Hypercomplex numbers

The extension of vector methods to more dimensions suggests the definition of related hypercomplex numbers. When the multiplication of two three-dimensional vectors is performed without defining the mathematical properties of the unit vectors i, j, k, the formal result is... 

In terms of the Hamilton formalism a hypercomplex number of unit norm can now be defined in the form... 

These numbers do not obey all of the laws of the algebra of complex numbers. They add like complex numbers, but their multiplication is not commutative. The general rules of multiplication of n-dimensional hypercomplex numbers were investigated by Grassmann who found a number of laws of multiplication, including Hamilton s rule. These methods still await full implimentation in physical theory. 

In 1843, during a flash of inspiration while walking with his wife, Hamilton realized that it took four (not three) numbers to accomplish a 3-D transformation of one vector into another. In that instant, Hamilton saw that one number was needed to adjust the length, another to specify the amount of rotation, and two more to specify the plane in which rotation takes place. This physical insight led Hamilton to study hypercomplex numbers (or quaternions) with four components, sometimes written with the form Q = + ad + a + aji where the as are... 

The concept of complex numbers can be generalized to hypercomplex numbers, with the next level being a 4-vector, called a quartemion, i.e. q = qo + iq + jqi + kq, with o, Qi, Qi, Qi being real numbers. A quartemion has a real part, q, and the three imaginary components qi, qi, q. The latter can be considered as a vector in a three-dimensional space, where each of the unit vectors have the property f = = -l. 

For the benefit of those readers who are not familiar with hypercomplex numbers and quaternions, an elementary introduction is provided as an Appendix A to this chapter. 

This hypercomplex number is given in matrix form by... 

Large relativistic effects in group 11 compounds. Hypercomplex number... 

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