Prerequisites
If you are not familiar with vectors this subject you may like to look at the following page first:
If you are not familiar with vectors this subject you may like to look at the following page first:
For 4 dimensions can be generated by 4 basis vectors, e_{1}, e_{2}, e_{3} and e_{4} this means that a 4D multivector is represented by 16 scalar numbers:
The pages below this explain in detail:
Choice of bivector and trivector basis  Here I have described how the order of the indexes was chosen. For instance, why don't we use e214 instead of e124? Unlike 3D multivectors there is no correspondence with cross multiplication as cross multiplication does not apply in 4D. Also I cant work out how to make the multiplication table symmetrical. Also its not possible to make all the e_{1234} terms on the bottom left to top right diagonal positive as some of these terms anticommute. 
Transform  
Arithmetic  This gives the rules for addition, subtraction and multiplication of 4D multivectors. 
Functions  This shows the dual, reverse and conjugate functions of of 4D multivectors. 
isometries  
code  
pauli  An alternative notation using a matrix to represent each multvector 
The following table explains the notation for each of the 16 elements as follows:
grade  base value  numerical value  

full 
shortened 

0=unit scalar  1  e  
1=unit length base vectors  e_{1}  e1  
e_{2}  e2  
e_{3}  e3  
e_{4}  e4  
2=unit length base bivectors  e_{1}^ e_{2}  e_{12}  e12 
e_{1}^ e_{3}  e_{13}  e13  
e_{1}^ e_{4}  e_{14}  e14  
e_{2}^ e_{3}  e_{23}  e23  
e_{4}^ e_{2}  e_{42}  e42  
e_{3}^ e_{4}  e_{34}  e34  
3=unit length base trivector  e_{3}^ e_{2}^ e_{1}  e_{321}  e321 
e_{1}^ e_{2}^ e_{4}  e_{124}  e124  
e_{4}^ e_{3}^ e_{1}  e_{431}  e431  
e_{2}^ e_{3}^ e_{4}  e_{234}  e234  
4=unit length base trivector  e_{1}^ e_{2}^ e_{3}^ e_{4}  e_{1234}  e1234 
So in this case the number of dimensions is:
In this case the number of scalar values in the multivector is 16 = (1+4+6+4+1)
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see also:  
Correspondence about this page 

Book Shop  Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. 
Geometric Computing for Perception Action Systems: Concepts, Algorithms, and Scientific Applications (Hardcover). This is the only book I have so far come across that has a reasonable explanation of 'motors' and why it is useful to use 4D Geometric algebra to represent kinematics of solid bodies (in chapter 2). The book is quite a slim volume considering that it covers both fundamental concepts and practical applications. Therefore I think you will need to have a good understanding of Geometric Algebra before starting on this book. 
Terminology and Notation Specific to this page here: 

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