Geometric numbers comprise the real numbers and two new elements a,b with special properties.
The new geometric numbers a,b also called g-numbers have the following properties:
(i) a2=0=b2 The new g-numbers are called nilpotent or null vectors.
(ii) 2a‧b =ab+ba=1
The g-Numbers Expressed as Matrices
The canonic elements a and b expressed as matrices:
Squaring of the canonic matrices will result in the zero matrix:
a2 = 0;
b2 = 0;
Building the product a and b and the product b and a.
The products ab and ba expressed as matrices:
ab + ba = 1
ab - ba = -h
(ab)(ba) = 0 = (ba)(ab)
From the above properties the multiplication table for basic g-numbers a, b, ab, ba can be dereived.
The basic g-numbers can be written in a matrix form:
We have the following matrices for a, b, ab and ba:
The general form of a g-matrix is as follows:
A geometric number g can know be expressed as follows.
Conjugation and Inversion
Each g-number can be seen as the sum of two parts g = go + ge
the odd part go and the even part ge.
The odd part of a g-number looks as follows:
The even part of a g-number looks as follows:
The reverse of a g-number
With (ab)† = ba the reverse operation for the odd and the even part are defined as follows:
go† = go
ge† = g11ab + g22ba
The reverse of a g-number has the following property:
(go + ge)† =
go† + ge† =
go + ge†
Hence we have for g†
The inverse of a g-number
The invers of a g-numger is named g- and has the following properties:
g-o = -g12b - g21a; for the odd part.
g-e = ge; for the even part.
The sum of the even and the odd part will give g-
g- = (go + ge)- = g-o + g-e
= -go + ge
Hence we have for g-
The combination of the operations of reversion and inversion of a g-number will give the mixed conjugation g*.
g* = (g†)- = (go + ge†)-
= -go + ge†
Geometrie of a and b
Construction of the geometric algebra 𝔾2 with the nilpotent 2x2 matrices a and b.
The relations between these matrices can be found by the matrix representation of the geometric algebra 𝔾2