert uv\right\Vert =\left\Vert u\right\Vert\left\Vert v\right\Vert }$.

Real division algebras are highly constrained: all have dimension 1, 2, 4, or 8; the commutative ones all have dimension 1 or 2; the only associative ones are ${\mathbb{R}}$, ${\mathbb{C}}$, and ${\mathbb{H}}$; and the only normed ones are ${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{H}}$, and ${\mathbb{O}}$. Here we review these division algebras:

• ${\mathbb{C}}$, the complex numbers, has basis ${\left\{ 1,i\right\} }$ where ${i^{2}\equiv-1}$
• ${\mathbb{H}}$, the quaternions, has basis ${\left\{ 1,i,j,k\right\} }$ where ${i^{2}=j^{2}=k^{2}=ijk\equiv-1}$
• ${\mathbb{O}}$, the octonions, has basis ${\left\{ 1,i,j,k,l,li,lj,lk\right\} }$, all anti-commuting square roots of ${-1}$; we will not describe the full multiplication table here

We can define the quaternionic conjugate by reversing the sign of the ${i}$, ${j}$, and ${k}$ components, with the octonionic conjugate defined similarly. The norm is then defined by ${\left\Vert v\right\Vert =\sqrt{vv^{*}}}$ in these algebras, as it is in ${\mathbb{C}}$. Note that this implies two sided inverses for all normed real division algebras, namely ${v^{-1}=v^{*}/\left\Vert v\right\Vert ^{2}}$.

We lose a property of the real numbers each time we increase dimension in the above algebras: ${\mathbb{C}}$ is not ordered; ${\mathbb{H}}$ is not commutative; and ${\mathbb{O}}$ is not associative. It turns out that any subalgebra of ${\mathbb{O}}$ generated by two elements, however, is in fact associative.

${\mathbb{C}}$ is a field (ignoring real scalar multiplication) and so can be used as the scalars in a vector space ${\mathbb{C}^{n}}$. One could imagine then trying to find a multiplication on ${\mathbb{C}^{n}}$ to obtain complex division algebras, but the only finite-dimensional complex division algebra is ${\mathbb{C}}$ itself.

The quaternions form a non-commutative ring, and so can be used as the scalars in a left module ${\mathbb{H}^{n}}$, but there is no obvious definition of ${\mathbb{O}^{n}}$ since the octonions are not associative. However, we can form all of the algebras ${\mathbb{R}\left(n\right)}$, ${\mathbb{C}\left(n\right)}$, ${\mathbb{H}\left(n\right)}$, and ${\mathbb{O}\left(n\right)}$, where ${\mathbb{K}\left(n\right)}$ denotes the algebra of ${n\times n}$ matrices with entries in ${\mathbb{K}}$. ${\mathbb{H}\left(n\right)}$ can even be viewed as the group of linear transformations on ${\mathbb{H}^{n}}$, if ${\mathbb{H}^{n}}$ is defined as a right module while matrix multiplication takes place from the left as usual.