3.7 Extension of Arithmetics

Quaternion

One can of course continue the extension of notion of a number from numbers to number pairs. The next step would be quaternions, which consist of the 4 real numbers. Quaternions can be used for calculations in for dimensional spaces; for example relativistic physical systems with spin can be described using quaternions. We here refer to the subject literature.

When defining applicable rules for arithmetic operations care is taken, that the complex numbers constitute a subset. However for these higher dimensional numbers possibly not all fundamental rules, that were given at the beginning of this chapters and remained valid up to the complex numbers, will hold, for example the rule of the commutativity of operations. GroupTheory

The group theory finally disassociates itself totally from the concept of the number, and defines arithmetic rules for elements, that can be numbers, but do not have to numbers. A group(set) of elements is defined, that the rules for this group are defined in such a way, that the application to elements of the group always yields a member of the group.

The rules applicable to groups are similar to those, that we had discussed at the beginning of our discussion of numbers. However the earlier implicitly assumed role of unity (the neutral element) will be explicitly defined. For the example of the multiplicative composition we shall assume by definition.

  1. The composition of two elements a,b of the Group G is again an element of the same group (closedness)a × b = c G.
  2. The sequence of operations is unimportant as long as the order is preserved:a ×b × c = a × b × c (Associativity)
  3. There is a neutral element e in the group G, for which a × e = e × a = a.
  4. For every element a in G, an inverse element(mirror image) a* exists, with the property to yield, when combined with a the neutral element:
    a × a* = a*× a = e.

A group is called Abelian or commutative, if one is allowed to commute the operands: i.e. a × b = b × a (Commutativity)

The set of integers Z with addition as the operation and zero as the neutral element is an Abelian Group. Somewhat more complicated is the situation with the set of rational numbers Q , multiplication as the operation, with 1 as the neutral element; here 0 would not have an inverse element.

The definition of group rules makes it possible also to have other objects than numbers as members of a group, as long as they satisfy the requested properties.

An example for such a group is the set of symmetry transformations rotation,reflection and inversion, via which a topological objects such as a polygon is mapped to itself; compositions are then transformations that are applied consecutively. This example is a non-Abelian group; rotation before reflection yields another result as reflection before rotation. This is visualized in the next simulation shown in Figure 3.12. It shows the consecutive operations rotation and reflection and reflection and rotation applied to an equilateral triangle, whose initial orientation can be adjusted.

Group theory is one of the foundations of the arithmetics used in quantum theory; you are encouraged to study the discussion in the chapter Physics/Symmetry and the essay by Schopper 3


PIC


Figure 3.12: Rotation and reflection of a pink equilateral triangle A. In the left window rotation (R) happens before mirroring (M); in the right window mirroring happens before rotation. The final result of the operations is coloured in blue, the intermediate result in green. The initial orientation of A and the rotation angle applied can be chosen with sliders.

End of chapter 3