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Just like how complex numbers generalize real numbers, Quaternions further generalize complex numbers to 4 dimensions, with 3 “imaginary” components. A quaternion is defined as a 4-vector:

\[q := a + bi +cj + dk\]The first dimension is called the *real part* and the latter three dimensions are called
the *vector part*. Just like how complex numbers allow interaction between the real
and imaginary components, quaternions similarly allow interactions via the following definitions:

- \[i^2 = j^2 = k^2 = -1\]
- \(i\) times \(j\) (i.e. \(ij\)) produces \(k\)
- \(j\) times \(k\) (i.e. \(jk\)) produces \(i\)
- \(k\) times \(i\) (i.e. \(ki\)) produces \(j\)
- The previous three rules, in the opposite order, require a multiplicative factor of \(-1\)

This seems hard to remember until you realize these rules function in a circle:

\[i \rightarrow j \rightarrow k \rightarrow i\]Going in the reverse order requires multiplying by \(-1\).

Just as we can write all unit complex numbers as

\(z = \cos (\theta) + \sin (\theta) i\),

We can similarly write any quarternion in as

\[q = \cos(\theta) + \sin (\theta) (ai + bj + ck)\]- Double cover: we can produce any desired rotation as \(q v \overbar{q} = (-q) v (- \overbar{q})\), meaning there are two ways to perform any rotation.