The Tree of Knowledge by Claudio Ronchi

Author:Claudio Ronchi
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where one can see that there are distinct th-roots of , having the same module and different arguments, corresponding to .

The fact that a complex number, z, has p roots of order p is a consequence of the definition of the product of two complex numbers. This property is of fundamental importance because it removes asymmetries and exceptions from algebra, which appear in the extraction of roots in the field of real numbers, an importance that would have scarcely been imaginable a priori. This most remarkable result is condensed in the Fundamental Theorem of Algebra which says:

An algebraic equation of n-degree possesses n complex solutions, some of which can be real numbers, depending on the values of the equation coefficients.

If we think how the limits and the intricate casuistry come together in solving algebraic equations in a real field, where solutions do or do not exist depending on the degree of the equation and on the numerical values of its coefficients, we immediately understand that complex numbers offer algebra a basis from which it can deploy its intrinsic completeness.

The following step consists in considering complex functions of complex variables, which introduce even greater operative advantages to the theory of differential equations than those obtained for algebraic equations. One of the more important ones is due to the fact that their differentials have the properties of vectors and, therefore, are characterised by a direction and amplitude. Due to this feature, it is possible to choose integration or derivation paths and “circumnavigate” singular points, offering new, powerful integration methods for differential equations.

If we consider the long arduous path from the first, tenuous steps made by Cardano to completely defining the property of the field of complex numbers, we must necessarily conclude that this field, which opens the door to a perfect fulfilment of algebra, existed, in one sense, already for itself, as a mathematical object, and, therefore, could only be “discovered” and not arbitrarily “created” by man.

On the other hand, however, when one began to formulate physical laws using complex numbers one rightly asked what they really represented. The demonstration, obtained later on, that a complex field represents a completion (called “algebraic closure”) of a real field and that this completion is unique, supplies the most important argument for justifying the use of complex numbers in describing physical phenomena. However, a rigorous and essential restriction is that, fundamentally, physical quantities must be represented only by complex numbers with an imaginary part exactly equal to zero, even though, in an operative context, a physical quantity can be decomposed into a real and an imaginary part.

Today in nearly all fields of physics complex numbers are used owing to the possibility of solving complicated differential equations using suitable variable transformations. There are mathematical procedures that are currently being used without questioning the precise physical significance of the variables in question. This is the case, for instance, of wave functions of quantum mechanics, where physical quantities are obtained with mathematical procedures defined in a field of complex numbers.

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