The Real Representations And Elementary Functions Of Quaternion Numbers

The quaternion was invented by the Irish mathematician, William Rowan Hamilton in1843. For more than one hundred years, many scientists have studied quaternion widely and deeply, and quaternion is applied widely to electromagnetics, cybernetics, aeronautics, spaceflight, artificial intelligence etc.This paper is divided into five chapters. In Chapter1, we introduce the definition and calculational rules of quaternion. In Chapter2, we define three different real representations of quaternion by the concepts and calculational rules. These real representations are extended to vectors and matrices on the quaternion field. By using these real representations alternately, we can commute the multiplication of quaternion numbers, vectors and matrices in the representative form. In this way, the calculational difficulties by the noncommutative multiplication will be reduced reasonably. In Chapter3, some algebraic expressions on quaternion field can be represented the equivalent expressions on real number field by using these real representations, some calculational problems of quaternion matrix can be solved on real number field.In Chapter4, we define some elementary functions on quaternion field with the form of power series. These functions are according to the Maclaurin expansions of some complex functions and the similarity between complex numbers and quaternion numbers. By compare with the complex functions, it is easy to get the basic properties of these functions on quaternion field. In Chapter5, acording to the main results in Chapter2,3,4, we compile some MATLAB programs. These programs help us to save and compute quaternion numbers, vectors and matrices by computers, and it is the final subject of this paper.