How are Quaternions derived from Complex numbers or Real numbers?

Question

I understand how complex numbers are derived from real numbers. Namely when you have a sqrt of a negative number you must have an answer of some kind, but this answer cannot be in the real number system, therefore you need another number system which you call complex.

What I do not understand is which problem in calculation, either calculation in real numbers or in complex numbers, you cannot solve within those systems, for which you need another system like the quaternion. Thus I fail to see how the quaternion system can be derived from the complex or real number system from a perspective of calculation.

To me it therefore seemed that someone wanted a 3dimensional number system to represent problems in graphing. This person then found out that the 3rd dimension in the number system made no sense (the relation of this 3rd dimension with complex and real numbers is unclear) and therefore defined a 4th dimension in order to make sense of this 3rd dimension. What I think at this moment is that he has not derived quaternions from complex and real numbers but instead simply chose to define another system.

However: Since I am a noob and this guy was clearly a genius, I presume that I fail to see something. And the relevant thing that I fail to see is: How are quaternions derived from complex and real numbers from a calculation perspective. (Not from a geometrical perspective).

Answer 1

The "someone" you're describing was the Irish physicist and mathematician Willian Rowan Hamilton.

What he wanted was exactly a 3-dimensional number system -- not particularly for graphing as such, but more generally to get something that would make it possible to use algebraic techniques in space geometry as easy as the complex plane had by then already made it for plane geometry.

In modern terms one might say that what Hamilton was really looking for was what we know as three-dimensional vectors, but in those days there was a general feeling that you needed to have a well-behaved multiplication rule for your system before you were "allowed to" use algebraic notation for your calculations with it. The modern concept of a vector space (where we're quite comfortable with having things that can be added, but don't have a multiplication that satisfies the same rules as multiplication of real numbers) did not exist yet.

Hamilton later wrote that he had tried for a long time to find a multiplication rule that would work for three-dimensional quantities, but without luck (later it was proved to be impossible). Then in a sudden flash of inspiration he realized that a four-dimensional system would be possible. There's a plaque at the exact spot in Dublin where he said this insight occurred.

It is therefore not so much a matter of "deriving" the quaternions, as of thinking about the problem for a long time (and by that time Hamilton no doubt had a keen intuitive experience with the ways a multiplication rule can fail to work) and then suddenly noticing that this particular rule happens to work. Once it works, it doesn't need to have a neat story of how you found it -- though Frobenius later proved that the quaternions are the only finite-dimensional associative division algebra over $\mathbb R$ other than the real and complex numbers, so sooner or later someone would have found it.