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I have seen many times before that Hamilton started off believing he would need a three-dimensional system over the reals in order to describe 3D rotations. He considered numbers of the form $a + bi + cj$, defined summation component-wise and tried to find a suitable multiplication.

My question is: how did he conclude these numbers could not be divided and how did he, from this point, conclude there would need to be four dimensions for the quaternions to work? Any references for further reading are also greatly appreciated!

Thanks in advance!

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    $\begingroup$ Because a 3 dimensional ring with required properties does not exist. Probably Hamilton did not have a proof of this, but he convinced himself in this fact after long trials. Then he found a 4-dimensional one. $\endgroup$ Jun 17 '21 at 23:28
  • $\begingroup$ @AlexandreEremenko Why does a 3D ring with required properties not exist? $\endgroup$
    – Gauss
    Jun 18 '21 at 0:54
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    $\begingroup$ It is a simple theorem: there is no 3-dimensional division algebra over reals. See, for example in the book of Kantor and Solodovnikov, Hypercomlex numbers. $\endgroup$ Jun 18 '21 at 2:35
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    $\begingroup$ There's a joke I like. Hamilton wants to extend the complex numbers to higher dimensions, so he introduces $j$, with $j^2=-1$, and he forms numbers like $a+bi+cj$ just as you've written them. But of course, one needs a rule for $i$ times $ j$. He tries to figure out what it will be, subject to obeying various nice laws such as associativity and distributivity and so on, but nothing works (the long trials in the comment of @AlexandreEremenko). And one day, walking over a Dublin bridge that crosses the river Liffey, it hits him: $i$ times $j$ is $k$!!!!! $\endgroup$
    – Lee Mosher
    Jun 19 '21 at 3:02

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