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I recently started reading about sets above the complex set (the set of quaternions, the set of octonions, etc...) and since I already had a lot of difficulty understanding why complex numbers were needed in the first place, I was wondering why sets even more complicated than the complex one would ever be needed, and how they would simplify calculations. Can someone please shed some light on this (give an example or a generic explanation)? Also, what's the highest set that's really useful?

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    $\begingroup$ Depends what you call higher set. It seems you are mainly interested in number-like sets. In that case, octonions are the largest division algebra over $\Bbb R$ (so with that restriction there is no "etc." beyond them). On our way to get there, we already sacrificed linear order ($\Bbb C$), commutativity of multiplication ($\Bbb H$), and finally even our most beloved associativity ($\Bbb O$). So it is more a question of how much pain you can endure. $\endgroup$ – Hagen von Eitzen Dec 12 '15 at 11:20
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You are making many assumptions in your question. Among others

  • "number sets" have to be useful

  • a way for number systems to be useful is to simplify calculations

  • there should be a convenient and definitive list to check

But most of the time, mathematical objects are named and manipulated long before most of their properties or relations with others are understood, accepted, put in perspective, routinely used and transformed into standard teaching material.

That's what happened for complex numbers over a course of about four centuries from the italian renaissance algebraists to the 20th century.

Quaternions, octonions, sedenions and other hypercomplex systems do not enjoy the same wide technical and mathematical notoriety complex numbers have but their applicability and interest depend more on our culture and our ingenuity (or lack of it) than you may suppose. Quaternions have had a few false starts in that respects during the second half of the 19th century (lookup Hamilton, P.-G. Tait, O. Heaviside).

Please note that number systems and their natural family (rings and algebras with at least two internal operations) do not form a line starting from the integers and going to complex numbers by successive extensions. They are a diverse forest, with many (very useful trees). Among the most famous:

  • Real and complex vector spaces (point geometry, equation systems, n-dimensional spaces, ...). This is a part you will certainly study soon. It is often called Linear Algebra.

  • Boolean algebras (logic calculus, circuit design, complexity theory, ...)

  • Lie algebras (symmetries of differential equations, and much more)

  • Hopf algebras (algebraic topology, non commutative geometry, conformal field theory, ...)

and Jordan algebras, C*-algebras, general rings, tropical algebra, ...

Each time with at least one use in physics, chemistry, engineering, computer science, biosciences, in addition to their intrinsic interest and their relations with other domains of mathematics.

I must stop there because doing justice to all these algebraic structures would be equivalent to present most of today's mathematics in full detail.

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Quaternions are used in (amongst other areas) 3D computer graphics, because one can represent 3D rotation in a convenient way using them. I'll give a link (which states even more applications), as I'm not too read-up on the details.

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

In fact, the idea of representing 3D-rotations by numbers, just like one can represent 2D-rotation by complex numbers, was one of the main reasons quaternions were even introduced and considered interesting in the first place.

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  • $\begingroup$ Then again, isn't that application just fancy-speak for using matrices? $\endgroup$ – Hagen von Eitzen Dec 12 '15 at 11:22
  • $\begingroup$ The following quote regarding unit quaternions is in the link "Compared to rotation matrices they are more numerically stable and may be more efficient.", so I guess it does make a difference in applications. I have to stress again that I am not too familiar with using quaternions as of yet...Historically speaking though, I believe the main focus was on compactly representing geometric operations by direct operations (addition, multiplication, division, subtraction) on number constructs (points on the line, in the plane, in space...) rather than by matrices or other operators. $\endgroup$ – A.Sh Dec 12 '15 at 11:33

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