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Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real.

Did Hamilton consider such a case?

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    $\begingroup$ No, there is no real division algebra of dimension $3$, see here. But Hamilton tried for a long time to find such numbers. $\endgroup$ Sep 1, 2015 at 14:42
  • $\begingroup$ So Hamilton knew this fact? $\endgroup$
    – kaiser
    Sep 1, 2015 at 14:44
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    $\begingroup$ Yes, Hamilton sort of knew it, i.e., realized it the hard way. $\endgroup$ Sep 1, 2015 at 14:44

2 Answers 2

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In fact Hamilton originally tried to construct precisely such a theory -- a three-dimensional analogue for the two-dimensional representation of the complex numbers. In order to define multiplication in such a system, you have to specify what the product of $i$ and $j$ should be. One obvious choice is to say that $ij=0$, but then you have a problem: the system you are working with is not a field (nor even an integral domain), and Hamilton wanted to construct a three-dimensional field. Hamilton tried various combinations, but in none of them did the geometric interpretation of the 3-vectors match up properly with the algebraic computations. He finally concluded that the only way to make everything work out was to introduce a fourth "direction", $k$, and to make multiplication non-commutative.

A good reference for this is "Concepts and the Mangle of Practice Constructing Quaternions" by Andrew Pickering, which is Chapter 15 of the book 18 Unconventional Essays On the Nature of Mathematics, edited by Reuben Hersh.

But of course you don't need to feel tied down to Hamilton's criteria; in particular, if you don't care about whether your algebraic structure is an integral domain then you can define lots of different multiplicative structures on the set of objects of the form $a + bi + cj$, corresponding to different choices of the products $ij$ and $ji$.

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Compare $b$i+$c$j+$d$k with the vector $(b,c,d)$ and for multiplication of such entities compare quaternion multiplication with the vector product. You will see that there is a clear relationship between the two.

Hidden within the quaternions is this structure which does relate to three dimensions.

So though you can't do everything you want with three components, you can get somewhere.

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