Geometric Numbers
Geometric numbers comprise the real numbers and two new elements a,b with special properties.
The new geometric numbers a,b also called gnumbers have the following properties:
(i) a^{2}=0=b^{2} The new gnumbers are called nilpotent or null vectors.
(ii) 2a‧b =ab+ba=1
The gNumbers Expressed as Matrices
The canonic elements a and b expressed as matrices:
Squaring of the canonic matrices will result in the zero matrix:
a^{2} = 0;
b^{2} = 0;
Building the product a and b and the product b and a.
ab
ba
The products ab and ba expressed as matrices:
ab + ba = 1
ab  ba = h
aba=(ab)a=(1ba)a=aba^{2}=a
bab=(ba)b=(1ab)b=bab^{2}=b
(ab)^{2}=(1ba)ab=ab+ba^{2}b=ab
(ba)^{2}=(1ab)ba=ba+ab^{2}a=ba
(ab)(ba) = 0 = (ba)(ab)
From the above properties the multiplication table for basic gnumbers a, b, ab, ba can be dereived.

a 
b 
ab 
ba 
a 
0 
ab 
0 
a 
b 
ba 
0 
b 
0 
ab 
a 
0 
ab 
0 
ba 
0 
b 
0 
ba 
The basic gnumbers can be written in a matrix form:
We have the following matrices for a, b, ab and ba:
The general form of a gmatrix is as follows:
A geometric number g can know be expressed as follows.
Conjugation and Inversion
Each gnumber can be seen as the sum of two parts g = g_{o} + g_{e}
the odd part g_{o} and the even part g_{e}.
The odd part of a gnumber looks as follows:
The even part of a gnumber looks as follows:
The reverse of a gnumber
With (ab)^{†} = ba the reverse operation for the odd and the even part are defined as follows:
g_{o}^{†} = g_{o}
g_{e}^{†} = g_{11}ab + g_{22}ba
The reverse of a gnumber has the following property:
g^{†} =
(g_{o} + g_{e})^{†} =
g_{o}^{†} + g_{e}^{†} =
g_{o} + g_{e}^{†}
Hence we have for g^{†}
The inverse of a gnumber
The invers of a gnumger is named g^{} and has the following properties:
g^{}_{o} = g_{12}b  g_{21}a; for the odd part.
g^{}_{e} = g_{e}; for the even part.
The sum of the even and the odd part will give g^{}
g^{} = (g_{o} + g_{e})^{} = g^{}_{o} + g^{}_{e}
= g_{o} + g_{e}
Hence we have for g^{}
Mixed conjugation
The combination of the operations of reversion and inversion of a gnumber will give the mixed conjugation g^{*}.
g^{*} = (g^{†})^{} = (g_{o} + g_{e}^{†})^{}
= g_{o} + g_{e}^{†}
Geometrie of a and b
Construction of the geometric algebra 𝔾^{2} with the nilpotent 2x2 matrices a and b.
The relations between these matrices can be found by the matrix representation of the geometric algebra 𝔾^{2}
Odd Number Plane

Even Number Plane
