deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 2fc2002..2cfaaf7 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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j\cdot k=k\cdot j=0  \end{equation}  This is similar to having 3 independant imaginary numbers or basis vectors. If this were an inner product then they are orthoganal but antiparallell with themselves. They form a NON-ASSOCIATIVE 'group' under a product with elements $ {0,1,i,j,k,-1,-i,-j,-k} $, this is an Abelian relationship as the non-Abelian properties of the quaternions was removed with the cross interactions. For example, \begin{equation}  (i \cdot i) \cdot j = -1 \cdot j = -j \\  i \cdot ( i \cdot j ) = i \cdot 0 = 0 \\  (a \cdot b) \cdot c \ne a \cdot (b \cdot c)  \end{equation}