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)."
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