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  "body": "\n## Example\n\nThe quaternions $\\mathbb{H}$ are the real vector space with basis $1, i, j, k$.\nEach quaternion is $a + b i + c j + d k$ with $a, b, c, d \\in \\mathbb{R}$.\nThe multiplication is $\\mathbb{R}$-bilinear, $1$ is the identity, and the basis elements obey\n$$i^2 = j^2 = k^2 = i j k = -1.$$\nThese rules give $i j = k = -j i$, $j k = i = -k j$ and $k i = j = -i k$.\nWith this multiplication, $\\mathbb{H}$ is a [[division ring|0WZXK05ME7]] that is not commutative.\n\n## Remarks\n\nThe conjugate of $q = a + b i + c j + d k$ is $\\bar{q} = a - b i - c j - d k$, and $q \\bar{q} = a^2 + b^2 + c^2 + d^2$.\nSo a nonzero $q$ has the inverse $q^{-1} = \\bar{q} / (a^2 + b^2 + c^2 + d^2)$.\nThe quaternions are not a field, because $i j \\neq j i$.\n\n## Explanations\n\n### L1 human\n\nThe quaternions extend the complex numbers with two more square roots of $-1$, called $j$ and $k$.\nYou can divide by every nonzero quaternion.\nBut the order of multiplication matters: $i j = k$ and $j i = -k$.\n",
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