We will confine ourselves here to algebraic complex numbers - that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra.

In the modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors.

The importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers.

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But an indefinite number of definitions of the product of two complex numbers yield interesting results.

Kronecker, on Complex Numbers and Modular Systems, Berl.

Running through these volumes in order, we have in the second the memoir, Summatio quarundam serierum singularium, the memoirs on the theory of biquadratic residues, in which the notion of complex numbers of the form a--bi was first introduced into the theory of numbers; and included in the Nachlass are some valuable tables.