The quaternions

Example

The quaternions $\mathbb{H}$ are the real vector space with basis $1, i, j, k$. Each quaternion is $a + b i + c j + d k$ with $a, b, c, d \in \mathbb{R}$. The multiplication is $\mathbb{R}$-bilinear, $1$ is the identity, and the basis elements obey $$i^2 = j^2 = k^2 = i j k = -1.$$ These rules give $i j = k = -j i$, $j k = i = -k j$ and $k i = j = -i k$. With this multiplication, $\mathbb{H}$ is a division ring that is not commutative.

Remarks

The conjugate of $q = a + b i + c j + d k$ is $\bar{q} = a - b i - c j - d k$, and $q \bar{q} = a^2 + b^2 + c^2 + d^2$. So a nonzero $q$ has the inverse $q^{-1} = \bar{q} / (a^2 + b^2 + c^2 + d^2)$. The quaternions are not a field, because $i j \neq j i$.

Explanations

L1 human

The quaternions extend the complex numbers with two more square roots of $-1$, called $j$ and $k$. You can divide by every nonzero quaternion. But the order of multiplication matters: $i j = k$ and $j i = -k$.