Hypercomplex Numbers

This book deals with various systems of "numbers" that can be constructed by adding "imaginary units" to the real numbers. The complex numbers are a classical example of such a system. One of the most important properties of the complex numbers is given by the identity (1) Izz'l = Izl·Iz'I· It says, roughly, that the absolute value of a product is equal to the product of the absolute values of the factors. If we put z = al + a2i, z' = b+ bi, 1 2 then we can rewrite (1) as The last identity states that "the product of a sum of two squares by a sum of two squares is a sum of two squares. " It is natural to ask if there are similar identities with more than two squares, and how all of them can be described. Already Euler had given an example of an identity with four squares. Later an identity with eight squares was found. But a complete solution of the problem was obtained only at the end of the 19th century. It is substantially true that every identity with n squares is linked to formula(1), except that z and z' no longer denote complex numbers but more general "numbers" where i,j, . . . , I are imaginary units. One of the main themes of this book is the establishing of the connection between identities with n squares and formula (1).

I Hypercomplex Numbers.- 1 Complex Numbers.- 1.1 Introduction.- 1.2 Operations on Complex Numbers.- 1.3 The Operation of Conjugation.- 1.4 The Absolute Value of a Complex Number: An Identity with Two Squares.- 1.5 Division of Complex Numbers.- 2 Alternate Arithmetics on the Numbers a + bi.- 2.1 Formulation of the Problem.- 2.2 Reduction to Three Systems.- 3 Quaternions.- 3.1 Preliminaries.- 3.2 The Definition of Quaternions.- 3.3 Associativity of Multiplication of Quaternions.- 3.4 Conjugation of Quaternions.- 3.5 The Quaternions as a Division System.- 3.6 Absolute Value of a Product.- 3.7 The Four-Square Identity. General Formulation of the Problem of the Sum of Squares.- 4 Quaternions and Vector Algebra.- 4.1 The Number and Vector Parts of a Quaternion.- 4.2 Scalar Product of Vectors.- 4.3 Cross Product of Vectors.- 4.4 The Geometric Interpretation of the Multiplication of a Quaternion by a Pure Vector Quaternion.- 4.5 Representation of an Arbitrary Rotation in Space by Means of Quaternions.- 4.6 The Problem of “Composition” of Rotations.- 5 Hypercomplex Numbers.- 5.1 Definition of a Hypercomplex Number System.- 5.2 Commutative Systems, Associative Systems, and Division Systems.- 6 The Doubling Procedure. Cayley Numbers.- 6.1 Another Approach to the Definition of the Quaternions.- 6.2 The Doubling of a Hypercomplex System. Definition of the Cayley Numbers.- 6.3 The Multiplication Table of the Cayley Numbers.- 6.4 Conjugation of Cayley Numbers. Absolute Values of Cayley Numbers.- 6.5 The Absolute Value of the Product of Cayley Numbers.- 6.6 The Eight-Square Identity.- 6.7 The Non-associativity of Cayley Numbers. The Alternative Property.- 6.8 The Cayley Numbers Are a Division System.- 7 Algebras.- 7.1 Heuristic Considerations.- 7.2 Definition of an Algebra.- 7.3 A Hypercomplex System as a Special Case of an Algebra.- 7.4 Commutative Algebras, Associative Algebras, and Division Algebras.- 7.5 Examples.- 7.6 An Important Example: The Algebra of n×n Matrices.- 7.7 Characterization of Multiplication in an Arbitrary Algebra.- II N-Dimensional Vectors.- 8 The N-Dimensional Vector Space An.- 8.1 Basic Definitions.- 8.2 The Concept of Linear Dependence.- 8.3 Another Definition of Linear Dependence.- 8.4 The Initial Basis.- 9 A Basis of The Space An.- 9.1 Definition of a Basis.- 9.2 Obtaining Other Bases.- 9.3 The Number of Basis Vectors.- 9.4 The Number of Vectors in a Linearly Independent System.- 9.5 A Consequence of Theorem 9.2 Pertaining to Algebras.- 9.6 Coordinates of a Vector Relative to a Basis.- 10 Subspaces.- 10.1 Definition of a Subspace.- 10.2 Examples.- 11 Lemma on Homogeneous Systems of Equations.- 12 Scalar Products.- 12.1 The Scalar Product of Geometric Vectors.- 12.2 General Definition of the Scalar Product.- 12.3 One Way of Introducing a Scalar Product.- 12.4 Length of a Vector. Orthogonal Vectors.- 12.5 Expressing a Scalar Product in Terms of Coordinates.- 12.6 Existence of a Vector Orthogonal to p Given Vectors, p < n.- 12.7 Decomposition of a Vector into Two Components.- 13 Orthonormal Basis. Orthogonal Transformation.- 13.1 Definition of an Orthonormal Basis.- 13.2 Existence of Orthonormal Bases.- 13.3 A Method for Obtaining All Orthonormal Bases.- 13.4 Orthogonal Transformations.- 13.5 The Inverse of an Orthogonal Transformation.- 13.6 “How Many” Different Orthogonal Transformations Are There?.- III The Exceptional Position of Four Algebras.- 14 Isomorphic Algebras.- 15 Subalgebras.- 16 Translation of the “Problem of the Sum of Squares” into the Language of Algebras. Normed Algebras.- 16.1 The Connection between (!) and a Certain Algebra A.- 16.2 The Possibility of Introducing a Norm in the Algebra A.- 16.3 Conclusion.- 17 Normed Algebras with an Identity. Hurwitz’s Theorem.- 17.1 Formulation of Hurwitz’s Theorem.- 17.2 Sketch of the Proof of Hurwitz’s Theorem.- 17.3 Two Lemmas.- 17.4 Conclusion of the Proof.- 18 A Method for Constructing All Normed Algebras and Its Implications for the Problem of the Sum of Squares.- 18.1 A Method for Constructing New Normed Algebras.- 18.2 Construction of All Normed Algebras.- 18.3 The Number n in the Identity (!).- 18.4 Survey of All Identities (!).- 18.5 Examples of 2- and 4-Dimensional Algebras and of the As sociated Identities (!).- 19 Frobenius’ Theorem.- 19.1 Formulation of Frobenius’ Theorem.- 19.2 Three Properties of Associative Division Algebras.- 19.3 Proof of Frobenius’ Theorem.- 19.4 Proof of the Three Assertions.- 19.5 Proof of the Generalized Frobenius Theorem Based on Hur witz’s Theorem.- 20 Commutative Division Algebras.- 20.1 Formulation of the Main Result.- 20.2 The Connection Between Multiplication in the Algebra A and Multiplication of Complex Numbers.- 20.3 Determination of the Algebra A(?, ?, ?) that is Isomorphic to the Algebra A.- 21 Conclusion.- 22 Notes.