Complex-numbers Sentence Examples
Such a number is a "one-many" relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2..
But an indefinite number of definitions of the product of two complex numbers yield interesting results.
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.
Each definition gives rise to a corresponding algebra of higher complex numbers.
The product of two complex numbers of the second order - namely, l e l +x 2 e 2 and y i e l +y 2 e 2, is in this case defined to mean the complex (x i y i - x 2 y 2)e i +(x i y 2 +x 2 y 1)e 2.Advertisement
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.
Even at an elementary level one can often use complex numbers to solve problems which are otherwise not very tractable.
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.
Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of Cauchy, Gauss and others, the properties of the expression a + b I were developed into the immense and most important subject now called the theory of complex numbers (see Number).
This should be used on systems where complex numbers are not available in quadruple precision.Advertisement
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.