We have already looked at complex numbers of the form: 3 + i 4. The 'i' operator has several functions here,
- It creates a new dimension (it tells us to keep 3 and 4 separate and not add to give 7).
- It tells us that when multiplied by itself we get -1 in other words we have i*i=-1.
- It is an operator which can be manipulated in the equation.
We took this further with quaternions and octions by adding extra dimensions that also square to a negative number (i,j,k…).
However we can generalise this further by separating out the above concepts and allowing dimensions which square to positive or zero in addition to negative values.
- i² = -1.
- D² = +1.
- ε² = 0.
These numbers have a form like:
a = b + ω c
where:
- ω is a general operator, which could be i, D or ε.
Can we define other algebras with similar properties?
Double Numbers
Double numbers are similar to this except that:
D2 = 1
So in this case ω=D, the double operator.
Another name for this is a split-complex number. This is a two dimensional algebraic quantity, in other words elements in this algebra contain two scalar values. Both these dimensions square to +1 (so there is no square root of -1)
As with most algebras, addition is defined by adding corresponding terms and the flavor of the algebra comes from the multiplication rules. If we denote the two dimensions by 1 and D then a '1' term is given by multiplying the same types together, a 'D' term is given by multiplying different terms. There are no sign changes. So we can define the algebra by the following table:
| a*b | b.1 | b.D |
| a.1 | 1 | D |
| a.D | D | 1 |
More information about double numbers here.
Dual Numbers
Again dual numbers are similar to the form for complex numbers above except that:
ε2 = 0
So in this case ω=ε, the dual operator.
Algebras which have this property are known as grassmann Algebras, in Clifford Algebra terms, this is the outer or exterior product. Vector cross products also have the same property.
It may seem strange that the square root of a non-zero quantity is zero, although it makes some kind of sense, because the square of a very small quantity is a very, very small quantity. However this is a multidimensional value which has different rules. For example the vector cross product, if we take a 3D vector 'v' then v x v = 0 because the cross product of parallel vectors is zero.
As with the other algebras here, addition is defined by adding corresponding terms and the flavor of the algebra comes from the multiplication rules. If we denote the two dimensions by 1 and ε then a '1' term is given by multiplying the same types together, a 'ε' term is given by multiplying different terms. There are no sign changes. So we can define the algebra by the following table:
| a*b | b.1 | b.ε |
| a.1 | 1 | ε |
| a.ε | ε | 0 |
More information about dual numbers here.
The Dual of a Quaternion
These can be used to form a 'motor' as explaind on this page. A motor is an abbreviation of "moment and vector" (to represent sums of spins) It can represent a combination of translation and rotation, known as screw motion or rigid motion.
The arithmatic of dual quaternions is explained on this page.
Definitions of Dual and Double
These terms are both heavily used, so we need to define how we are using these terms:
dual vector space: example, in 3D, vectors and bivectors
hodge dual
clifford dual
