Complex Rings Quaternion Rings and Octonion Rings

Complex numbers describe the two dimensional geometry. William Rowan Hamilton (1805-1865) researched numbers which represent the three dimensional geometry. However this study was extremely difficult. After Hamilton had spent more than ten years desperately studying in 1843 while walking beside the Royal Canal he suddenly discovered the quaternion algebra over real numbers with a basis {1 i j k}. He stopped walking and carved equations on a stone nearby a bridge; i2 = j2 = k2 = ijk = −1With Hamilton's discovery both Arthur Cayley (1821-1895) and John T. Graves (1806-1870) independently discovered the octonion numbers. Through Frobenius Wedderburn and other mathematicians quaternion numbers connected to the modern algebra of the 20th century. Today quaternion numbers are applied to science in particular to the space industry and the CG industry. The objective of this book is to introduce and describe complex rings quaternion rings and octonion rings. Several structure theorems are stated from ring theoretic view points. In presenting of this book we hope the reader will be able to feel the atmosphere of quaternion numbers.