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I'm trying to solve the Quaternion function: $i^2 = j^2 =k^2 =ijk=-1$

I'm trying to solve it by myself since I found it quite interesting even though its far above and beyond Advanced Level syllabus, does anyone here have any advice on how to go about it? I can't really ask those around me as they have up to Advanced Level Knowledge.

Edit: my main stumbling block is proving $ijk=-1$

Andrew1024
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2 Answers2

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$$ \begin{array}{ccccc} & & i \\ & \nearrow & & \searrow \\ k & & \leftarrow & & j \end{array} $$ \begin{align} \hline \text{following} & \text{ the arrows} & \text{opposite} & \text{ direction} \\[6pt] \hline ij & = k & ji & = -k \\ jk & = i & kj & = -i \\ ki & = j & ik & = -j \\ \hline \end{align} The above is the multiplication table.

$$ ijk = (ij)k = (k)k = -1 \quad \text{or}\quad ijk = i(jk) = i(i) = -1. $$

Michael Hardy
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What you have here is a definition of the $4$ dimensional algebra of the Quaternions. By definition, \begin{align} \mathbb{H} &= \{a + bi + cj + dk : a, b, c, d ∈ \mathbb{R}\} \\ \end{align} Here, $i, j$ and $k$ are all square roots of $−1$ with relations

$$ij = k = −ji, jk = i = −kj, ki = j = −ik$$

These all follow from $$i^2=j^2=k^2=ijk=-1$$

And you can check these yourself.