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As I studied at high school there are 5 Number systems :

Natural numbers (N) Integers (Z) Rational numbers (Q) Real numbers (R) Complex numbers (C).

I remember one time our teacher told us that there is a sixth number system which called H , and it is used by video games developers, is it right that there is such number system ?

and if it's true, are they other number systems ? and why we haven't studied them at university ?

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Quaternions. –  MJD Apr 30 '13 at 0:44
Since the Wikipedia page is exceptionally abstruse on this point, a few simple details: Quaternions are used particularly to represent the orientations of bodies in game development - their 3-dimensional 'roation' - and they're used because the multplication of quaternions provides a natural representation for composing rotations (i.e., 'first twist like SO, then twist like SO'). They're not the only means of representing rotations, but they have many mathematically desirable qualities. –  Steven Stadnicki Apr 30 '13 at 1:49
The article on quaternions and spatial rotation is somewhat less abstruse, at least in some parts. –  Ilmari Karonen Apr 30 '13 at 4:59
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6 Answers 6

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The five systems you have listed are the most commonly seen systems of numbers, but there are indeed others. The $\Bbb{H}$ your teacher mentioned is most likely the quaternions, which is a way of extending the complex numbers: rather than elements that look like $a + bi$, you have things of the form $a + bi + cj + dk$, where $i^2 = j^2 = k^2 = -1$, and there are specific rules for multiplying any two of these "imaginary units." (You do lose commutativity of multiplication in the process of creating $\Bbb H$, though: that is, $ab$ might not be the same as $ba$ for $a,b\in\Bbb H$). Another example of a number system you might not have seen before is the $p$-adic numbers ($\Bbb{Q}_p$), which is important in number theory. These numbers are created by completing the rational numbers ($\Bbb Q$) with respect to a different absolute value that has to do with how many times a prime $p$ divides the numerator and denominator of your rational number. Many of these number systems are studied at university, but you have to take the right courses! Number theory will introduce you to $\Bbb H$ and $\Bbb{Q}_p$, and abstract algebra will also give you some insight into $\Bbb{H}$. Other systems of numbers add infinities and infinitesimals to the real numbers $\Bbb R$, and those are encountered in non-standard analysis.

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The set denoted $\mathbb H$ could also refer to the hyperreals (the infinities and infinitesimals). These are critical to the theories behind differential and integral calculus (a high-school subject in many locales), if not necessarily the practice. Calculus is in turn critical in game development in a number of ways (physics, lighting, etc) but the hyperreals themselves less so, so quaternions (which can describe rotations in N dimensions) is probably the better bet. –  KeithS Apr 30 '13 at 3:18
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Your teacher was probably talking about Quaternions, but there are many, many more number systems. They are studied in Modern Algebra, or more specifically Ring Theory.

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The quaternions are a special collection of numbers that generate a number system called the quaternion algebra and denoted $\mathbb{H}$.

If you take a course on abstract algebra, then you will surely encounter the quaternions. My first encounter with them was in the study of groups, although you may also see them in a physics course where they are denoted $\hat{i},\hat{j},\hat{k}$ and not typically called by name.

There are many more number systems than those one encounters in high school or beginning university mathematics. For example, there are finite number systems referred to as modular arithmetic, many examples between $\mathbb{Q}$ and $\mathbb{C}$ called number fields, and for each prime number $p$ there are the $p$-adic numbers $\mathbb{Q}_p$. If one broadens their definition of number, then there is a vast supply of examples called fields, rings, and groups (descending in order of abstraction.)

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Undoubtedly your teacher was referring to the quaternions. The use of $\mathbb{H}$ to represent them is due to Rowan Hamilton being the original discoverer. The following anecdote from Wikipedia's article on the history of quaternions is associated with Hamilton's discovery:

On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:

        i^2 = j^2 = k^2 = ijk = -1
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There are also quaternions and octonions. See the book by Conway and Smith on this subject. The former corresponds to four-dimensional numbers, the latter to eight-dimensional numbers.

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There are many more number systems than the ones people are normally taught:

  • $(1) \mathbb{P}$ (the primes)
  • $(2) \mathbb{A}$ (the algebraic numbers)
  • $(3) \mathbb{T}$ (the transcendental numbers)
  • $(4) \mathbb{I}$ (the imaginary numbers)
  • $ (5) \mathbb{J}$ (the irrationals)
  • $ (6) \mathbb{H}$ (the quaternions: 4-dimensional)
  • $(7)\mathbb{O}$ (the octonions: 8-dimensional)
  • $(8) \mathbb{S}$ (the sedenions: 16-dimensional)
  • $(9)\mathbb{L}$ (the pathions: 32-dimensional)
  • $(10) \mathbb{X}$ (the chingons: 64-dimensional)
  • $(11) \mathbb{U}$ (the routons: 128-dimensional)
  • $(12) \mathbb{V}$ (the voudons: 256-dimensional)

Numbers $(6)$ to $(12)$ are hypercomplex number systems, formed by extending the set of complex numbers. The last four are not useful (I'm guessing because not enough research has gone into them, and they have very strange properties).

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I would disagree with (1), (3), (4), and (5). Since they aren't closed under the usual operations, they aren't really number systems. –  goblin Jun 14 at 18:15
@user18921 Where, in the definition of a number system, does it mention closure under $+, \times, \div, -$? –  alexqwx Jun 14 at 18:40
@user18921 By that reasoning, $\mathbb{N}$ is not a number system since it's not closed under subtraction. –  alexqwx Jun 14 at 18:41
I mean, look. Each of $\mathbb{P},\mathbb{T},$ and $\mathbb{J}$ are very important subsets of other number systems. But as stand alone algebraic systems in their own right? They aren't closed under addition. Nor are they closed under multiplication. So they can only really be equipped with unary operations. How boring is that? –  goblin Jun 14 at 20:25
Now consider $\mathbb{I}.$ This is a little better, since it is closed under addition, which, being a binary operation, is a little more interesting. But $(\mathbb{I},0,+)$ is isomorphic to $(\mathbb{R},0,+).$ So although $\mathbb{I}$ is an important subset of $\mathbb{C}$, nonetheless, as a stand-alone algebraic structure, there's really nothing to be said about it. –  goblin Jun 14 at 20:26
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