The problem of finding all hypercomplex number system remains unsolved. The authors consider some variants of this problem. They begin with examples including the quaternions and Cayley numbers; the second (auxiliary) part is an elementary exposition of linear algebra; the third part explains the unique role of quaternions, complex- and Cayley numbers in the problem. Neither index nor bibliography.

Table of Contents:
I. Hypercomplex Numbers
1. Complex Numbers
2. Alternate Arithmetics on the Numbers a + bi
3. Quaternions
4. Quaternions and Vector Algebra
5. Hypercomplex Numbers
6. The Doubling Procedure. Cayley Numbers
7. Algebras

II. N-Dimensional Vectors
8. The N-Dimensional Vector Space An
9. A Basis of the Space An
10. Subspaces
11. Lemma on Homogeneous Systems of Equations
12. Scalar Products
13. Orthonormal Basis. Orthogonal Transformation

III. The Exceptional Position of Four Algebras
14. Isomorphic Algebra
15. Subalgebras
16. Translation of the "Problem of the Sum of Squares" into the Language of Algebras. Normed Algebras
17. Normed Algebras with an Identity. Hurwitz's Theorem
18. A Method for Constructing All Normed Algebras and Its Implications for the Problm of the Sum of Squares
19. Frobenius' Theorem
20. Commutative Division Algebras
21. Conclusion
22. Notes