Linear Electronics in Control Systems by Robert C. Baker

By Robert C. Baker

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Example text

His theory has been accorded a prominent place in mathematics only in the last forty years, and it is hardly known at all to physicists. Grassmann himself was the only one to use it during the first two decades after it was published. There are several reasons for this. The most important one arises from the fact that Grassmann’s understanding of the abstract nature of mathematics was far ahead of his time. He was the first person to arrive at the modern conception of algebra as a system of rules relating otherwise undefined entities.

This provides a geometric interpretation for the equation when i and -1 are both regarded as operators (by multiplication) on vectors. 1. Notice that multiplication by i rotates vectors counterclockwise by a right angle.

All the geometrical facts needed to discover such a product have been mentioned above. It has already been noted that the inner and outer products seem to complement one another by describing independent geometrical relations. This circumstance deserves the most careful study. The simplest approach is to entertain the possibility of introducing a new kind of product ab by the equation Here the scalar has been added to the bivector At first sight it may seem absurd to add two directed numbers with different grades.

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